10.E: Fixed-Axis Rotation Introduction (Exercises)
Conceptual Questions
10.1 Rotational Variables
- A clock is mounted on the wall. As you look at it, what is the direction of the angular velocity vector of the second hand?
- What is the value of the angular acceleration of the second hand of the clock on the wall?
- A baseball bat is swung. Do all points on the bat have the same angular velocity? The same tangential speed?
- The blades of a blender on a counter are rotating clockwise as you look into it from the top. If the blender is put to a greater speed what direction is the angular acceleration of the blades?
10.2 Rotation with Constant Angular Acceleration
- If a rigid body has a constant angular acceleration, what is the functional form of the angular velocity in terms of the time variable?
- If a rigid body has a constant angular acceleration, what is the functional form of the angular position?
- If the angular acceleration of a rigid body is zero, what is the functional form of the angular velocity?
- A massless tether with a masses tied to both ends rotates about a fixed axis through the center. Can the total acceleration of the tether/mass combination be zero if the angular velocity is constant?
10.3 Relating Angular and Translational Quantities
- Explain why centripetal acceleration changes the direction of velocity in circular motion but not its magnitude.
- In circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction. Explain your answer.
- Suppose a piece of food is on the edge of a rotating microwave oven plate. Does it experience nonzero tangential acceleration, centripetal acceleration, or both when: (a) the plate starts to spin faster? (b) The plate rotates at constant angular velocity? (c) The plate slows to a halt?
10.4 Moment of Inertia and Rotational Kinetic Energy
- What if another planet the same size as Earth were put into orbit around the Sun along with Earth. Would the moment of inertia of the system increase, decrease, or stay the same?
- A solid sphere is rotating about an axis through its center at a constant rotation rate. Another hollow sphere of the same mass and radius is rotating about its axis through the center at the same rotation rate. Which sphere has a greater rotational kinetic energy?
10.5 Calculating Moments of Inertia
- If a child walks toward the center of a merry-go-round, does the moment of inertia increase or decrease?
- A discus thrower rotates with a discus in his hand before letting it go. (a) How does his moment of inertia change after releasing the discus? (b) What would be a good approximation to use in calculating the moment of inertia of the discus thrower and discus?
- Does increasing the number of blades on a propeller increase or decrease its moment of inertia, and why?
- The moment of inertia of a long rod spun around an axis through one end perpendicular to its length is \(\frac{mL^{2}}{3}\). Why is this moment of inertia greater than it would be if you spun a point mass m at the location of the center of mass of the rod (at \(\frac{L}{2}\)) (that would be \(\frac{mL^{2}}{4}\))
- Why is the moment of inertia of a hoop that has a mass M and a radius R greater than the moment of inertia of a disk that has the same mass and radius?
10.6 Torque
- What three factors affect the torque created by a force relative to a specific pivot point?
- Give an example in which a small force exerts a large torque. Give another example in which a large force exerts a small torque.
- When reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction in the mass of the bicycle’s frame?
- Can a single force produce a zero torque?
- Can a set of forces have a net torque that is zero and a net force that is not zero?
- Can a set of forces have a net force that is zero and a net torque that is not zero?
- In the expression \(\vec{r} \times \vec{F}\) can \(|\vec{r}|\) ever be less than the lever arm? Can it be equal to the lever arm?
10.7 Newton’s Second Law for Rotation
- If you were to stop a spinning wheel with a constant force, where on the wheel would you apply the force to produce the maximum negative acceleration?
- A rod is pivoted about one end. Two forces \(\vec{F}\) and \(− \vec{F}\) are applied to it. Under what circumstances will the rod not rotate?
Problems
10.1 Rotational Variables
- Calculate the angular velocity of Earth.
- A track star runs a 400-m race on a 400-m circular track in 45 s. What is his angular velocity assuming a constant speed?
- A wheel rotates at a constant rate of 2.0 x 10 3 rev/min. (a) What is its angular velocity in radians per second? (b) Through what angle does it turn in 10 s? Express the solution in radians and degrees.
- A particle moves 3.0 m along a circle of radius 1.5 m. (a) Through what angle does it rotate? (b) If the particle makes this trip in 1.0 s at a constant speed, what is its angular velocity? (c) What is its acceleration?
- A compact disc rotates at 500 rev/min. If the diameter of the disc is 120 mm, (a) what is the tangential speed of a point at the edge of the disc? (b) At a point halfway to the center of the disc?
- Unreasonable results . The propeller of an aircraft is spinning at 10 rev/s when the pilot shuts off the engine. The propeller reduces its angular velocity at a constant 2.0 rad/s 2 for a time period of 40 s. What is the rotation rate of the propeller in 40 s? Is this a reasonable situation?
- A gyroscope slows from an initial rate of 32.0 rad/s at a rate of 0.700 rad/s 2 . How long does it take to come to rest?
- On takeoff, the propellers on a UAV (unmanned aerial vehicle) increase their angular velocity for 3.0 s from rest at a rate of \(\omega\) = (25.0t) rad/s where t is measured in seconds. (a) What is the instantaneous angular velocity of the propellers at t = 2.0 s? (b) What is the angular acceleration?
- The angular position of a rod varies as 20.0t 2 radians from time t = 0. The rod has two beads on it as shown in the following figure, one at 10 cm from the rotation axis and the other at 20 cm from the rotation axis. (a) What is the instantaneous angular velocity of the rod at t = 5 s? (b) What is the angular acceleration of the rod? (c) What are the tangential speeds of the beads at t = 5 s? (d) What are the tangential accelerations of the beads at t = 5 s? (e) What are the centripetal accelerations of the beads at t = 5 s?
10.2 Rotation with Constant Angular Acceleration
- A wheel has a constant angular acceleration of 5.0 rad/s 2 . Starting from rest, it turns through 300 rad. (a) What is its final angular velocity? (b) How much time elapses while it turns through the 300 radians?
- During a 6.0-s time interval, a flywheel with a constant angular acceleration turns through 500 radians that acquire an angular velocity of 100 rad/s. (a) What is the angular velocity at the beginning of the 6.0 s? (b) What is the angular acceleration of the flywheel?
- The angular velocity of a rotating rigid body increases from 500 to 1500 rev/min in 120 s. (a) What is the angular acceleration of the body? (b) Through what angle does it turn in this 120 s?
- A flywheel slows from 600 to 400 rev/min while rotating through 40 revolutions. (a) What is the angular acceleration of the flywheel? (b) How much time elapses during the 40 revolutions?
- A wheel 1.0 m in radius rotates with an angular acceleration of 4.0 rad/s 2 . (a) If the wheel’s initial angular velocity is 2.0 rad/s, what is its angular velocity after 10 s? (b) Through what angle does it rotate in the 10-s interval? (c) What are the tangential speed and acceleration of a point on the rim of the wheel at the end of the 10-s interval?
- A vertical wheel with a diameter of 50 cm starts from rest and rotates with a constant angular acceleration of 5.0 rad/s 2 around a fixed axis through its center counterclockwise. (a) Where is the point that is initially at the bottom of the wheel at t = 10 s? (b) What is the point’s linear acceleration at this instant?
- A circular disk of radius 10 cm has a constant angular acceleration of 1.0 rad/s 2 ; at t = 0 its angular velocity is 2.0 rad/s. (a) Determine the disk’s angular velocity at t = 5.0 s . (b) What is the angle it has rotated through during this time? (c) What is the tangential acceleration of a point on the disk at t = 5.0 s?
- The angular velocity vs. time for a fan on a hovercraft is shown below. (a) What is the angle through which the fan blades rotate in the first 8 seconds? (b) Verify your result using the kinematic equations.
- A rod of length 20 cm has two beads attached to its ends. The rod with beads starts rotating from rest. If the beads are to have a tangential speed of 20 m/s in 7 s, what is the angular acceleration of the rod to achieve this?
10.3 Relating Angular and Translational Quantities
- At its peak, a tornado is 60.0 m in diameter and carries 500 km/h winds. What is its angular velocity in revolutions per second?
- A man stands on a merry-go-round that is rotating at 2.5 rad/s. If the coefficient of static friction between the man’s shoes and the merry-go-round is \(\mu\)S = 0.5, how far from the axis of rotation can he stand without sliding?
- An ultracentrifuge accelerates from rest to 100,000 rpm in 2.00 min. (a) What is the average angular acceleration in rad/s 2 ? (b) What is the tangential acceleration of a point 9.50 cm from the axis of rotation? (c) What is the centripetal acceleration in m/s 2 and multiples of g of this point at full rpm? (d) What is the total distance traveled by a point 9.5 cm from the axis of rotation of the ultracentrifuge?
- A wind turbine is rotating counterclockwise at 0.5 rev/s and slows to a stop in 10 s. Its blades are 20 m in length. (a) What is the angular acceleration of the turbine? (b) What is the centripetal acceleration of the tip of the blades at t = 0 s? (c) What is the magnitude and direction of the total linear acceleration of the tip of the blades at t = 0 s?
- What is (a) the angular speed and (b) the linear speed of a point on Earth’s surface at latitude 30° N. Take the radius of the Earth to be 6309 km. (c) At what latitude would your linear speed be 10 m/s?
- A child with mass 30 kg sits on the edge of a merrygo-round at a distance of 3.0 m from its axis of rotation. The merry-go-round accelerates from rest up to 0.4 rev/s in 10 s. If the coefficient of static friction between the child and the surface of the merry-go-round is 0.6, does the child fall off before 5 s?
- A bicycle wheel with radius 0.3m rotates from rest to 3 rev/s in 5 s. What is the magnitude and direction of the total acceleration vector at the edge of the wheel at 1.0 s?
- The angular velocity of a flywheel with radius 1.0 m varies according to \(\omega\)(t) = 2.0t. Plot a c (t) and a t (t) from t = 0 to 3.0 s for r = 1.0 m. Analyze these results to explain when a c >> a t and when a c << a t for a point on the flywheel at a radius of 1.0 m.
10.4 Moment of Inertia and Rotational Kinetic Energy
- A system of point particles is shown in the following figure. Each particle has mass 0.3 kg and they all lie in the same plane. (a) What is the moment of inertia of the system about the given axis? (b) If the system rotates at 5 rev/s, what is its rotational kinetic energy?
- (a) Calculate the rotational kinetic energy of Earth on its axis. (b) What is the rotational kinetic energy of Earth in its orbit around the Sun?
- Calculate the rotational kinetic energy of a 12-kg motorcycle wheel if its angular velocity is 120 rad/s and its inner radius is 0.280 m and outer radius 0.330 m.
- A baseball pitcher throws the ball in a motion where there is rotation of the forearm about the elbow joint as well as other movements. If the linear velocity of the ball relative to the elbow joint is 20.0 m/s at a distance of 0.480 m from the joint and the moment of inertia of the forearm is 0.500 kg-m 2 , what is the rotational kinetic energy of the forearm?
- A diver goes into a somersault during a dive by tucking her limbs. If her rotational kinetic energy is 100 J and her moment of inertia in the tuck is 9.0 kg • m 2 , what is her rotational rate during the somersault?
- An aircraft is coming in for a landing at 300 meters height when the propeller falls off. The aircraft is flying at 40.0 m/s horizontally. The propeller has a rotation rate of 20 rev/s, a moment of inertia of 70.0 kg • m 2 , and a mass of 200 kg. Neglect air resistance. (a) With what translational velocity does the propeller hit the ground? (b) What is the rotation rate of the propeller at impact?
- If air resistance is present in the preceding problem and reduces the propeller’s rotational kinetic energy at impact by 30%, what is the propeller’s rotation rate at impact?
- A neutron star of mass 2 x 10 30 kg and radius 10 km rotates with a period of 0.02 seconds. What is its rotational kinetic energy?
- An electric sander consisting of a rotating disk of mass 0.7 kg and radius 10 cm rotates at 15 rev/s. When applied to a rough wooden wall the rotation rate decreases by 20%. (a) What is the final rotational kinetic energy of the rotating disk? (b) How much has its rotational kinetic energy decreased?
- A system consists of a disk of mass 2.0 kg and radius 50 cm upon which is mounted an annular cylinder of mass 1.0 kg with inner radius 20 cm and outer radius 30 cm (see below). The system rotates about an axis through the center of the disk and annular cylinder at 10 rev/s. (a) What is the moment of inertia of the system? (b) What is its rotational kinetic energy?
10.5 Calculating Moments of Inertia
- While punting a football, a kicker rotates his leg about the hip joint. The moment of inertia of the leg is 3.75 kg • m 2 and its rotational kinetic energy is 175 J. (a) What is the angular velocity of the leg? (b) What is the velocity of tip of the punter’s shoe if it is 1.05 m from the hip joint?
- Using the parallel axis theorem, what is the moment of inertia of the rod of mass m about the axis shown below?
- Find the moment of inertia of the rod in the previous problem by direct integration.
- A uniform rod of mass 1.0 kg and length 2.0 m is free to rotate about one end (see the following figure). If the rod is released from rest at an angle of 60° with respect to the horizontal, what is the speed of the tip of the rod as it passes the horizontal position?
- A pendulum consists of a rod of mass 2 kg and length 1 m with a solid sphere at one end with mass 0.3 kg and radius 20 cm (see the following figure). If the pendulum is released from rest at an angle of 30°, what is the angular velocity at the lowest point?
- A solid sphere of radius 10 cm is allowed to rotate freely about an axis. The sphere is given a sharp blow so that its center of mass starts from the position shown in the following figure with speed 15 cm/s. What is the maximum angle that the diameter makes with the vertical?
- Calculate the moment of inertia by direct integration of a thin rod of mass M and length L about an axis through the rod at L/3, as shown below. Check your answer with the parallel-axis theorem.
10.6 Torque
- Two flywheels of negligible mass and differ3ent radii are bonded together and rotate about a common axis (see below). The smaller flywheel of radius 30 cm has a cord that has a pulling force of 50 N on it. What pulling force needs to be applied to the cord connecting the larger flywheel of radius 50 cm such that the combination does not rotate?
- The cylinder head bolts on a car are to be tightened with a torque of 62.0 N·m. If a mechanic uses a wrench of length 20 cm, what perpendicular force must he exert on the end of the wrench to tighten a bolt correctly?
- (a) When opening a door, you push on it perpendicularly with a force of 55.0 N at a distance of 0.850 m from the hinges. What torque are you exerting relative to the hinges? (b) Does it matter if you push at the same height as the hinges? There is only one pair of hinges.
- When tightening a bolt, you push perpendicularly on a wrench with a force of 165 N at a distance of 0.140 m from the center of the bolt. How much torque are you exerting in newton-meters (relative to the center of the bolt)?
- What hanging mass must be placed on the cord to keep the pulley from rotating (see the following figure)? The mass on the frictionless plane is 5.0 kg. The inner radius of the pulley is 20 cm and the outer radius is 30 cm.
- A simple pendulum consists of a massless tether 50 cm in length connected to a pivot and a small mass of 1.0 kg attached at the other end. What is the torque about the pivot when the pendulum makes an angle of 40° with respect to the vertical?
- Calculate the torque about the z-axis that is out of the page at the origin in the following figure, given that F 1 = 3 N, F 2 = 2 N, F 3 = 3 N, F 4 = 1.8 N.
- A seesaw has length 10.0 m and uniform mass 10.0 kg and is resting at an angle of 30° with respect to the ground (see the following figure). The pivot is located at 6.0 m. What magnitude of force needs to be applied perpendicular to the seesaw at the raised end so as to allow the seesaw to barely start to rotate?
- A pendulum consists of a rod of mass 1 kg and length 1 m connected to a pivot with a solid sphere attached at the other end with mass 0.5 kg and radius 30 cm. What is the torque about the pivot when the pendulum makes an angle of 30° with respect to the vertical?
- A torque of 5.00 x 10 3 N • m is required to raise a drawbridge (see the following figure). What is the tension necessary to produce this torque? Would it be easier to raise the drawbridge if the angle \(\theta\) were larger or smaller?
- A horizontal beam of length 3 m and mass 2.0 kg has a mass of 1.0 kg and width 0.2 m sitting at the end of the beam (see the following figure). What is the torque of the system about the support at the wall?
- What force must be applied to end of a rod along the x-axis of length 2.0 m in order to produce a torque on the rod about the origin of 8.0\(\hat{k}\) N • m?
- What is the torque about the origin of the force (5.0 \(\hat{i}\) − 2.0 \(\hat{j}\) + 1.0\(\hat{k}\)) N if it is applied at the point whose position is: \(\vec{r}\) = (−2.0 \(\hat{i}\) + 4.0\(\hat{j}\)) m?
10.7 Newton’s Second Law for Rotation
- You have a grindstone (a disk) that is 90.0 kg, has a 0.340-m radius, and is turning at 90.0 rpm, and you press a steel axe against it with a radial force of 20.0 N. (a) Assuming the kinetic coefficient of friction between steel and stone is 0.20, calculate the angular acceleration of the grindstone. (b) How many turns will the stone make before coming to rest?
- Suppose you exert a force of 180 N tangential to a 0.280-m-radius, 75.0-kg grindstone (a solid disk). (a)What torque is exerted? (b) What is the angular acceleration assuming negligible opposing friction? (c) What is the angular acceleration if there is an opposing frictional force of 20.0 N exerted 1.50 cm from the axis?
- A flywheel (I = 50 kg • m 2 ) starting from rest acquires an angular velocity of 200.0 rad/s while subject to a constant torque from a motor for 5 s. (a) What is the angular acceleration of the flywheel? (b) What is the magnitude of the torque?
- A constant torque is applied to a rigid body whose moment of inertia is 4.0 kg • m 2 around the axis of rotation. If the wheel starts from rest and attains an angular velocity of 20.0 rad/s in 10.0 s, what is the applied torque?
- A torque of 50.0 N • m is applied to a grinding wheel (I = 20.0 kg • m 2 ) for 20 s. (a) If it starts from rest, what is the angular velocity of the grinding wheel after the torque is removed? (b) Through what angle does the wheel move through while the torque is applied?
- A flywheel (I = 100.0 kg • m 2 ) rotating at 500.0 rev/ min is brought to rest by friction in 2.0 min. What is the frictional torque on the flywheel?
- A uniform cylindrical grinding wheel of mass 50.0 kg and diameter 1.0 m is turned on by an electric motor. The friction in the bearings is negligible. (a) What torque must be applied to the wheel to bring it from rest to 120 rev/min in 20 revolutions? (b) A tool whose coefficient of kinetic friction with the wheel is 0.60 is pressed perpendicularly against the wheel with a force of 40.0 N. What torque must be supplied by the motor to keep the wheel rotating at a constant angular velocity?
- Suppose when Earth was created, it was not rotating. However, after the application of a uniform torque after 6 days, it was rotating at 1 rev/day. (a) What was the angular acceleration during the 6 days? (b) What torque was applied to Earth during this period? (c) What force tangent to Earth at its equator would produce this torque?
- A pulley of moment of inertia 2.0 kg • m 2 is mounted on a wall as shown in the following figure. Light strings are wrapped around two circumferences of the pulley and weights are attached. What are (a) the angular acceleration of the pulley and (b) the linear acceleration of the weights? Assume the following data: r 1 = 50 cm, r 2 = 20 cm, m 1 = 1.0 kg, m 2 = 2.0 kg.
- A block of mass 3 kg slides down an inclined plane at an angle of 45° with a massless tether attached to a pulley with mass 1 kg and radius 0.5 m at the top of the incline (see the following figure). The pulley can be approximated as a disk. The coefficient of kinetic friction on the plane is 0.4. What is the acceleration of the block?
- The cart shown below moves across the table top as the block falls. What is the acceleration of the cart? Neglect friction and assume the following data: m 1 = 2.0 kg, m 2 = 4.0 kg, I = 0.4 kg • m 2 , r = 20 cm.
- A uniform rod of mass and length is held vertically by two strings of negligible mass, as shown below. (a) Immediately after the string is cut, what is the linear acceleration of the free end of the stick? (b) Of the middle of the stick?
- A thin stick of mass 0.2 kg and length L = 0.5 m is attached to the rim of a metal disk of mass M = 2.0 kg and radius R = 0.3 m. The stick is free to rotate around a horizontal axis through its other end (see the following figure). (a) If the combination is released with the stick horizontal, what is the speed of the center of the disk when the stick is vertical? (b) What is the acceleration of the center of the disk at the instant the stick is released? (c) At the instant the stick passes through the vertical?
10.8 Work and Power for Rotational Motion
- A wind turbine rotates at 20 rev/min. If its power output is 2.0 MW, what is the torque produced on the turbine from the wind?
- A clay cylinder of radius 20 cm on a potter’s wheel spins at a constant rate of 10 rev/s. The potter applies a force of 10 N to the clay with his hands where the coefficient of friction is 0.1 between his hands and the clay. What is the power that the potter has to deliver to the wheel to keep it rotating at this constant rate?
- A uniform cylindrical grindstone has a mass of 10 kg and a radius of 12 cm. (a) What is the rotational kinetic energy of the grindstone when it is rotating at 1.5 x 10 3 rev/min? (b) After the grindstone’s motor is turned off, a knife blade is pressed against the outer edge of the grindstone with a perpendicular force of 5.0 N. The coefficient of kinetic friction between the grindstone and the blade is 0.80. Use the work energy theorem to determine how many turns the grindstone makes before it stops.
- A uniform disk of mass 500 kg and radius 0.25 m is mounted on frictionless bearings so it can rotate freely around a vertical axis through its center (see the following figure). A cord is wrapped around the rim of the disk and pulled with a force of 10 N. (a) How much work has the force done at the instant the disk has completed three revolutions, starting from rest? (b) Determine the torque due to the force, then calculate the work done by this torque at the instant the disk has completed three revolutions? (c) What is the angular velocity at that instant? (d) What is the power output of the force at that instant?
- A propeller is accelerated from rest to an angular velocity of 1000 rev/min over a period of 6.0 seconds by a constant torque of 2.0 x 10 3 N • m. (a) What is the moment of inertia of the propeller? (b) What power is being provided to the propeller 3.0 s after it starts rotating?
- A sphere of mass 1.0 kg and radius 0.5 m is attached to the end of a massless rod of length 3.0 m. The rod rotates about an axis that is at the opposite end of the sphere (see below). The system rotates horizontally about the axis at a constant 400 rev/min. After rotating at this angular speed in a vacuum, air resistance is introduced and provides a force 0.15 N on the sphere opposite to the direction of motion. What is the power provided by air resistance to the system 100.0 s after air resistance is introduced?
- A uniform rod of length L and mass M is held vertically with one end resting on the floor as shown below. When the rod is released, it rotates around its lower end until it hits the floor. Assuming the lower end of the rod does not slip, what is the linear velocity of the upper end when it hits the floor?
- An athlete in a gym applies a constant force of 50 N to the pedals of a bicycle at a rate of the pedals moving 60 rev/min. The length of the pedal arms is 30 cm. What is the power delivered to the bicycle by the athlete?
- A 2-kg block on a frictionless inclined plane at 40° has a cord attached to a pulley of mass 1 kg and radius 20 cm (see the following figure). (a) What is the acceleration of the block down the plane? (b) What is the work done by the cord on the pulley?
- Small bodies of mass m 1 and m 2 are attached to opposite ends of a thin rigid rod of length L and mass M. The rod is mounted so that it is free to rotate in a horizontal plane around a vertical axis (see below). What distance d from m 1 should the rotational axis be so that a minimum amount of work is required to set the rod rotating at an angular velocity \(\omega\)?
Additional Problems
- A cyclist is riding such that the wheels of the bicycle have a rotation rate of 3.0 rev/s. If the cyclist brakes such that the rotation rate of the wheels decrease at a rate of 0.3 rev/s 2 , how long does it take for the cyclist to come to a complete stop?
- Calculate the angular velocity of the orbital motion of Earth around the Sun.
- A phonograph turntable rotating at 33\(\frac{1}{3}\) rev/min slows down and stops in 1.0 min. (a) What is the turntable’s angular acceleration assuming it is constant? (b) How many revolutions does the turntable make while stopping?
- With the aid of a string, a gyroscope is accelerated from rest to 32 rad/s in 0.40 s under a constant angular acceleration. (a) What is its angular acceleration in rad/s 2 ? (b) How many revolutions does it go through in the process?
- Suppose a piece of dust has fallen on a CD. If the spin rate of the CD is 500 rpm, and the piece of dust is 4.3 cm from the center, what is the total distance traveled by the dust in 3 minutes? (Ignore accelerations due to getting the CD rotating.)
- A system of point particles is rotating about a fixed axis at 4 rev/s. The particles are fixed with respect to each other.  (a) What is the moment of inertia of the system? (b) What is the rotational kinetic energy of the system?
- Calculate the moment of inertia of a skater given the following information. (a) The 60.0-kg skater is approximated as a cylinder that has a 0.110-m radius. (b) The skater with arms extended is approximated by a cylinder that is 52.5 kg, has a 0.110-m radius, and has two 0.900-m-long arms which are 3.75 kg each and extend straight out from the cylinder like rods rotated about their ends.
- A stick of length 1.0 m and mass 6.0 kg is free to rotate about a horizontal axis through the center. Small bodies of masses 4.0 and 2.0 kg are attached to its two ends (see the following figure). The stick is released from the horizontal position. What is the angular velocity of the stick when it swings through the vertical?
- A pendulum consists of a rod of length 2 m and mass 3 kg with a solid sphere of mass 1 kg and radius 0.3 m attached at one end. The axis of rotation is as shown below. What is the angular velocity of the pendulum at its lowest point if it is released from rest at an angle of 30°?
- Calculate the torque of the 40-N force around the axis through O and perpendicular to the plane of the page as shown below.
- Two children push on opposite sides of a door during play. Both push horizontally and perpendicular to the door. One child pushes with a force of 17.5 N at a distance of 0.600 m from the hinges, and the second child pushes at a distance of 0.450 m. What force must the second child exert to keep the door from moving? Assume friction is negligible.
- The force of 20 \(\hat{j}\) N is applied at \(\vec{r}\) = (4.0 \(\hat{i}\) − 2.0 \(\hat{j}\)) m. What is the torque of this force about the origin? 119. An automobile engine can produce 200 N • m of torque. Calculate the angular acceleration produced if 95.0% of this torque is applied to the drive shaft, axle, and rear wheels of a car, given the following information. The car is suspended so that the wheels can turn freely. Each wheel acts like a 15.0-kg disk that has a 0.180-m radius. The walls of each tire act like a 2.00-kg annular ring that has inside radius of 0.180 m and outside radius of 0.320 m. The tread of each tire acts like a 10.0-kg hoop of radius 0.330 m. The 14.0-kg axle acts like a rod that has a 2.00-cm radius. The 30.0-kg drive shaft acts like a rod that has a 3.20-cm radius.
- A grindstone with a mass of 50 kg and radius 0.8 m maintains a constant rotation rate of 4.0 rev/s by a motor while a knife is pressed against the edge with a force of 5.0 N. The coefficient of kinetic friction between the grindstone and the blade is 0.8. What is the power provided by the motor to keep the grindstone at the constant rotation rate?
Challenge Problems
- The angular acceleration of a rotating rigid body is given by \(\alpha\) = (2.0 − 3.0t) rad/s 2 . If the body starts rotating from rest at t = 0, (a) what is the angular velocity? (b) Angular position? (c) What angle does it rotate through in 10 s? (d) Where does the vector perpendicular to the axis of rotation indicating 0° at t = 0 lie at t = 10 s?
- Earth’s day has increased by 0.002 s in the last century. If this increase in Earth’s period is constant, how long will it take for Earth to come to rest?
- A disk of mass m, radius R, and area A has a surface mass density \(\sigma = \frac{mr}{AR}\) (see the following figure). What is the moment of inertia of the disk about an axis through the center?
- Zorch, an archenemy of Rotation Man, decides to slow Earth’s rotation to once per 28.0 h by exerting an opposing force at and parallel to the equator. Rotation Man is not immediately concerned, because he knows Zorch can only exert a force of 4.00 x 10 7 N (a little greater than a Saturn V rocket’s thrust). How long must Zorch push with this force to accomplish his goal? (This period gives Rotation Man time to devote to other villains.)
- A cord is wrapped around the rim of a solid cylinder of radius 0.25 m, and a constant force of 40 N is exerted on the cord shown, as shown in the following figure. The cylinder is mounted on frictionless bearings, and its moment of inertia is 6.0 kg • m 2 . (a) Use the work energy theorem to calculate the angular velocity of the cylinder after 5.0 m of cord have been removed. (b) If the 40-N force is replaced by a 40-N weight, what is the angular velocity of the cylinder after 5.0 m of cord have unwound?
Contributors and Attributions
Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a
Creative Commons Attribution License (by 4.0)
.


1.7: Abstraction- Absurd AND Necessary
You thought we were done talking about abstraction. Sorry – one last discussion. It is that important.  They have an ample supply of canned food, but alas, no can opener.
The chemist suggests that they should light a fire under the cans so that they would burst open.
The physicist suggests dropping the cans from the cliff to its rocky bottom to smash them open.
The economist declares: “Let’s assume we have a can opener.”
Financial and economic theory makes, what may appear at first, some absurd assumptions. But that is only because the social world, the world inhabited by people, is far more complex, in many ways, than the real world, the world of the hard sciences.
Are the chemist’s molecules motivated by fear and greed? Will ambition or altruism affect the trajectory of the falling cans? Economists cannot keep track of every alternative that the human mind may consider, so it abstracts by looking into the outcomes or choices that are usually independent of human foibles, or dare I say, are “logical.” Without abstraction, economists would never arrive at any generalizations. We would therefore learn nothing! Zilch.
Suppose you just arrived in New York City for the first time. If you wish to find Times Square, would you use a map (imagine that there is no GPS or Waze) that details all the streets, or just the main arteries? No! You would not be concerned with all the confusing, and mostly useless, details.
A burgeoning field, Behavioral Economics and Finance, deals with the human condition and its interaction with traditional, “objective” economics. However, first things first.

-Benjamin Franklin
Instruct me and I shall be silent. Make me understand where I have erred.
-Book of Job (6:24) (Artscroll translation)


26.11.3: Figuring for Yourself
Figuring for Yourself
According to Hubble’s law, what is the recessional velocity of a galaxy that is 10 8 light-years away from us? (Assume a Hubble constant of 22 km/s per million light-years.)
28.
A cluster of galaxies is observed to have a recessional velocity of 60,000 km/s. Find the distance to the cluster. (Assume a Hubble constant of 22 km/s per million light-years.)
29.
Suppose we could measure the distance to a galaxy using one of the distance techniques listed in Table 26.2 and it turns out to be 200 million light-years. The galaxy’s redshift tells us its recessional velocity is 5000 km/s. What is the Hubble constant?
30.
Calculate the mass-to-light ratio for a globular cluster with a luminosity of 10 6 L Sun and 10 5 stars. (Assume that the average mass of a star in such a cluster is 1 M Sun .)
31.
Calculate the mass-to-light ratio for a luminous star of 100 M Sun having the luminosity of 10 6 L Sun .


1.4: Non-Natural Powers
Having defined the exponential and logarithm, we have the tools needed to address the issue raised earlier, i.e. how to define non-natural power operations. First, observe that \[\textrm{For}\;\,y \in \mathbb{N}, \;\;\;\ln(x^y) = \underbrace{\ln(x)\ln(x)\cdots\ln(x)}_{y\;\text{times}} = y \ln(x).\] Hence, by applying the exponential to each side of the above equation, \[x^y = \exp[y \ln(x)] \quad \mathrm{for} \;\,y \in \mathbb{N}.\] We can generalize the above equation so that it holds for any positive \(x\) and real \(y\) , not just \(y \in \mathbb{N}\) . In other words, we treat this as our definition of the power operation for non-natural powers: \[x^y \equiv \exp[y \ln(x)] \quad\; \mathrm{for}\;\, x \in \mathbb{R}^+, \;y \notin \mathbb{N}.\] By this definition, the power operation always gives a positive result. And for \(y \in \mathbb{N}\) , the result of the formula is consistent with the standard definition based on multiplying \(x\) by itself \(y\) times.
This generalization of the power operation leads to several important consequences:
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The zeroth power yield unity: \[\displaystyle x^0 = 1 \;\;\mathrm{for}\;\, x \in \mathbb{R}^+.\]
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Negative powers are reciprocals: \[x^{-y} = \exp[-y\ln(x)] = \exp[-\ln(x^y)] = \frac{1}{x^y}.\]
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The output of the exponential function is equivalent to a power operation: \[\exp(y) = e^y\] where \[e \equiv \exp(1) = 2.718281828459\!\dots\] (This follows by plugging in
\(x=e\)
and using the fact that
\(\ln(e) = 1\)
.)
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For
\(x \le 0\)
, the meaning of
\(x^y\)
for non-natural
\(y\)
is ill-defined, since the logarithm does not accept negative inputs.


7.6: Quantum States, Microstates, and Energy Spreading
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Learning Objectives
- Explain how entropy is a measure of the degree of the spreading and sharing of thermal energy within a system.
- Explain why the entropy of a substance increases with its molecular weight and complexity and with temperature.
- Explain why entropy increases as the pressure or concentration becomes smaller.
- Explain why entropies of gases are much larger than those of condensed phases.
Entropy is a measure of the degree of spreading and sharing of thermal energy within a system. This “spreading and sharing” can be spreading of the thermal energy into a larger volume of space or its sharing amongst previously inaccessible microstates of the system. Entropy is an extensive quantity ; that is, it is proportional to the quantity of matter in a system; thus 100 g of metallic copper has twice the entropy of 50 g at the same temperature. This makes sense because the larger piece of copper contains twice as many quantized energy levels able to contain the thermal energy.
Thermal energy is the portion of a molecule's energy that is proportional to its temperature , and thus relates to motion at the molecular scale. What kinds of molecular motions are possible? For monatomic molecules, there is only one: actual movement from one location to another, which we call translation . Since there are three directions in space, all molecules possess three modes of translational motion .
For polyatomic molecules, two additional kinds of motions are possible. One of these is rotation ; a linear molecule such as CO 2 in which the atoms are all laid out along the x-axis can rotate along the y- and z-axes, while molecules having less symmetry can rotate about all three axes. Thus linear molecules possess two modes of rotational motion, while non-linear ones have three rotational modes. Finally, molecules consisting of two or more atoms can undergo internal vibrations . For freely moving molecules in a gas, the number of vibrational modes or patterns depends on both the number of atoms and the shape of the molecule, and it increases rapidly as the molecule becomes more complicated.
The relative populations of the quantized translational, rotational, and vibrational energy states of a typical diatomic molecule are depicted by the thickness of the lines in this schematic (not-to-scale!) diagram. The colored shading indicates the total thermal energy available at a given temperature. The numbers at the top show order-of-magnitude spacings between adjacent levels. It is readily apparent that virtually all the thermal energy resides in translational states.
Notice the greatly different spacing of the three kinds of energy levels in Figure \(\PageIndex{1}\). This is extremely important because it determines the number of energy quanta that a molecule can accept, and, as Figure \(\PageIndex{2}\) demonstrates, the number of different ways this energy can be distributed amongst the molecules.
The spacing of molecular energy states becomes closer as the mass and number of bonds in the molecule increases, so we can generally say that the more complex the molecule, the greater the density of its energy states.
At the atomic and molecular level, all energy is quantized ; each particle possesses discrete states of kinetic energy and is able to accept thermal energy only in packets whose values correspond to the energies of one or more of these states. Polyatomic molecules can store energy in rotational and vibrational motions, and all molecules (even monatomic ones) will possess translational kinetic energy (thermal energy) at all temperatures above absolute zero. The energy difference between adjacent translational states is so minute that translational kinetic energy can be regarded as continuous (non-quantized) for most practical purposes.
The number of ways in which thermal energy can be distributed amongst the allowed states within a collection of molecules is easily calculated from simple statistics, but we will confine ourselves to an example here. Suppose that we have a system consisting of three molecules and three quanta of energy to share among them. We can give all the kinetic energy to any one molecule, leaving the others with none, we can give two units to one molecule and one unit to another, or we can share out the energy equally and give one unit to each molecule. All told, there are ten possible ways of distributing three units of energy among three identical molecules as shown here:
Each of these ten possibilities represents a distinct microstate that will describe the system at any instant in time. Those microstates that possess identical distributions of energy among the accessible quantum levels (and differ only in which particular molecules occupy the levels) are known as configurations . Because all microstates are equally probable, the probability of any one configuration is proportional to the number of microstates that can produce it. Thus in the system shown above, the configuration labeled ii will be observed 60% of the time, while iii will occur only 10% of the time.
As the number of molecules and the number of quanta increases, the number of accessible microstates grows explosively; if 1000 quanta of energy are shared by 1000 molecules, the number of available microstates will be around 10 600 — a number that greatly exceeds the number of atoms in the observable universe! The number of possible configurations (as defined above) also increases, but in such a way as to greatly reduce the probability of all but the most probable configurations. Thus for a sample of a gas large enough to be observable under normal conditions, only a single configuration (energy distribution amongst the quantum states) need be considered ; even the second-most-probable configuration can be neglected.
The bottom line : any collection of molecules large enough in numbers to have chemical significance will have its thermal energy distributed over an unimaginably large number of microstates. The number of microstates increases exponentially as more energy states ("configurations" as defined above) become accessible owing to
- Addition of energy quanta (higher temperature),
- Increase in the number of molecules (resulting from dissociation, for example).
- the volume of the system increases (which decreases the spacing between energy states, allowing more of them to be populated at a given temperature.)
Spontaneous Process: Heat Death
Energy is conserved ; if you lift a book off the table, and let it fall, the total amount of energy in the world remains unchanged. All you have done is transferred it from the form in which it was stored within the glucose in your body to your muscles, and then to the book (that is, you did work on the book by moving it up against the earth’s gravitational field). After the book has fallen, this same quantity of energy exists as thermal energy (heat) in the book and table top.
What has changed, however, is the availability of this energy. Once the energy has spread into the huge number of thermal microstates in the warmed objects, the probability of its spontaneously (that is, by chance) becoming un-dispersed is essentially zero. Thus although the energy is still “there”, it is forever beyond utilization or recovery. The profundity of this conclusion was recognized around 1900, when it was first described at the “heat death” of the world. This refers to the fact that every spontaneous process (essentially every change that occurs) is accompanied by the “dilution” of energy. The obvious implication is that all of the molecular-level kinetic energy will be spread out completely, and nothing more will ever happen.
Spontaneous Process: Gases Expansions
Everybody knows that a gas, if left to itself, will tend to expand and fill the volume within which it is confined completely and uniformly. What “drives” this expansion? At the simplest level it is clear that with more space available, random motions of the individual molecules will inevitably disperse them throughout the space. But as we mentioned above, the allowed energy states that molecules can occupy are spaced more closely in a larger volume than in a smaller one. The larger the volume available to the gas, the greater the number of microstates its thermal energy can occupy. Since all such states within the thermally accessible range of energies are equally probable, the expansion of the gas can be viewed as a consequence of the tendency of thermal energy to be spread and shared as widely as possible. Once this has happened, the probability that this sharing of energy will reverse itself (that is, that the gas will spontaneously contract) is so minute as to be unthinkable.
Imagine a gas initially confined to one half of a box (Figure \(\PageIndex{4}\)). The barrier is then removed so that it can expand into the full volume of the container. We know that the entropy of the gas will increase as the thermal energy of its molecules spreads into the enlarged space. In terms of the spreading of thermal energy, Figure Figure \(\PageIndex{5}\) may be helpful. The tendency of a gas to expand is due to the more closely-spaced thermal energy states in the larger volume .
Spontaneous Process: Mixing and dilution
Mixing and dilution really amount to the same thing, especially for ideal gases. Replace the pair of containers shown above with one containing two kinds of molecules in the separate sections (Figure \(\PageIndex{6}\)). When we remove the barrier, the "red" and "blue" molecules will each expand into the space of the other. (Recall Dalton's Law that "each gas is a vacuum to the other gas"). However, notice that although each gas underwent an expansion, the overall process amounts to what we call "mixing".
What is true for gaseous molecules can, in principle, apply also to solute molecules dissolved in a solvent. But bear in mind that whereas the enthalpy associated with the expansion of a perfect gas is by definition zero, Δ H 's of mixing of two liquids or of dissolving a solute in a solvent have finite values which may limit the miscibility of liquids or the solubility of a solute.
But what's really dramatic is that when just one molecule of a second gas is introduced into the container ( in Figure \(\PageIndex{5}\)), an unimaginably huge number of new configurations become possible, greatly increasing the number of microstates that are thermally accessible (as indicated by the pink shading above).
Spontaneous Process: Thermal Equilibrium
Just as gases spontaneously change their volumes from “smaller-to-larger”, the flow of heat from a warmer body to a cooler one always operates in the direction “warmer-to-cooler” because this allows thermal energy to populate a larger number of energy microstates as new ones are made available by bringing the cooler body into contact with the warmer one; in effect, the thermal energy becomes more “diluted”.
When the bodies are brought into thermal contact ( b ), thermal energy flows from the higher occupied levels in the warmer object into the unoccupied levels of the cooler one until equal numbers are occupied in both bodies, bringing them to the same temperature. As you might expect, the increase in the amount of energy spreading and sharing is proportional to the amount of heat transferred q , but there is one other factor involved, and that is the temperature at which the transfer occurs. When a quantity of heat q passes into a system at temperature T , the degree of dilution of the thermal energy is given by
\[\dfrac{q}{T} \label{eq70}\]
To understand why we have to divide by the temperature, consider the effect of very large and very small values of \(T\) in the denominator of Equation \ref{eq70}. If the body receiving the heat is initially at a very low temperature, relatively few thermal energy states are initially occupied, so the amount of energy spreading into vacant states can be very great. Conversely, if the temperature is initially large, more thermal energy is already spread around within it, and absorption of the additional energy will have a relatively small effect on the degree of thermal disorder within the body.
Summary
Entropy is a measure of the degree of spreading and sharing of thermal energy within a system. This “spreading and sharing” can be spreading of the thermal energy into a larger volume of space or its sharing amongst previously inaccessible microstates of the system. The following box shows how this concept applies to a number of common processes.
Observations and Explanations in terms of Entropy changes
A deck of cards is shuffled, or 100 coins, initially heads up, are randomly tossed.
- Source of Entropy Increase of System
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This has nothing to do with entropy because macro objects are unable to exchange thermal energy with the surroundings within the time scale of the process
Two identical blocks of copper, one at 20°C and the other at 40°C, are placed in contact.
- Source of Entropy Increase of System
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The cooler block contains more unoccupied microstates, so heat flows from the warmer block until equal numbers of microstates are populated in the two blocks.
A gas expands isothermally to twice its initial volume.
- Source of Entropy Increase of System
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A constant amount of thermal energy spreads over a larger volume of space
1 mole of water is heated by 1C°.
- Source of Entropy Increase of System
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The increased thermal energy makes additional microstates accessible. (The increase is by a factor of about 10 20,000,000,000,000, 000,000,000 .)
Equal volumes of two gases are allowed to mix.
- Source of Entropy Increase of System
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The effect is the same as allowing each gas to expand to twice its volume; the thermal energy in each is now spread over a larger volume.
One mole of dihydrogen, H 2 , is placed in a container and heated to 3000K.
- Source of Entropy Increase of System
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Some of the \(\ce{H2}\) dissociates to \(\ce{H}\) atoms because at this temperature there are more thermally accessible microstates in the 2 moles of \(\ce{H}\) atoms.
The above reaction mixture is cooled to 300 K.
- Source of Entropy Increase of System
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The composition shifts back to virtually all \(\ce{H2}\) because this molecule contains more thermally accessible microstates at low temperatures.


10: Building and Maintaining Relationships
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Learning Objectives
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Identify and understand different types of interpersonal relationships, needs, roles, and norms
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Discuss how interpersonal communication changes as we move through various relationship stages
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Explain the influence that context, identity, and culture have on our interpersonal communication interactions and relationships
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Apply interpersonal communication theories and strategies when discussing and analyzing relationships
Interpersonal communication is vital to building, maintaining, and even ending our relationships. In this chapter, we discuss the foundations of relationships, relational stages, and the tools and strategies to achieve the relational outcomes we desire. We talk about healthy and unhealthy communication in relationships, and how to deal with relationship termination.
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10.1: Introduction to Building and Maintaining Relationships
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A short vignette and reflection questions are used to introduce the Building and Maintaining Relationships chapter.
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10.2: Foundations of Relationships
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Types of relationships, boundaries, norms and rules are discussed. We examine which relationships are voluntary, involuntary, social or personal, and explore dialectical tensions that exist in all relationships.
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10.3: Common Relationship Types
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Family, friendships, work relationships, and romantic relationships are common relationship types discussed in this section. Additionally, we look at Family Types, based on how Conversation and Conformity Orientations interact in different ways.
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10.4: Relationship Stages
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A review of Knapp and Vangelisti's (2009) Model of Relationship Stages.
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10.5: The Role of Self-Disclosure
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Self-Disclosure plays an important role in the development of our relationships. The Johari Window is a tool we can use to examine how we handle both self-disclosure, as well as feedback from others. Uncertainty Reduction Theory speaks to the strategies we use to increase predictability about other people.
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10.6: Couple Communication
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Focusing on evidence-based theories of couple communication, including the Gottman Method and Emotionally-Focused Therapy, we examine strategies for healthy communication in love relationships. We also explore how jealousy and sexual communication play a role in our love relationships.
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10.7: Cycle of Abuse and Coming Apart
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The Cycle of Abuse describes the unhealthy relationship communication that is often present in abusive or violent relationships. This section also discusses healthy versus unhealthy ways relationships end or terminate.
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10.8: Summary and Review
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10.9: References
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Thumbnail:
Sisters Laughing Together
by
Alex Starnes
in
Public Domain


3.5: Liar's Poker
In this section, we will continue to explore the ideas of a mixed strategy equilibrium. We have seen several examples of finding an equilibrium. We began with games which had a pure strategy equilibrium and then moved to games with mixed strategy equilibrium. We saw two different methods for finding an equilibrium. The first employed graphs in order to understand and find the maximin and minimax strategies, and hence the equilibrium mixed strategy. The second method employed the ideas of expected value to find the equilibrium strategy. We will continue to solidify these ideas with another game, a simplified variation of poker.
We begin with a deck of cards which has \(50 \%\) aces (A) and \(50 \%\) kings (K). Aces rank higher than kings. Player 1 is dealt one card, face down. Player 1 can look at the card, but does not show the card to Player 2. Player 1 then says “ace” or “king” depending on what his card is. Player 1 can either tell the truth and say what the card is (T), or he can bluff and say that he has a higher ranking card (B). Note that if Player 1 has an ace, he must tell the truth since there are no higher-ranking cards. However, if he is dealt a king, he can bluff, by saying he has an ace. If Player 1 says “king” the game ends and both players break even. If Player 1 says “ace” then Player 2 can either call (C) or fold (F). If Player 2 folds, then Player 1 wins \($0.50\). If Player 2 calls and Player 1 does not have an ace, then Player 2 wins \($1\). If Player 2 calls and Player 1 does have an ace, then Player 1 wins \($1\).
Choose an opponent and play Liar's Poker several times. Be sure to play the game as Player 1 and as Player 2. This is important for understanding the game. Keep track of the outcomes.
Just from playing Liar's Poker several times, can you suggest a strategy for Player 1? What about for Player 2? Does this game seem fair, or does one of the players seem to have an advantage? Explain your answers.
In order to formally analyze Liar's Poker, we should find the payoff matrix. Do your best to find the payoff matrix. In a single hand of Liar's Poker, what are the possible strategies for Player 1? What are the possible strategies for Player 2? Determine any payoffs that you can.
Finding the payoff matrix in Exercise \(3.5.3\) is probably more challenging than it appears. Eventually, we want to employ the same method for finding the payoff matrix that we used in One-Card Stud Poker from Example \(2.4.1\) in Chapter 2 , but first we need to understand each player's strategies, and the resulting payoffs. We begin with the fact that Player 1 can be dealt an ace or a king.
Assume Player 1 is dealt an ace. What can Player 1 do? What can Player 2 do? What is the payoff for each situation?
Assume Player 1 is dealt a king. What can Player 1 do? What can Player 2 do? What is the payoff for each situation?
Since Player 1 must say “ace” when dealt an ace, he only has a choice of strategy when dealt a king. So we can define his strategy independent of the deal. One strategy is to say “ace” when dealt an ace and say “ace” when dealt a king; call this the bluffing strategy (B). The other strategy is to say “ace” when dealt an ace and say “king” when dealt a king; call this the truth strategy (T). Recall that the only time Player 2 has a choice is when Player 1 says “ace.” In this case, Player 2 can call (C) or fold (F). Since we need to determine the payoff matrix, we first need to determine the payoffs for pure strategies. This is similar to what we did for the One-Card Stud game.
Consider Player 1's pure strategy of always bluffing when dealt a king (B) and Player 2's pure strategy of always calling (C). Determine the expected value for Player 1. What is Player 2's expected value?
- Hint
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You need to consider each possible deal.
Similarly, determine the expected value for Player 1 for the pure strategy pair [B, F]. What is Player 2's expected value?
Determine the expected value for Player 1 for the pure strategy pair [T, C]. What is Player 2's expected value?
Determine the expected value for Player 1 for the pure strategy pair [T, F]. What is Player 2's expected value?
Using the expected values you calculated in Exercise \(3.5.6\), Exercise \(3.5.7\), Exercise \(3.5.8\), and Exercise \(3.5.9\), set up the \(2 \times 2\) payoff matrix for Liar's Poker.
Once you have determined the payoff matrix for Liar's Poker, you can use either the graphical method or expected value method to solve the game. But before using either of these methods always check for a pure strategy equilibrium!
Using the payoff matrix you found in Exercise \(3.5.10\), does Liar's Poker have a pure strategy equilibrium? Explain.
Use any method you have learned to find a mixed strategy equilibrium for Liar's Poker. Give the mixed strategy for Player 1 and the mixed strategy for Player 2.
Compare your solution from Exercise \(3.5.12\) to your conjectured strategy from Exercise \(3.5.2\).
What is the expected value of the game for each player? How much would Player 1 expect to win if she played \(15\) games using the equilibrium mixed strategy?
Is this game fair? Explain. Again, compare your answer to your conjecture in Exercise \(3.5.2\).
Congratulations! You can now set up matrices for simple games of chance and solve for a mixed strategy equilibrium. Before solving a more complicated game, let's get the help of technology for solving larger matrix games.


8.19: Outcome- Developing Positioning Statements
What you’ll learn to do: develop and evaluate positioning statements based on defined criteria
After marketers work through the process of homing in on the best positioning strategy, they arrive at the final step: the positioning statement. The positioning statement reflects everything you’ve learned up to that point about how your product, service, or brand can best reach your target segment. As a document, it explains exactly how you plan to provide value to those target customers. In effect, it’s a short, persuasive argument.
In this next section, you’ll learn a simple formula for creating effective positioning statements. You’ll also learn how to evaluate existing positioning statements and decide whether and how they might be improved.
The specific things you’ll learn in this section include:
- Describe a standard structure for positioning statements
- Outline criteria for a strong positioning statement
- Recognize good examples of positioning statements
- Create a positioning statement aligned with a value proposition and target audience
Learning Activities
The learning activities for this section include the following:
- Reading: Developing Positioning Statements
- Self Check: Developing Positioning Statements
Contributors and Attributions
- Outcome: Developing Positioning Statements. Provided by : Lumen Learning. License : CC BY: Attribution


4.3: Role of Banks
What you’ll learn to do: explain the role of banks in the U.S. monetary system
In this section you’ll learn how banks serve as intermediaries for the flow of money between lenders and borrowers. You’ll also learn about the system in which which banks operate: the Federal Reserve System.
Learning Objectives
- Explain how banks act as intermediaries between savers and borrowers
- Explain how the Federal Reserve System implements monetary policy
- Explain how the Federal Reserve System implements monetary policy
Measuring and Tracking the Money Supply
Now that you have a good understanding of money, what qualifies as money, and how money facilitates exchanges between buyers and sellers, we need to look at how money evolves from a medium of exchange to a system. There was a time in the United States when there was no monetary system, and buyers and sellers who traveled from state to state had to carry multiple currencies. The Confederate States of America dollar was issued by the newly formed confederacy just before the outbreak of the American Civil War. It wasn’t backed by hard assets (i.e., commodities) but simply by a promise to pay the bearer after the war, on the prospect of Southern victory and independence. [1] Alabama, Arkansas, Florida, Georgia, Louisiana, Mississippi, South Carolina, Tennessee, Texas, and Virginia each printed and circulated currency that had value only within the state. It was not until 1863 when President Lincoln signed the National Banking Act that the federal dollar was established as the sole currency in the United States.
What about other kinds of currency? Cash in your wallet certainly serves as money, but how about checks or credit cards? Are they money, too? Rather than trying to determine a single way of measuring money, economists offer broader definitions of money based on liquidity. Liquidity refers to how quickly a financial asset can be used to buy a good or service. For example, cash is very liquid. Your $10 bill can easily be used to buy a hamburger at lunchtime. However, $10 that you have in your savings account is not so easy to use. You must go to the bank or ATM machine and withdraw that cash to buy your lunch. Thus, $10 in your savings account is less liquid.
The Federal Reserve Bank, which is the central bank of the United States, is a bank regulator. It’s responsible for monetary policy, and it defines money according to its liquidity. You will learn more about the Federal Reserve System in the next section. There are two definitions of money: M1 and M2 money supply. M1 money supply includes those monies that are very liquid such as cash, checkable (demand) deposits, and traveler’s checks. M2 money supply is less liquid in nature and includes M1 monies plus savings and time deposits, certificates of deposits, and money market funds.
M1 money supply includes coins and currency in circulation —the coins and bills that circulate in an economy that are not held by the U.S. Treasury, at the Federal Reserve Bank, or in bank vaults. Closely related to currency are checkable deposits, also known as demand deposits . These are the amounts held in checking accounts. They are called demand deposits or checkable deposits because the banking institution must give the deposit holder his money “on demand” when a check is written or a debit card is used. These items together—currency, and checking accounts in banks—comprise the money defined as M1, which is measured daily by the Federal Reserve System. Traveler’s checks are also included in M1 but have recently decreased in use.
M2 is a broader category of money. It includes everything in M1 but also adds other types of deposits. For example, M2 includes savings deposits in banks, which are bank accounts on which you cannot write a check directly, but from which you can easily withdraw the money at an automatic teller machine or bank. Many banks and other financial institutions also offer a chance to invest in money market funds , where the deposits of many individual investors are pooled together and invested in a safe way, such as in short-term government bonds. Another portion of M2 are the relatively small (that is, less than about $100,000) certificates of deposit (CDs) or time deposits , which are accounts that the depositor has committed to leaving in the bank for a certain period of time, ranging from a few months to a few years, in exchange for a higher interest rate. In short, all these types of M2 are money that you can withdraw and spend, but which require a greater effort to do so than the items in M1. Figure 1, below, should help you visualize the relationship between M1 and M2. Note that M1 is included in the M2 calculation.
The Federal Reserve System is responsible for tracking the amounts of M1 and M2 and prepares a weekly release of information about the money supply. For example, according to the Federal Reserve Bank’s measure of the U.S. money stock, at the end of February 2015, M1 in the United States was $3 trillion, while M2 was $11.8 trillion. A breakdown of the portion of each type of money that comprised M1 and M2 in February 2015, as reported by the Federal Reserve Bank, is provided in Table 1.
| Components of M1 in the U.S. (February 2015, seasonally adjusted) | $ billions |
|---|---|
| Currency | $1,271.8 |
| Traveler’s checks | $2.9 |
| Demand deposits and other checking accounts | $1,713.5 |
| Total M1 | $2,988.2 (or $3 trillion) |
| Components of M2 in the U.S. (February 2015, seasonally adjusted) | $ billions |
| M1 money supply | $2,988.2 |
| Savings accounts | $7,712.1 |
| Time deposits | $509.2 |
| Individual money market mutual fund balances | $610.8 |
| Total M2 | $11,820.3 (or $11.8 trillion) |
The lines separating M1 and M2 can become a little blurry. Sometimes elements of M1 are not treated alike; for example, some businesses will not accept personal checks for large amounts but will accept traveler’s checks or cash. Changes in banking practices and technology have made the savings accounts in M2 more similar to the checking accounts in M1. For example, some savings accounts will allow depositors to write checks, use automatic teller machines, and pay bills over the Internet, which has made it easier to access savings accounts. As with many other economic terms and statistics, the important point is to know the strengths and limitations of the various definitions of money, not to believe that such definitions are as clear-cut to economists as, say, the definition of nitrogen is to chemists.
Where does “plastic money” like debit cards, credit cards, and smart money fit into this picture? A debit card, like a check, is an instruction to the user’s bank to transfer money directly and immediately from your bank account to the seller. It is important to note that in our definition of money, it’s checkable deposits that are money, not the paper check or the debit card. Although you can make a purchase with a credit card, it is not considered money but rather a short term loan from the credit card company to you. When you make a purchase with a credit card, the credit card company immediately transfers money from its checking account to the seller, and at the end of the month, the credit card company sends you a bill for what you have charged that month. Until you pay the credit card bill, you have effectively borrowed money from the credit card company. With a smart card, you can store a certain value of money on the card and then use the card to make purchases. Some “smart cards” used for specific purposes, like long-distance phone calls or making purchases at a campus bookstore and cafeteria, are not really all that smart, because they can only be used for certain purchases or in certain places.
In short, credit cards, debit cards, and smart cards are different ways to move money when a purchase is made. But having more credit cards or debit cards does not change the quantity of money in the economy, any more than having more checks printed increases the amount of money in your checking account.
One key message here is that counting and tracking the money in a modern economy doesn’t just involve paper bills and coins; instead, money is closely linked to bank accounts. Indeed, the macroeconomic policies concerning money are largely conducted through the banking system. The next section explains how banks function as an intermediary to financial transactions.
Banks As Financial Intermediaries
The late bank robber named Willie Sutton was once asked why he robbed banks. He answered: “That’s where the money is.” While this may have been true at one time, from the perspective of modern economists, Sutton is both right and wrong. He is wrong because the overwhelming majority of money in the economy is not in the form of currency sitting in vaults or drawers at banks, waiting for a robber to appear. Most money is in the form of bank accounts, which exist only as electronic records on computers. From a broader perspective, however, the bank robber was more right than he may have known. Banking is intimately interconnected with money and, consequently, with the broader economy.
Banks make it far easier for a complex economy to carry out the extraordinary range of transactions that occur in goods, labor, and financial capital markets. Imagine for a moment what the economy would be like if all payments had to be made in cash. When shopping for a large purchase or going on vacation, you might need to carry hundreds of dollars in a pocket or purse. Even small businesses would need stockpiles of cash to pay workers and to purchase supplies. A bank allows people and businesses to store this money in either a checking account or savings account, for example, and then withdraw this money as needed through the use of a direct withdrawal, writing a check, or using a debit card.
Banks are a critical intermediary in what is called the payment system, which helps an economy exchange goods and services for money or other financial assets. Also, people with extra money that they’d like to save can store their money in a bank rather than look for an individual who is willing to borrow it from them and then repay them at a later date. Those who want to borrow money can go directly to a bank rather than trying to find someone to lend them cash. Thus, banks act as financial intermediaries—they bring savers and borrowers together.
An intermediary is one who stands between two other parties. Banks are a financial intermediary —that is, an institution that operates between a saver who deposits money in a bank and a borrower who receives a loan from that bank. All the funds deposited are mingled in one big pool, which is then loaned out. Figure 2 illustrates the position of banks as financial intermediaries, with deposits flowing into a bank and loans flowing out.
For some concrete examples of what banks do, watch the following video from Paul Solman’s Making Sense of Financial News .
The Federal Reserve System
Figure \(\PageIndex{3}\). Marriner S. Eccles Federal Reserve Headquarters, Washington, DC. Some of the most influential decisions regarding monetary policy in the United States are made behind these doors.
Money, loans, and banks are all tied together. Money is deposited in bank accounts, which is then loaned to businesses, individuals, and other banks. When the interlocking system of money, loans, and banks works well, economic transactions in goods and labor markets happen smoothly, and savers are connected with borrowers. If the money and banking system does not operate smoothly, the economy can either fall into recession or suffer prolonged inflation.
The government of every country has public policies that support the system of money, loans, and banking. But these policies do not always work perfectly. In this section we will explore how monetary policy works and what may prevent it from working perfectly.
In making decisions about the money supply—that is, the total amount of monetary assets available in an economy at a specific time—a central bank decides whether to raise or lower interest rates and, in this way, to influence macroeconomic policy, whose goal is low unemployment and low inflation. The central bank is also responsible for regulating all or part of the nation’s banking system to protect bank depositors and insure the health of the bank’s finances.
The organization responsible for conducting monetary policy and ensuring that a nation’s financial system operates smoothly is called the central bank . Most nations have central banks or currency boards. Some prominent central banks around the world include the European Central Bank, the Bank of Japan, and the Bank of England. In the United States, the central bank is called the Federal Reserve —often abbreviated as “the Fed.” This section explains the organization of the U.S. Federal Reserve and identifies the major responsibilities of a central bank.
Structure/Organization of the Federal Reserve
Unlike most central banks, the Federal Reserve is semi-decentralized, mixing government appointees with representation from private-sector banks. At the national level, it is run by a board of governors, consisting of seven members appointed by the president of the United States and confirmed by the Senate. Appointments are for fourteen-year terms and they are arranged so that one term expires January 31 of every even-numbered year. The purpose of the long and staggered terms is to insulate the board of governors as much as possible from political pressure so that policy decisions can be made based only on their economic merits. In addition, except when filling an unfinished term, each member only serves one term, further insulating decision-making from politics. Policy decisions of the Fed do not require congressional approval, and the president cannot ask for the resignation of a Federal Reserve governor as the president can with cabinet positions.
One member of the board of governors is designated as the chair. For example, from 1987 until early 2006, the chair was Alan Greenspan. From 2006 until 2014, Ben Bernanke held the post. The current chair, Janet Yellen, has made many headlines already. Why? See the following feature to find out.
Who has the most immediate economic power in the world
With a PhD in economics from Yale University, Yellen has taught macroeconomics at Harvard, the London School of Economics, and most recently at the University of California at Berkeley. From 2004–2010, Yellen was president of the Federal Reserve Bank of San Francisco.
Not an ivory-tower economist, Yellen became one of the few economists who warned about a possible bubble in the housing market, more than two years before the financial crisis occurred. Yellen served on the board of governors of the Federal Reserve twice, most recently as vice chair. She also spent two years as chair of the President’s Council of Economic Advisors. If experience and credentials mean anything, Yellen is likely to be an effective Fed chair.
The Fed chair is first among equals on the board of governors. While he or she has only one vote, the chair controls the agenda, and is the public voice of the Fed, so he or she has more power and influence than one might expect.
The Federal Reserve is more than the board of governors. The Fed also includes twelve regional Federal Reserve banks , each of which is responsible for supporting the commercial banks and economy generally in its district. The Federal Reserve districts and the cities where their regional headquarters are located are shown in Figure \(\PageIndex{5}\). The commercial banks in each district elect a board of directors for each regional Federal Reserve bank, and that board chooses a president for each regional Federal Reserve district. Thus, the Federal Reserve System includes both federally and private-sector appointed leaders.
What Does a Central Bank Do?
The Federal Reserve, like most central banks, is designed to perform the following three important functions:
- To conduct monetary policy
- To promote stability of the financial system
- To provide banking services to commercial banks and other depository institutions, and to provide banking services to the federal government
The Federal Reserve provides many of the same services to banks as banks provide to their customers. For example, all commercial banks have an account at the Fed where they deposit reserves, and they can obtain loans from the Fed through the “discount window,” which will be discussed in the next reading. The Fed is also responsible for check processing. When you write a check to buy groceries, for example, the grocery store deposits the check in its bank account. Then, the physical check (or an image of that actual check) is returned to your bank, after which funds are transferred from your bank account to the account of the grocery store. The Fed is responsible for how these transactions are handled once the check leaves the cash register and is deposited into the store’s bank account. Does that mean that your check to the grocery store goes all the way to Washington, DC.? No. Instead the regulations that govern how banks handle checks, deposits, withdrawals are regulated by The Federal Reserve Act. This act is the reason that a bank must start paying you interest on a savings deposit the day it is received. It’s also the reason that if you deposit a large check, your bank may tell you that the funds will not be available for three to five business days.
On a more mundane level, the Federal Reserve ensures that enough currency and coins are circulating through the financial system to meet public demands. For example, each year the Fed increases the amount of currency available in banks around the Christmas shopping season and reduces it again in January.
Finally, the Fed is responsible for assuring that banks are in compliance with a wide variety of consumer protection laws. For example, banks are forbidden from discriminating on the basis of age, race, sex, or marital status. Banks are also required to publicly disclose information about the loans they make for buying houses and how those loans are distributed geographically, as well as by sex and race of the loan applicants.
How a Central Bank Executes Monetary Policy
The most important function of the Federal Reserve is to conduct the nation’s monetary policy. Article I, Section 8 of the U.S. Constitution gives Congress the power “to coin money” and “to regulate the value thereof.” As part of the 1913 legislation that created the Federal Reserve, Congress delegated these powers to the Fed. Monetary policy involves managing interest rates and credit conditions, which influence the level of economic activity, as described in more detail below.
A central bank has the following three traditional tools to implement monetary policy in the economy:
- Open market operations
- Changing reserve requirements
- Changing the discount rate
In discussing how these three tools work, it is useful to think of the central bank as a “bank for banks”—that is, each private-sector bank has its own account at the central bank. We will discuss each of these monetary policy tools in the sections below.
Open Market Operations
The most commonly used tool of monetary policy in the U.S. is open market operations. Open market operations take place when the central bank sells or buys U.S. Treasury bonds in order to influence the quantity of bank reserves and the level of interest rates. The specific interest rate targeted in open market operations is the federal funds rate. The name is a bit of a misnomer since the federal funds rate is the interest rate charged by commercial banks making overnight loans to other banks. As such, it is a very short-term interest rate, but one that reflects credit conditions in financial markets very well.
The Federal Open Market Committee (FOMC) makes the decisions regarding these open market operations. The FOMC is made up of the seven members of the Federal Reserve’s board of governors. It also includes five voting members who are drawn, on a rotating basis, from the regional Federal Reserve Banks. The New York district president is a permanent voting member of the FOMC and the other four spots are filled on a rotating, annual basis, from the other eleven districts. The FOMC typically meets every six weeks, but it can meet more frequently if necessary. The FOMC tries to act by consensus; however, the chairman of the Federal Reserve has traditionally played a very powerful role in defining and shaping that consensus. For the Federal Reserve, and for most central banks, open market operations have, over the last few decades, been the most commonly used tool of monetary policy. The following video explains how these operations work.
Is it a sale of bonds by the central bank that increases bank reserves and lowers interest rates, or is it a purchase of bonds by the central bank? The easy way to keep track of this is to treat the central bank as being outside the banking system. When a central bank buys bonds, money is flowing from the central bank to individual banks in the economy, increasing the supply of money in circulation. When a central bank sells bonds, then money from individual banks in the economy is flowing into the central bank—reducing the quantity of money in the economy.
Changing Reserve Requirements
A second method of conducting monetary policy is for the central bank to raise or lower the reserve requirement, which is the percentage of each bank’s deposits that it is legally required to hold either as cash in their vault or on deposit with the central bank. If banks are required to hold a greater amount in reserves, they have less money available to lend out. If banks are allowed to hold a smaller amount in reserves, they will have a greater amount of money available to lend out. The following video will explain how changing the reserve requirement alters the money supply.
In early 2015, the Federal Reserve required banks to hold reserves equal to 0% of the first $14.5 million in deposits, then to hold reserves equal to 3% of the deposits up to $103.6 million, and 10% of any amount above $103.6 million. Small changes in the reserve requirements are made almost every year. For example, the $103.6 million dividing line is sometimes bumped up or down by a few million dollars. In practice, large changes in reserve requirements are rarely used to execute monetary policy. A sudden demand that all banks increase their reserves would be extremely disruptive and difficult to comply with, while loosening requirements too much would create a danger of banks being unable to meet the demand for withdrawals.
Changing the Discount Rate
The Federal Reserve was founded in the aftermath of the Financial Panic of 1907 when many banks failed as a result of bank runs. As mentioned earlier, since banks make profits by lending out their deposits, no bank, can withstand a bank run. As a result of the Panic, the Federal Reserve was founded to be the “lender of last resort.” In the event of a bank run, sound banks could borrow as much cash as they needed from the Fed’s discount “window” to cover the bank run. The interest rate banks pay for such loans is called the discount rate . They are so named because loans are made against the bank’s outstanding loans “at a discount” of their face value. Once depositors became convinced that the bank would be able to honor their withdrawals, they no longer had a reason to make a run on the bank. In short, the Federal Reserve was originally intended to provide credit passively, but in the years since its founding, the Fed has taken on a more active role with monetary policy.
So, the third traditional method for conducting monetary policy is to raise or lower the discount rate. If the central bank raises the discount rate, then commercial banks will reduce their borrowing of reserves from the Fed, and instead call in loans to replace those reserves. Since fewer loans are available, the money supply falls and market interest rates rise. If the central bank lowers the discount rate it charges to banks, the process works in reverse.
The following video explains the impact of changes to the Fed’s discount rate.
In recent decades, the Federal Reserve has made relatively few discount loans. Before a bank borrows from the Federal Reserve to fill out its required reserves, the bank is expected to first borrow from other available sources, like other banks. This is encouraged by the Fed charging a higher discount rate than the federal funds rate.  More important, the Fed has found from experience that open market operations are a more precise and powerful means of executing any desired monetary policy.
Chair the Fed
Achieve Low Unemployment and Low Inflation Rates
We have now seen that the Fed has three primary goals and a set of tools at its disposal to help it achieve these goals. If you were the chairperson of the Federal Reserve, do you think that you could accomplish these goals? Let’s find out!
After reading the following information, click on the Chair the Fed link below, which will take you to the Federal Reserve Bank of San Francisco Web site, where YOU will act as the Chair of the Fed. By manipulating the fed funds rate, you will try to keep inflation and unemployment at target rates.
Instructions for Playing the Game
The game puts the player in the role of setting monetary policy as Chair of the Fed. The goals are as follows: inflation (2 percent) and unemployment (5 percent). Remember that the fed funds rate is the primary tool for monetary policy and is shown on the game screen (green line in the chart area is initially set at 4 percent rate).
Record the starting levels for inflation, unemployment, and the fed funds rate (2.11 percent, 4.68 percent, and 4.00 percent, respectively) in your notes by making a small table with four columns labeled: Quarters Remaining, Inflation, Unemployment, and Fed Funds Rate.
Review the “rules” and functions of the simulation by clicking on “ YOUR JOB .”
Start the game by clicking on the “ Go ” button. Once the first quarter is completed (fifteen quarters remaining), record all three rates. Using the “raise” and “cut” buttons, make adjustments to the fed funds rate. The information in the headline reflects changes in the levels of inflation and unemployment.
Work through the remaining fifteen quarters, pausing to review each headline and record the new values of inflation and unemployment.
The game ends on an announcement screen indicating “Congratulations” if the Chair has kept the economy on track (close to the goals for inflation and unemployment) or “Sorry” if the goals have not been met.
Good Luck!
Play the Chair the Fed Simulation


5.7: Chapter 5 Activity
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Let’s examine prioritizing in your life. Think about the last week or seven days. Use the space below to list all of the primary activities you did over those seven days (e.g. work, school, errands, childcare, etc.).
Now list smaller activities including meal prep, self-care, commuting/driving, etc.
Look at each item and analyze the importance of each activity in relation to your life goals plus consider how urgent each activity was. Next to each item listed above, write a “I,” “II,” “III,” and “IV” to indicate into which Quadrant each activity falls. Finally list how many activities are in each Quadrant.
Quadrant I =
Quadrant II =
Quadrant III =
Quadrant IV =
Where are you spending your time? Are you stressed and overwhelmed because everything is falling into Quadrant I? Are you feeling pretty much in control because most activities are planned and important in II? Are you responding to others’ demands in III? Are you spending a lot of unproductive time in IV? What changes do you want to make based on this analysis?


33.9: Where is American Literature Today?- The Answer Might Surprise You
Since the dawn of the country, works been contributed to the vast group of texts considered American literature. From the works detailing the first colonial encounter all the way to poems hastily created within two-hundred and eighty characters on Twitter, all can fall under the umbrella. The multitude of works today were all influenced by their predecessors, earlier American texts. Often touching on early topics such as the colonial encounter, expansion, or slavery, these works make up the basis for what is deemed American literature today. American literature has undoubtedly changed and evolved a lot since then in its content, distribution, medium, and reaction. As we work to fit new molds into existing ideas of American literature, the very definition and notions held must be abolished if we are to be able to examine a new age of American literature.
“This is America” by Childish Gambino took the world by a storm when it was first released in May of 2018. It made a political commentary on a topic that people had been all too happy to sweep under the rug—racism in America. Gambino’s work was praised for being revolutionary and bringing to the forefront issues that have been mostly ignored since the civil rights movement. However, what Gambino was doing was not as new or revolutionary as the media reported it to be. Gambino had a long line of history that influenced multiple aspects in his music and video.
https://www.youtube.com/watch?v=VYOjWnS4cMY&w=560&h=315
“This is America” is loud and proud about its message. This song attempts to display the current climate surrounding race relations in America. Using many visual images from contemporary black history and contrasting them with earlier ones, the video is adamant about the fact that race relations today may not be much different from the time of early black advocates.  However, these are contrasted by the much older images. A person riding by on a horse conjures images of the KKK. Gambino dons confederate army trousers through the entirety of the video. Gambino is seen striking an exaggerated pose before shooting the guitarist in the beginning of the video. This is a reference to Jim Crow, the system for oppression used on slaves in the south. The scene at the end where Gambino is running terrified at the audience can easily be compared to enslaved people as they raced to freedom and the north. Gambino used his song as a call to action, to raise awareness for the state of black people living in America today.
Gambino utilizes a style that is raw and in-your-face. The video showcases multiple scenes that are violent and brutal; nothing is censored for the audience. Gambino puts on display death, murder, persecution, mass shootings, poverty, and supremacy to name just a few. They audience is expected to sit through four minutes of the authentic black experience in America. He utilizes complex scenes with most of the action taking place in the background. The bitter realities he attempts to show us are hidden behind his upbeat dancing. This parallels real life where issues that need to be addressed are often hidden behind “fluff” and entertainment.
So, what does a text like this have to do with the study of early American literature? This video connects strongly to many canonical texts in early American literature, specifically slave narratives. Gambino was doing with his music video what other activists have been doing since the dawn of slavery. Texts like Uncle Tom’s Cabin, The Heroic Slave, and Incidents in the Life of a Slave Girl all showcase the heart and message captured by Gambino in “This is America.” The Heroic Slave in particular shares an affinity with this song. Frederick Douglass, like Gambino, believes strongly in the cause to bring about change to the lives of black people. While he was advocating for the end to enslavement, Gambino calls to end black oppression, the modern-day slavery. “This is America” calls on images of slavery, the KKK and the running Gambino, to highlight to connection between the modern-day oppression of blacks and slavery. There is an interconnectedness to the two topics that Gambino did not ignore. Black people living in America today and slaves had a similar struggle, the context and means of the struggle is what has changed over the years. The authors of these early slave texts paved a path for Gambino to speak openly of his struggle today.
Undeniably different is the medium with which Gambino portrays his message from his predecessors. In the time of early American literature, texts had to be printed and distributed to their intended audiences. This took time and resources to make. These texts had to cost money because of the material and labor needed to make them, making them only available to people who could pay. Distribution is a key issue with this method for spreading information. Some people lived isolated on farms and received limited knowledge about what is happening in the real world; recent print material would be hard to come by for them. Additionally, many people were not literate at the time, even with access to the texts they would be unable to understand the message. The spread of knowledge, and conversation around these key texts, was also significantly different in the time or early American literature. Ideas were mostly spread through conversation. The types of insight gained through a conversation with someone is limited greatly to the people you are discussing with. The knowledge people have on the topic and their own personal opinions drastically change the trajectory of their interpretations and impact of the message. This differs drastically when compared to today. Today people primarily consume media through the internet. Modern texts can be found for free and can be watched infinite amount of times. While mediums like a high production music video like “This is America,” still takes time and money to make, it can be shared instantly to a widespread audience. The way in which people convey their messages has also changed profoundly. Today people are much more likely to opt for a song, video, or social media post. These are far more accessible to all people and lead to them being more widely appreciated. Conversations surrounding a piece have changed drastically as well, gone are the days a few people in a room constituted a sound dialogue for a text. Today texts can be talked about between a diverse range of people. The news might pick up a story and spread it. Conversations sparked on social media proliferates the message and sparks conversation between a widespread group of people. This contributes to a stronger overall understanding of a text.
Modern technologies undeniably make a text more accessible and allow for a more widespread conversation sparked by a multitude of viewpoints to increase understanding. These new ways of portraying a message can also contribute to a sounder method for delivery. There are things that can be portrayed in a video that simply cannot be brought through in a written text. For instance, “This is America” creates a chaotic message through intense visuals. How could all of this be effectively portrayed through writing alone? Social media allows for more interaction with a text across demographic, geographic, and socio-economic backgrounds. Thus, people can get closer to what the text actually means. With the accessibility of technology, a text can actually reach its intended audience. This contributes to an increased power to spark change compared to print media which is timely and hard to circulate.
The modern era has brought about big changes in the study of literature. People today are less likely to pick up canonical works outside of an educational environment. This contributes to a decrease in the overall readings of these pieces. Although people may have a better understanding for the ones that they do read. People spend a lot more time consuming digital media. However, are the works we consume today really disconnected from early American literature? Early American literature gave us the framework for many topics, subjects, stylistic element, grammar and modern struggles currently written about. When people do pick up a text today, they are more likely to read it on a device. This changes significantly how we interact with a text. Technology makes it a lot easier to share our ideas on what we read, watch, hear or see. The study of literature is no longer a solitary pursuit but a collaborative one. Programs like hypothes.is turn the reading on a text into a collaborative venture with people from across the globe. Collaboration contributes to a better overall understanding and interest in a subject. Open anthologies bring the study and conversations surrounding literature into an accessible, easy to use format. The internet in general serves us a place where information can be complied in vast quantities. Thus, allowing access to works that have not entered the canon.
The residual impact of early American literature is all around us in nearly all media we consume. It has infiltrated every area of our society. While the method for consuming these age-old messages has changed the desire for people to speak out on injustices and record the important events they see happening around them has not. Childish Gambino found a way to capitalize on the past and created the powerfully political work “This is America.”


1.7: Russell’s Paradox
As the ideas for set theory were explored, there were attempts to define sets as broadly as possible. It was hoped that any collection of mathematical objects that could be defined by a formula would qualify as a set. This belief was known as the General Comprehension Principle (GCP). Unfortunately, the GCP gave rise to conclusions which were unacceptable for mathematics. Consider the collection defined by the following simple formula: \[V=\{x \mid x \text { is a set and } x=x\} .\] If \(V\) is considered as a set, then since \(V=V\) , \[V \in V \text {. }\] If this is not an inconsistency, it is at least unsettling. Unfortunately, it gets worse. Consider the collection \[X=\{x \mid x \notin x\} .\] Then \[X \in X \text { if and only if } X \notin X \text {. }\] This latter example is called Russell’s paradox, and showed that the GCP is false. Clearly there would have to be some control over which definitions give rise to sets. Axiomatic set theory was developed to provide rules for rigorously defining sets. We give a brief discussion in Appendix B.