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Conceptual Questions
-
Give one example of a transverse wave and one example of a longitudinal wave, being careful to note the relative directions of the disturbance and wave propagation in each.
-
A sinusoidal transverse wave has a wavelength of 2.80 m. It takes 0.10 s for a portion of the string at a position x to move from a maximum position of y = 0.03 m to the equilibrium position y = 0. What are the period, frequency, and wave speed of the wave?
-
What is the difference between propagation speed and the frequency of a mechanical wave? Does one or both affect wavelength? If so, how?
-
Consider a stretched spring, such as a slinky. The stretched spring can support longitudinal waves and transverse waves. How can you produce transverse waves on the spring? How can you produce longitudinal waves on the spring?
-
Consider a wave produced on a stretched spring by holding one end and shaking it up and down. Does the wavelength depend on the distance you move your hand up and down?
-
A sinusoidal, transverse wave is produced on a stretched spring, having a period T. Each section of the spring moves perpendicular to the direction of propagation of the wave, in simple harmonic motion with an amplitude A. Does each section oscillate with the same period as the wave or a different period? If the amplitude of the transverse wave were doubled but the period stays the same, would your answer be the same?
-
An electromagnetic wave, such as light, does not require a medium. Can you think of an example that would support this claim?
-
If you were to shake the end of a taut spring up and down 10 times a second, what would be the frequency and the period of the sinusoidal wave produced on the spring?
-
If you shake the end of a stretched spring up and down with a frequency f, you can produce a sinusoidal, transverse wave propagating down the spring. Does the wave number depend on the frequency you are shaking the spring?
-
Does the vertical speed of a segment of a horizontal taut string through which a sinusoidal, transverse wave is propagating depend on the wave speed of the transverse wave?
-
In this section, we have considered waves that move at a constant wave speed. Does the medium accelerate?
-
If you drop a pebble in a pond you may notice that several concentric ripples are produced, not just a single ripple. Why do you think that is?
-
If the tension in a string were increased by a factor of four, by what factor would the wave speed of a wave on the string increase?
-
Does a sound wave move faster in seawater or fresh water, if both the sea water and fresh water are at the same temperature and the sound wave moves near the surface?$$\rho_{w} \approx 1000\; kg/m^{3}, \rho_{s} \approx 1030\; kg/m^{3}, B_{w} = 2.15 \times 10^{9}\; Pa, B_{s} = 2.34 \times 10^{9}\; Pa$$
-
Guitars have strings of different linear mass density. If the lowest density string and the highest density string are under the same tension, which string would support waves with the higher wave speed?
-
Shown below are three waves that were sent down a string at different times. The tension in the string remains constant. (a) Rank the waves from the smallest wavelength to the largest wavelength. (b) Rank the waves from the lowest frequency to the highest frequency.
-
Electrical power lines connected by two utility poles are sometimes heard to hum when driven into oscillation by the wind. The speed of the waves on the power lines depend on the tension. What provides the tension in the power lines?
-
Two strings, one with a low mass density and one with a high linear density are spliced together. The higher density end is tied to a lab post and a student holds the free end of the low-mass density string. The student gives the string a flip and sends a pulse down the strings. If the tension is the same in both strings, does the pulse travel at the same wave velocity in both strings? If not, where does it travel faster, in the low density string or the high density string?
-
Consider a string with under tension with a constant linear mass density. A sinusoidal wave with an angular frequency and amplitude produced by some external driving force. If the frequency of the driving force is decreased to half of the original frequency, how is the time-averaged power of the wave affected? If the amplitude of the driving force is decreased by half, how is the time-averaged power affected? Explain your answer.
-
Circular water waves decrease in amplitude as they move away from where a rock is dropped. Explain why.
-
In a transverse wave on a string, the motion of the string is perpendicular to the motion of the wave. If this is so, how is possible to move energy along the length of the string?
-
The energy from the sun warms the portion of the earth facing the sun during the daylight hours. Why are the North and South Poles cold while the equator is quite warm?
-
The intensity of a spherical waves decreases as the wave moves away from the source. If the intensity of the wave at the source is I
0
, how far from the source will the intensity decrease by a factor of nine?
-
An incident sinusoidal wave is sent along a string that is fixed to the wall with a wave speed of v. The wave reflects off the end of the string. Describe the reflected wave.
-
A string of a length of 2.00 m with a linear mass density of \(\mu\) = 0.006 kg/m is attached to the end of a 2.00-m-long string with a linear mass density of \(\mu\) = 0.012 kg/m. The free end of the higher-density string is fixed to the wall, and a student holds the free end of the low-density string, keeping the tension constant in both strings. The student sends a pulse down the string. Describe what happens at the interface between the two strings.
-
A long, tight spring is held by two students, one student holding each end. Each student gives the end a flip sending one wavelength of a sinusoidal wave down the spring in opposite directions. When the waves meet in the middle, what does the wave look like?
-
Many of the topics discussed in this chapter are useful beyond the topics of mechanical waves. It is hard to conceive of a mechanical wave with sharp corners, but you could encounter such a wave form in your digital electronics class, as shown below. This could be a signal from a device known as an analog to digital converter, in which a continuous voltage signal is converted into a discrete signal or a digital recording of sound. What is the result of the superposition of the two signals?
-
A string of a constant linear mass density is held taut by two students, each holding one end. The tension in the string is constant. The students each send waves down the string by wiggling the string. (a) Is it possible for the waves to have different wave speeds? (b) Is it possible for the waves to have different frequencies? (c) Is it possible for the waves to have different wavelengths?
-
A truck manufacturer finds that a strut in the engine is failing prematurely. A sound engineer determines that the strut resonates at the frequency of the engine and suspects that this could be the problem. What are two possible characteristics of the strut can be modified to correct the problem?
-
Why do roofs of gymnasiums and churches seem to fail more than family homes when an earthquake occurs?
-
Wine glasses can be set into resonance by moistening your finger and rubbing it around the rim of the glass. Why?
-
Air conditioning units are sometimes placed on the roof of homes in the city. Occasionally, the air conditioners cause an undesirable hum throughout the upper floors of the homes. Why does this happen? What can be done to reduce the hum?
-
Consider a standing wave modeled as y(x, t) = 4.00 cm sin (3 m
−1
x) cos (4 s
−1
t). Is there a node or an antinode at x = 0.00 m? What about a standing wave modeled as y(x, t) = 4.00 cm sin (3 m
−1
x + \(\frac{\pi}{2}\)) cos (4 s
−1
t)? Is there a node or an antinode at the x = 0.00 m position?
Problems
-
Storms in the South Pacific can create waves that travel all the way to the California coast, 12,000 km away. How long does it take them to travel this distance if they travel at 15.0 m/s?
-
Waves on a swimming pool propagate at 0.75 m/s. You splash the water at one end of the pool and observe the wave go to the opposite end, reflect, and return in 30.00 s. How far away is the other end of the pool?
-
Wind gusts create ripples on the ocean that have a wavelength of 5.00 cm and propagate at 2.00 m/s. What is their frequency?
-
How many times a minute does a boat bob up and down on ocean waves that have a wavelength of 40.0 m and a propagation speed of 5.00 m/s?
-
Scouts at a camp shake the rope bridge they have just crossed and observe the wave crests to be 8.00 m apart. If they shake the bridge twice per second, what is the propagation speed of the waves?
-
What is the wavelength of the waves you create in a swimming pool if you splash your hand at a rate of 2.00 Hz and the waves propagate at a wave speed of 0.800 m/s?
-
What is the wavelength of an earthquake that shakes you with a frequency of 10.0 Hz and gets to another city 84.0 km away in 12.0 s?
-
Radio waves transmitted through empty space at the speed of light (v = c = 3.00 x 10
8
m/s) by the Voyager spacecraft have a wavelength of 0.120 m. What is their frequency?
-
Your ear is capable of differentiating sounds that arrive at each ear just 0.34 ms apart, which is useful in determining where low frequency sound is originating from. (a) Suppose a low-frequency sound source is placed to the right of a person, whose ears are approximately 18 cm apart, and the speed of sound generated is 340 m/s. How long is the interval between when the sound arrives at the right ear and the sound arrives at the left ear? (b) Assume the same person was scuba diving and a low-frequency sound source was to the right of the scuba diver. How long is the interval between when the sound arrives at the right ear and the sound arrives at the left ear, if the speed of sound in water is 1500 m/s? (c) What is significant about the time interval of the two situations?
-
(a) Seismographs measure the arrival times of earthquakes with a precision of 0.100 s. To get the distance to the epicenter of the quake, geologists compare the arrival times of S- and P-waves, which travel at different speeds. If S- and P-waves travel at 4.00 and 7.20 km/s, respectively, in the region considered, how precisely can the distance to the source of the earthquake be determined? (b) Seismic waves from underground detonations of nuclear bombs can be used to locate the test site and detect violations of test bans. Discuss whether your answer to (a) implies a serious limit to such detection. (Note also that the uncertainty is greater if there is an uncertainty in the propagation speeds of the S- and P-waves.)
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A Girl Scout is taking a 10.00-km hike to earn a merit badge. While on the hike, she sees a cliff some distance away. She wishes to estimate the time required to walk to the cliff. She knows that the speed of sound is approximately 343 meters per second. She yells and finds that the echo returns after approximately 2.00 seconds. If she can hike 1.00 km in 10 minutes, how long would it take her to reach the cliff?
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A quality assurance engineer at a frying pan company is asked to qualify a new line of nonstick-coated frying pans. The coating needs to be 1.00 mm thick. One method to test the thickness is for the engineer to pick a percentage of the pans manufactured, strip off the coating, and measure the thickness using a micrometer. This method is a destructive testing method. Instead, the engineer decides that every frying pan will be tested using a nondestructive method. An ultrasonic transducer is used that produces sound waves with a frequency of f = 25 kHz. The sound waves are sent through the coating and are reflected by the interface between the coating and the metal pan, and the time is recorded. The wavelength of the ultrasonic waves in the coating is 0.076 m. What should be the time recorded if the coating is the correct thickness (1.00 mm)?
-
A pulse can be described as a single wave disturbance that moves through a medium. Consider a pulse that is defined at time t = 0.00 s by the equation y(x) = \(\frac{6.00\; m^{3}}{x^{2} + 2.00\; m^{2}}\) centered around x = 0.00 m. The pulse moves with a velocity of v = 3.00 m/s in the positive x-direction. (a) What is the amplitude of the pulse? (b) What is the equation of the pulse as a function of position and time? (c) Where is the pulse centered at time t = 5.00 s ?
-
A transverse wave on a string is modeled with the wave function y(x, t) = (0.20 cm) sin(2.00 m
−1
x − 3.00 s
−1
t + \(\frac{\pi}{16}\)). What is the height of the string with respect to the equilibrium position at a position x = 4.00 m and a time t = 10.00 s?
-
Consider the wave function y(x, t) = (3.00 cm) sin(0.4 m
−1
x + 2.00 s
−1
t + \(\frac{\pi}{10}\)). What are the period, wavelength, speed, and initial phase shift of the wave modeled by the wave function?
-
A pulse is defined as y(x, t) = \(e ^{−2.77 \left(\dfrac{2.00(x − 2.00\; m/s(t))}{5.00\; m}\right)^{2}}\). Use a spreadsheet, or other computer program, to plot the pulse as the height of medium y as a function of position x. Plot the pulse at times t = 0.00 s and t = 3.00 s on the same graph. Where is the pulse centered at time t = 3.00 s ? Use your spreadsheet to check your answer.
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A wave is modeled at time t = 0.00 s with a wave function that depends on position. The equation is y(x) = (0.30 m) sin(6.28 m
−1
x). The wave travels a distance of 4.00 meters in 0.50 s in the positive x-direction. Write an equation for the wave as a function of position and time.
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A wave is modeled with the function y(x, t) = (0.25 m) cos(0.30 m
−1
x − 0.90 s
−1
t + \(\frac{\pi}{3}\)). Find the (a) amplitude, (b) wave number, (c) angular frequency, (d) wave speed, (e) initial phase shift, (f) wavelength, and (g) period of the wave.
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A surface ocean wave has an amplitude of 0.60 m and the distance from trough to trough is 8.00 m. It moves at a constant wave speed of 1.50 m/s propagating in the positive x-direction. At t = 0, the water displacement at x = 0 is zero, and vy is positive. (a) Assuming the wave can be modeled as a sine wave, write a wave function to model the wave. (b) Use a spreadsheet to plot the wave function at times t = 0.00 s and t = 2.00 s on the same graph. Verify that the wave moves 3.00 m in those 2.00 s.
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A wave is modeled by the wave function y(x, t) = (0.30 m) sin\(\Big[ \frac{2 \pi}{4.50\; m} (x − 18.00\; m/s\; t) \Big]\). What are the amplitude, wavelength, wave speed, period, and frequency of the wave?
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A transverse wave on a string is described with the wave function y(x, t) = (0.50 cm) sin(1.57 m
−1
x − 6.28 s
−1
t). (a) What is the wave velocity of the wave? (b) What is the magnitude of the maximum velocity of the string perpendicular to the direction of the motion?
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A swimmer in the ocean observes one day that the ocean surface waves are periodic and resemble a sine wave. The swimmer estimates that the vertical distance between the crest and the trough of each wave is approximately 0.45 m, and the distance between each crest is approximately 1.8 m. The swimmer counts that 12 waves pass every two minutes. Determine the simple harmonic wave function that would describes these waves.
-
Consider a wave described by the wave function y(x, t) = 0.3 m sin(2.00 m
−1
x − 628.00 s
−1
t). (a) How many crests pass by an observer at a fixed location in 2.00 minutes? (b) How far has the wave traveled in that time?
-
Consider two waves defined by the wave functions y
1
(x, t) = 0.50 m sin\(\left(\dfrac{2 \pi}{3.00\; m}x + \dfrac{2 \pi}{4.00\; s}t \right)\) and y
2
(x, t) = 0.50 m sin\(\left(\dfrac{2 \pi}{6.00\; m}x − \dfrac{2 \pi}{4.00\; s}t \right)\). What are the similarities and differences between the two waves?
-
Consider two waves defined by the wave functions y
1
(x, t) = 0.20 m sin\(\left(\dfrac{2 \pi}{6.00\; m}x − \dfrac{2 \pi}{4.00\; s}t \right)\) and y
2
(x, t) = 0.20 m cos\(\left(\dfrac{2 \pi}{6.00\; m}x − \dfrac{2 \pi}{4.00\; s}t \right)\). What are the similarities and differences between the two waves?
-
The speed of a transverse wave on a string is 300.00 m/s, its wavelength is 0.50 m, and the amplitude is 20.00 cm. How much time is required for a particle on the string to move through a distance of 5.00 km?
-
Transverse waves are sent along a 5.00-m-long string with a speed of 30.00 m/s. The string is under a tension of 10.00 N. What is the mass of the string?
-
A copper wire has a density of \(\rho\) = 8920 kg/m
3
, a radius of 1.20 mm, and a length L. The wire is held under a tension of 10.00 N. Transverse waves are sent down the wire. (a) What is the linear mass density of the wire? (b) What is the speed of the waves through the wire?
-
A piano wire has a linear mass density of \(\mu\) = 4.95 x 10
−3
kg/m. Under what tension must the string be kept to produce waves with a wave speed of 500.00 m/s?
-
A string with a linear mass density of \(\mu\) = 0.0060 kg/m is tied to the ceiling. A 20-kg mass is tied to the free end of the string. The string is plucked, sending a pulse down the string. Estimate the speed of the pulse as it moves down the string.
-
A cord has a linear mass density of \(\mu\) = 0.0075 kg/m and a length of three meters. The cord is plucked and it takes 0.20 s for the pulse to reach the end of the string. What is the tension of the string?
-
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
-
A sound wave travels through a column of nitrogen at STP. Assuming a density of \(\rho\) = 1.25 kg/m
3
and a bulk modulus of \(\beta\) = 1.42 x 10
5
Pa, what is the approximate speed of the sound wave?
-
What is the approximate speed of sound traveling through air at a temperature of T = 28°C ?
-
Transverse waves travel through a string where the tension equals 7.00 N with a speed of 20.00 m/s. What tension would be required for a wave speed of 25.00 m/s?
-
Two strings are attached between two poles separated by a distance of 2.00 m as shown below, both under the same tension of 600.00 N. String 1 has a linear density of \(\mu_{1}\) = 0.0025 kg/m and string 2 has a linear mass density of \(\mu_{2}\) = 0.0035 kg/m. Transverse wave pulses are generated simultaneously at opposite ends of the strings. How much time passes before the pulses pass one another?
-
Two strings are attached between two poles separated by a distance of 2.00 meters as shown in the preceding figure, both strings have a linear density of \(\mu_{1}\) = 0.0025 kg/m, the tension in string 1 is 600.00 N and the tension in string 2 is 700.00 N. Transverse wave pulses are generated simultaneously at opposite ends of the strings. How much time passes before the pulses pass one another?
-
The note E
4
is played on a piano and has a frequency of f = 393.88. If the linear mass density of this string of the piano is \(\mu\) = 0.012 kg/m and the string is under a tension of 1000.00 N, what is the speed of the wave on the string and the wavelength of the wave?
-
Two transverse waves travel through a taut string. The speed of each wave is v = 30.00 m/s. A plot of the vertical position as a function of the horizontal position is shown below for the time t = 0.00 s. (a) What is the wavelength of each wave? (b) What is the frequency of each wave? (c) What is the maximum vertical speed of each string?
-
A sinusoidal wave travels down a taut, horizontal string with a linear mass density of µ = 0.060 kg/m . The maximum vertical speed of the wave is vy max = 0.30 cm/s. The wave is modeled with the wave equation y(x, t) = A sin(6.00 m
−1
x − 24.00 s
−1
t) . (a) What is the amplitude of the wave? (b) What is the tension in the string?
-
The speed of a transverse wave on a string is v = 60.00 m/s and the tension in the string is F
T
= 100.00 N . What must the tension be to increase the speed of the wave to v = 120.00 m/s?
-
A string of length 5 m and a mass of 90 g is held under a tension of 100 N. A wave travels down the string that is modeled as y(x, t) = 0.01 m \(\sin\)(15.7 m
−1
x − 1170.12 s
−1
). What is the power over one wavelength?
-
Ultrasound of intensity 1.50 x 10
2
W/m
2
is produced by the rectangular head of a medical imaging device measuring 3.00 cm by 5.00 cm. What is its power output?
-
The low-frequency speaker of a stereo set has a surface area of A = 0.05 m
2
and produces 1 W of acoustical power. (a) What is the intensity at the speaker? (b) If the speaker projects sound uniformly in all directions, at what distance from the speaker is the intensity 0.1 W/m
2
?
-
To increase the intensity of a wave by a factor of 50, by what factor should the amplitude be increased?
-
A device called an insolation meter is used to measure the intensity of sunlight. It has an area of 100 cm
2
and registers 6.50 W. What is the intensity in W/m
2
?
-
Energy from the Sun arrives at the top of Earth’s atmosphere with an intensity of 1400 W/m
2
. How long does it take for 1.80 x 10
9
J to arrive on an area of 1.00 m
2
?
-
Suppose you have a device that extracts energy from ocean breakers in direct proportion to their intensity. If the device produces 10.0 kW of power on a day when the breakers are 1.20 m high, how much will it produce when they are 0.600 m high?
-
A photovoltaic array of (solar cells) is 10.0% efficient in gathering solar energy and converting it to electricity. If the average intensity of sunlight on one day is 70.00 W/m
2
, what area should your array have to gather energy at the rate of 100 W? (b) What is the maximum cost of the array if it must pay for itself in two years of operation averaging 10.0 hours per day? Assume that it earns money at the rate of 9.00 cents per kilowatt-hour.
-
A microphone receiving a pure sound tone feeds an oscilloscope, producing a wave on its screen. If the sound intensity is originally 2.00 x 10
−5
W/m
2
, but is turned up until the amplitude increases by 30.0%, what is the new intensity?
-
A string with a mass of 0.30 kg has a length of 4.00 m. If the tension in the string is 50.00 N, and a sinusoidal wave with an amplitude of 2.00 cm is induced on the string, what must the frequency be for an average power of 100.00 W?
-
The power versus time for a point on a string (\(\mu\) = 0.05 kg/m) in which a sinusoidal traveling wave is induced is shown in the preceding figure. The wave is modeled with the wave equation y(x, t) = A sin(20.93 m
−1
x − \(\omega\)t). What is the frequency and amplitude of the wave?
-
A string is under tension F
T1
. Energy is transmitted by a wave on the string at rate P
1
by a wave of frequency f
1
. What is the ratio of the new energy transmission rate P
2
to P
1
if the tension is doubled?
-
A 250-Hz tuning fork is struck and the intensity at the source is I1 at a distance of one meter from the source. (a) What is the intensity at a distance of 4.00 m from the source? (b) How far from the tuning fork is the intensity a tenth of the intensity at the source? 88. A sound speaker is rated at a voltage of P = 120.00 V and a current of I = 10.00 A. Electrical power consumption is P = IV. To test the speaker, a signal of a sine wave is applied to the speaker. Assuming that the sound wave moves as a spherical wave and that all of the energy applied to the speaker is converted to sound energy, how far from the speaker is the intensity equal to 3.82 W/m
2
?
-
The energy of a ripple on a pond is proportional to the amplitude squared. If the amplitude of the ripple is 0.1 cm at a distance from the source of 6.00 meters, what was the amplitude at a distance of 2.00 meters from the source?
-
Consider two sinusoidal waves traveling along a string, modeled as y
1
(x, t) = 0.3 m sin(4 m
−1
x + 3 s
−1
t) and y
2
(x, t) = 0.6 m sin(8 m
−1
x − 6 s
−1
t). What is the height of the resultant wave formed by the interference of the two waves at the position x = 0.5 m at time t = 0.2 s?
-
Consider two sinusoidal sine waves traveling along a string, modeled as y
1
(x, t) = 0.3 m sin(4 m
−1
x + 3 s
−1
t + \(\frac{\pi}{3}\)) and y
2
(x, t) = 0.6 m sin(8 m
−1
x − 6 s
−1
t). What is the height of the resultant wave formed by the interference of the two waves at the position x = 1.0 m at time t = 3.0 s?
 What is the wave function of the resulting wave? [Hint: Use the trig identity sin(u ± v) = sin u cos v ± cos u sin v]
-
Two sinusoidal waves are moving through a medium in the same direction, both having amplitudes of 3.00 cm, a wavelength of 5.20 m, and a period of 6.52 s, but one has a phase shift of an angle \(\phi\). What is the phase shift if the resultant wave has an amplitude of 5.00 cm? [Hint: Use the trig identity sin u + sin v = 2 sin\(\left(\dfrac{u + v}{2}\right)\) cos\(\left(\dfrac{u − v}{2}\right)\)]
-
Two sinusoidal waves are moving through a medium in the positive x-direction, both having amplitudes of 6.00 cm, a wavelength of 4.3 m, and a period of 6.00 s, but one has a phase shift of an angle \(\phi\) = 0.50 rad. What is the height of the resultant wave at a time t = 3.15 s and a position x = 0.45 m?
-
Two sinusoidal waves are moving through a medium in the positive x-direction, both having amplitudes of 7.00 cm, a wave number of k = 3.00 m−1 , an angular frequency of \(\omega\) = 2.50 s
−1
, and a period of 6.00 s, but one has a phase shift of an angle \(\phi\) = \(\frac{\pi}{12}\) rad. What is the height of the resultant wave at a time t = 2.00 s and a position x = 0.53 m?
-
Consider two waves y
1
(x, t) and y
2
(x, t) that are identical except for a phase shift propagating in the same medium. (a) What is the phase shift, in radians, if the amplitude of the resulting wave is 1.75 times the amplitude of the individual waves? (b) What is the phase shift in degrees? (c) What is the phase shift as a percentage of the individual wavelength?
-
Two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is y
R
(x, t) = 0.70 m sin(3.00 m
−1
x − 6.28 s
−1
t + \(\frac{\pi}{16}\) rad). What are the angular frequency, wave number, amplitude, and phase shift of the individual waves?
-
Two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is y
R
(x, t) = 0.35 cm sin(6.28 m
−1
x − 1.57 s
−1
t + \(\frac{\pi}{4}\)). What are the period, wavelength, amplitude, and phase shift of the individual waves?
-
Consider two wave functions, y
1
(x, t) = 4.00 m sin(\(\pi\) m
−1
x − \(\pi\) s
−1
t) and y
2
(x, t) = 4.00 m sin(\(\pi\) m
−1
x − \(\pi\) s
−1
t + \(\frac{\pi}{3}\)). (a) Using a spreadsheet, plot the two wave functions and the wave that results from the superposition of the two wave functions as a function of position (0.00 ≤ x ≤ 6.00 m) for the time t = 0.00 s. (b) What are the wavelength and amplitude of the two original waves? (c) What are the wavelength and amplitude of the resulting wave?
-
Consider two wave functions, y
2
(x, t) = 2.00 m sin(\(\frac{\pi}{2}\) m
−1
x − \(\frac{\pi}{3}\) s
−1
t) and y
2
(x, t) = 2.00 m sin(\(\frac{\pi}{2}\) m
−1
x − \(\frac{\pi}{3}\) s
−1
t + \(\frac{\pi}{6}\)). (a) Verify that y
R
= 2A cos\(\left(\dfrac{\phi}{2}\right)\) sin(kx − \(\omega\)t + \(\frac{\phi}{2}\)) is the solution for the wave that results from a superposition of the two waves. Make a column for x, y
1
, y
2
, y
1
+ y
2
, and y
R
= 2A cos\(\left(\dfrac{\phi}{2}\right)\) sin(kx − \(\omega\)t + \(\frac{\phi}{2}\)). Plot four waves as a function of position where the range of x is from 0 to 12 m.
-
Consider two wave functions that differ only by a phase shift, y
1
(x, t) = A cos(kx − \(\omega\)t) and y
2
(x, t) = A cos(kx − \(\omega\)t + \(\phi\)). Use the trigonometric identities cos u + cos v = 2 cos\(\left(\dfrac{u − v}{2}\right)\) cos \(\left(\dfrac{u + v}{2}\right)\) and cos(−\(\theta\)) = cos(\(\theta\)) to find a wave equation for the wave resulting from the superposition of the two waves. Does the resulting wave function come as a surprise to you?
-
A wave traveling on a Slinky® that is stretched to 4 m takes 2.4 s to travel the length of the Slinky and back again. (a) What is the speed of the wave? (b) Using the same Slinky stretched to the same length, a standing wave is created which consists of three antinodes and four nodes. At what frequency must the Slinky be oscillating?
-
A 2-m long string is stretched between two supports with a tension that produces a wave speed equal to v
w
= 50.00 m/s. What are the wavelength and frequency of the first three modes that resonate on the string?
-
Consider the experimental setup shown below. The length of the string between the string vibrator and the pulley is L = 1.00 m. The linear density of the string is \(\mu\) = 0.006 kg/m. The string vibrator can oscillate at any frequency. The hanging mass is 2.00 kg. (a)What are the wavelength and frequency of n = 6 mode? (b) The string oscillates the air around the string. What is the wavelength of the sound if the speed of the sound is v
s
= 343.00 m/s?
-
A cable with a linear density of \(\mu\) = 0.2 kg/m is hung from telephone poles. The tension in the cable is 500.00 N. The distance between poles is 20 meters. The wind blows across the line, causing the cable resonate. A standing waves pattern is produced that has 4.5 wavelengths between the two poles. The speed of sound at the current temperature T = 20 °C is 343.00 m/s. What are the frequency and wavelength of the hum?
-
Consider a rod of length L, mounted in the center to a support. A node must exist where the rod is mounted on a support, as shown below. Draw the first two normal modes of the rod as it is driven into resonance. Label the wavelength and the frequency required to drive the rod into resonance.
-
Consider two wave functions y(x, t) = 0.30 cm sin(3 m
−1
x − 4 s
−1
t) and y(x, t) = 0.30 cm sin(3 m
−1
x + 4 s
−1
t). Write a wave function for the resulting standing wave.
-
A 2.40-m wire has a mass of 7.50 g and is under a tension of 160 N. The wire is held rigidly at both ends and set into oscillation. (a) What is the speed of waves on the wire? The string is driven into resonance by a frequency that produces a standing wave with a wavelength equal to 1.20 m. (b) What is the frequency used to drive the string into resonance?
-
A string with a linear mass density of 0.0062 kg/m and a length of 3.00 m is set into the n = 100 mode of resonance. The tension in the string is 20.00 N. What is the wavelength and frequency of the wave?
-
A string with a linear mass density of 0.0075 kg/m and a length of 6.00 m is set into the n = 4 mode of resonance by driving with a frequency of 100.00 Hz. What is the tension in the string?
-
Two sinusoidal waves with identical wavelengths and amplitudes travel in opposite directions along a string producing a standing wave. The linear mass density of the string is \(\mu\) = 0.075 kg/m and the tension in the string is F
T
= 5.00 N. The time interval between instances of total destructive interference is \(\Delta\)t = 0.13 s. What is the wavelength of the waves?
-
A string, fixed on both ends, is 5.00 m long and has a mass of 0.15 kg. The tension if the string is 90 N. The string is vibrating to produce a standing wave at the fundamental frequency of the string. (a) What is the speed of the waves on the string? (b) What is the wavelength of the standing wave produced? (c) What is the period of the standing wave?
-
A string is fixed at both end. The mass of the string is 0.0090 kg and the length is 3.00 m. The string is under a tension of 200.00 N. The string is driven by a variable frequency source to produce standing waves on the string. Find the wavelengths and frequency of the first four modes of standing waves.
-
The frequencies of two successive modes of standing waves on a string are 258.36 Hz and 301.42 Hz. What is the next frequency above 100.00 Hz that would produce a standing wave?
-
A string is fixed at both ends to supports 3.50 m apart and has a linear mass density of \(\mu\) = 0.005 kg/m. The string is under a tension of 90.00 N. A standing wave is produced on the string with six nodes and five antinodes. What are the wave speed, wavelength, frequency, and period of the standing wave?
-
Sine waves are sent down a 1.5-m-long string fixed at both ends. The waves reflect back in the opposite direction. The amplitude of the wave is 4.00 cm. The propagation velocity of the waves is 175 m/s. The n = 6 resonance mode of the string is produced. Write an equation for the resulting standing wave.
Additional Problems
-
Ultrasound equipment used in the medical profession uses sound waves of a frequency above the range of human hearing. If the frequency of the sound produced by the ultrasound machine is f = 30 kHz, what is the wavelength of the ultrasound in bone, if the speed of sound in bone is v = 3000 m/s?
-
Shown below is the plot of a wave function that models a wave at time t = 0.00 s and t = 2.00 s. The dotted line is the wave function at time t = 0.00 s and the solid line is the function at time t = 2.00 s. Estimate the amplitude, wavelength, velocity, and period of the wave.
-
The speed of light in air is approximately v = 3.00 x 10
8
m/s and the speed of light in glass is v = 2.00 x 10
8
m/s. A red laser with a wavelength of \(\lambda\) = 633.00 nm shines light incident of the glass, and some of the red light is transmitted to the glass. The frequency of the light is the same for the air and the glass. (a) What is the frequency of the light? (b) What is the wavelength of the light in the glass?
-
A radio station broadcasts radio waves at a frequency of 101.7 MHz. The radio waves move through the air at approximately the speed of light in a vacuum. What is the wavelength of the radio waves?
-
A sunbather stands waist deep in the ocean and observes that six crests of periodic surface waves pass each minute. The crests are 16.00 meters apart. What is the wavelength, frequency, period, and speed of the waves?
-
A tuning fork vibrates producing sound at a frequency of 512 Hz. The speed of sound of sound in air is v = 343.00 m/s if the air is at a temperature of 20.00 °C. What is the wavelength of the sound?
-
A motorboat is traveling across a lake at a speed of v
b
= 15.00 m/s. The boat bounces up and down every 0.50 s as it travels in the same direction as a wave. It bounces up and down every 0.30 s as it travels in a direction opposite the direction of the waves. What is the speed and wavelength of the wave?
-
Use the linear wave equation to show that the wave speed of a wave modeled with the wave function y(x, t) = 0.20 m sin(3.00 m
−1
x + 6.00 s
−1
t) is v = 2.00 m/s. What are the wavelength and the speed of the wave?
-
Given the wave functions y
1
(x, t) = A sin(kx − \(\omega\)t) and y
2
(x, t) = A sin(kx − \(\omega\)t + \(\phi\)) with \(\phi \neq \frac{\pi}{2}\), show that y
1
(x, t) + y
2
(x, t) is a solution to the linear wave equation with a wave velocity of v = \(\frac{\omega}{k}\).
-
A transverse wave on a string is modeled with the wave function y(x, t) = 0.10 m sin(0.15 m
−1
x + 1.50 s
−1
t + 0.20). (a) Find the wave velocity. (b) Find the position in the y-direction, the velocity perpendicular to the motion of the wave, and the acceleration perpendicular to the motion of the wave, of a small segment of the string centered at x = 0.40 m at time t = 5.00 s.
-
A sinusoidal wave travels down a taut, horizontal string with a linear mass density of \(\mu\) = 0.060 kg/m. The magnitude of maximum vertical acceleration of the wave is ay max = 0.90 cm/s
2
and the amplitude of the wave is 0.40 m. The string is under a tension of F
T
= 600.00 N . The wave moves in the negative x-direction. Write an equation to model the wave.
-
A transverse wave on a string (\(\mu\) = 0.0030 kg/m) is described with the equation y(x, t) = 0.30 m sin\(\left(\dfrac{2 \pi}{4.00\; m}(x − 16.00\; m/s\; t)\right)\). What is the tension under which the string is held taut?
-
A transverse wave on a horizontal string (\(\mu\) = 0.0060 kg/m) is described with the equation y(x, t) = 0.30 m sin\(\left(\dfrac{2 \pi}{4.00\; m(x − v_{w}t)}\right)\). The string is under a tension of 300.00 N. What are the wave speed, wave number, and angular frequency of the wave?
-
A student holds an inexpensive sonic range finder and uses the range finder to find the distance to the wall. The sonic range finder emits a sound wave. The sound wave reflects off the wall and returns to the range finder. The round trip takes 0.012 s. The range finder was calibrated for use at room temperature T = 20 °C, but the temperature in the room is actually T = 23 °C. Assuming that the timing mechanism is perfect, what percentage of error can the student expect due to the calibration?
-
A wave on a string is driven by a string vibrator, which oscillates at a frequency of 100.00 Hz and an amplitude of 1.00 cm. The string vibrator operates at a voltage of 12.00 V and a current of 0.20 A. The power consumed by the string vibrator is P = IV. Assume that the string vibrator is 90% efficient at converting electrical energy into the energy associated with the vibrations of the string. The string is 3.00 m long, and is under a tension of 60.00 N. What is the linear mass density of the string?
-
A traveling wave on a string is modeled by the wave equation y(x, t) = 3.00 cm sin(8.00 m
−1
x + 100.00 s
−1
t). The string is under a tension of 50.00 N and has a linear mass density of \(\mu\) = 0.008 kg/m. What is the average power transferred by the wave on the string?
-
A transverse wave on a string has a wavelength of 5.0 m, a period of 0.02 s, and an amplitude of 1.5 cm. The average power transferred by the wave is 5.00 W. What is the tension in the string?
-
(a) What is the intensity of a laser beam used to burn away cancerous tissue that, when 90.0% absorbed, puts 500 J of energy into a circular spot 2.00 mm in diameter in 4.00 s? (b) Discuss how this intensity compares to the average intensity of sunlight (about) and the implications that would have if the laser beam entered your eye. Note how your answer depends on the time duration of the exposure.
-
Consider two periodic wave functions, y
1
(x, t) = A sin(kx − \(\omega\)t) and y
2
(x, t) = A sin(kx − \(\omega\)t + \(\phi\)). (a) For what values of \(\phi\) will the wave that results from a superposition of the wave functions have an amplitude of 2A? (b) For what values of \(\phi\) will the wave that results from a superposition of the wave functions have an amplitude of zero?
-
Consider two periodic wave functions, y
1
(x, t) = A sin(kx − \(\omega\)t) and y
2
(x, t) = A cos(kx − \(\omega\)t + \(\phi\)). (a) For what values of \(\phi\) will the wave that results from a superposition of the wave functions have an amplitude of 2A? (b) For what values of \(\phi\) will the wave that results from a superposition of the wave functions have an amplitude of zero?
 Small-amplitude surface water waves are produced from both ends of the trough by paddles oscillating in simple harmonic motion.  What is the wave function of the resulting wave after the waves reach one another and before they reach the end of the trough (i.e., assume that there are only two waves in the trough and ignore reflections)? Use a spreadsheet to check your results.  If they traveled the same path at constant wave speeds of v
S
= 4.00 km/s and v
P
= 7.50 km/s, how far away is the epicenter of the earthquake?
-
Consider what is shown below. A 20.00-kg mass rests on a frictionless ramp inclined at 45°. A string with a linear mass density of \(\mu\) = 0.025 kg/m is attached to the 20.00-kg mass. The string passes over a frictionless pulley of negligible mass and is attached to a hanging mass (m). The system is in static equilibrium. A wave is induced on the string and travels up the ramp. (a) What is the mass of the hanging mass (m)? (b) At what wave speed does the wave travel up the string?
-
Consider the superposition of three wave functions y(x, t) = 3.00 cm sin(2 m
−1
x − 3 s
−1
t), y(x, t) = 3.00 cm sin(6 m
−1
x + 3 s
−1
t), and y(x, t) = 3.00 cm sin(2 m
−1
x − 4 s
−1
t). What is the height of the resulting wave at position x = 3.00 m at time t = 10.0 s?
-
A string has a mass of 150 g and a length of 3.4 m. One end of the string is fixed to a lab stand and the other is attached to a spring with a spring constant of k
s
= 100 N/m. The free end of the spring is attached to another lab pole. The tension in the string is maintained by the spring. The lab poles are separated by a distance that stretches the spring 2.00 cm. The string is plucked and a pulse travels along the string. What is the propagation speed of the pulse?
-
A standing wave is produced on a string under a tension of 70.0 N by two sinusoidal transverse waves that are identical, but moving in opposite directions. The string is fixed at x = 0.00 m and x = 10.00 m. Nodes appear at x = 0.00 m, 2.00 m, 4.00 m, 6.00 m, 8.00 m, and 10.00 m. The amplitude of the standing wave is 3.00 cm. It takes 0.10 s for the antinodes to make one complete oscillation. (a) What are the wave functions of the two sine waves that produce the standing wave? (b) What are the maximum velocity and acceleration of the string, perpendicular to the direction of motion of the transverse waves, at the antinodes?
-
A string with a length of 4 m is held under a constant tension. The string has a linear mass density of \(\mu\) = 0.006 kg/m. Two resonant frequencies of the string are 400 Hz and 480 Hz. There are no resonant frequencies between the two frequencies. (a) What are the wavelengths of the two resonant modes? (b) What is the tension in the string?
Challenge Problems
-
A copper wire has a radius of 200 \(\mu\)m and a length of 5.0 m. The wire is placed under a tension of 3000 N and the wire stretches by a small amount. The wire is plucked and a pulse travels down the wire. What is the propagation speed of the pulse? (Assume the temperature does not change: (\(\rho\) = 8.96 g/cm
3
, Y = 1.1 x 10
11
N/m).
-
A pulse moving along the x axis can be modeled as the wave function y(x, t) = 4.00 m \(e^{− \left(\dfrac{x + (2.00\; m/s)t}{1.00\; m}\right)^{2}}\). (a) What are the direction and propagation speed of the pulse? (b) How far has the wave moved in 3.00 s? (c) Plot the pulse using a spreadsheet at time t = 0.00 s and t = 3.00 s to verify your answer in part (b).
-
A string with a linear mass density of \(\mu\) = 0.0085 kg/m is fixed at both ends. A 5.0-kg mass is hung from the string, as shown below. If a pulse is sent along section A, what is the wave speed in section A and the wave speed in section B?
-
Consider two wave functions y
1
(x, t) = A sin(kx − \(\omega\)t) and y
2
(x, t) = A sin(kx + \(\omega\)t + \(\phi\)). What is the wave function resulting from the interference of the two wave? [Hint: sin(\(\alpha \pm \beta\)) = sin \(\alpha\) cos \(\beta\) ± cos \(\alpha\) sin \(\beta\) and \(\phi = \frac{\phi}{2} + \frac{\phi}{2}\)).
-
The wave function that models a standing wave is given as yR (x, t) = 6.00 cm sin(3.00 m
−1
x + 1.20 rad) cos(6.00 s
−1
t + 1.20 rad). What are two wave functions that interfere to form this wave function? Plot the two wave functions and the sum of the sum of the two wave functions at t = 1.00 s to verify your answer.
-
Consider two wave functions y
1
(x, t) = A sin(kx − \(\omega\)t) and y
2
(x, t) = A sin(kx + \(\omega\)t + \(\phi\)). The resultant wave form when you add the two functions is y
R
= 2A sin(kx + \(\frac{\phi}{2}\)) cos(\omega t + \frac{\phi}{2}\)). Consider the case where A = 0.03 m
−1
, k = 1.26 m
−1
, \(\omega = \pi\) s
−1
, and \(\phi = \frac{\pi}{10}\). (a) Where are the first three nodes of the standing wave function starting at zero and moving in the positive x direction? (b) Using a spreadsheet, plot the two wave functions and the resulting function at time t = 1.00 s to verify your answer.


 His father John More (1451-1530) married Agnes Graunger, whose father was a wealthy merchant and Sheriff of London. John More was a successful lawyer, ultimately knighted and serving on the Court of King’s Bench. Thomas More received a gentleman’s education, studying at St. Anthony’s School in London before entering the household of John Morton (1420-1500), who was at that time both Archbishop of Canterbury and Lord Chancellor of England. Thomas More’s learning, humanist scholarship, and writing defined his character, as did his statesmanship. He studied for two years at Oxford before he acceded to his father’s wishes by entering New Inn to study law, being called to the Bar in 1502. His own wishes had a more spiritual bent, for the next year saw him living near a Carthusian monastery, following that order’s discipline, praying, fasting, self-flagellating, and abasing the flesh by wearing a hair shirt. According to his friend, the Humanist Desiderius Erasmus of Rotterdam (1466-1536), More considered joining the Franciscan order.
Instead, he entered Parliament, advocating government efficiency and denouncing heavy taxation. In 1505, he married Jane Colt; they had four children before she died in 1500. An exceptionally short thirty days later, he married Alice Harpur Middleton (1474-1546), a wealthy widow. Her daughter from her first marriage, along with More’s daughters Margaret, Elizabeth, and Cicely received a classical education, with Margaret later gaining renown for her knowledge of Latin. More balanced a contented domestic life with an active public one. A successful barrister, he served as under-sherrif of the City of London, as Master of the Court of Requests, and as a Privy Counsellor. He traveled to Flanders as a delegate for Henry VIII and Cardinal Wolsey (1473-1530)—who recommended him as Speaker of the House of Commons—then became Chancellor of the Duchy of Lancaster.
He gained influence as Henry VIII’s advisor, particularly through his condemnation of the Lutheran heresy (that is, the Reformation). Henry VIII granted More a pension, land, and sinecures. When Wolsey failed to obtain Henry VIII’s annulment from Catherine of Aragon, Wolsey was removed as Lord Chancellor of England and expelled from court. Henry VIII then named More Wolsey’s replacement as Lord Chancellor. In this position, More continued to support the Roman Catholic Church, possibly to the point of approving the torture and burning of Protestants. He certainly suppressed William Tyndale’s English translation of the New Testament (1526), one that turned directly to the Hebrew and Greek, as challenging the power of the Roman Catholic Church in England. Tyndale’s The Obedience of a Christian Man, and how Christian rulers ought to govern (1528) probably helped to encourage Henry VIII to become head of the Church in England, replacing the Pope, and breaking with the Roman Catholic Church. He then divorced Catherine of Aragon to marry Anne Boleyn and had several acts passed to secure the legitimacy of their issue. More resigned as Lord Chancellor and refused to take the Succession Oath. He was imprisoned then beheaded for high treason.
Seemingly far away from these political and religious controversies, More’s Utopia (“No Place” or “Nowhere,” in English) depicts a fictional good place located somewhere in the New World whose citizens practice a form of communism, welfare, and undenominational theism. To the classical model of Plato’s The Republic, More added the contemporary contrast of New World and Old World cultures. In Utopia, some women could be priests, and all priests are chosen for their holiness. All citizens work, for a limited time on farms, and then in various trades, such as carpentry and masonry; they use their leisure time for learning. The ethical basis of their lives satirically critiques the corruption and immorality of More’s society. It’s difficult to attribute any particular Utopian practice to More’s own held beliefs, especially considering their religious tolerance and his actions against Protestants. Their ethical behavior seems superior to England’s. But that may not be More’s main point. His main critique may be wonder at a society that can do so much good—without Christian revelation. And condemnation of the pride that keeps his society from emulating such goodness—despite having that revelation.


4.5: Re-Evaluating Policy
mala prohibita rather than mala in se , so creating laws and policies around them is as short-lived and fleeting as the new most popular meme. For example, the Eighteenth Amendment states (1917):
“After one year from the ratification of this article the manufacture, sale, or transportation of intoxicating liquors within, the importation thereof into, or the exportation thereof from the United States and all territory subject to the jurisdiction thereof for beverage purposes is hereby prohibited.”
[1] As you know, the experiment was an absolute failure. Crime actually increased and became organized, alcohol became more dangerous to drink because of the lack of regulations on bootlegged and black market production, corruption of public officials was rampant, and courts and prisons were stretched to capacity.
Conclusion
-
Thornton, M. (1991). Alcohol prohibition was a failure.
Policy Analysis, 157:
1-12. Retrieved object.cato.org/sites/cato.org/files/pubs/pdf/pa157.pdf%5D
↵
-
Isaac. J. (2009). The Wacky World of Laws. San Diego, CA: Lawyer In Blue Jeans Group Publishing. Reader’s Digest Editors (n.d.). Here are 50 of the dumbest laws in every state. Reader’s Digest. Retrieved:
https://www.rd.com/funny-stuff/dumbest-laws-america/
Triangle Realty. (2018). Unusual Laws in Texas. Retrieved
https://www.trianglerealtyllc.com/unusual-laws-in-texas/
↵


8.6: Education Debt- Paying for College
-
- Last updated
- Save as PDF
Questions to consider:
- What choices should you consider when taking on student debt?
- How do you match debt to postgraduate income?
- What types of financial aid are available?
- How do you apply for financial aid?
- What are the best repayment strategies?
“An investment in knowledge always pays the best interest.”
—Benjamin Franklin, The Way to Wealth : Ben Franklin on Money and Success
As you progress through your college experience, the cost of college can add up rapidly. Worse, your anxiety about the cost of college may rise faster as you hear about the rising costs of college and horror stories regarding the “student loan crisis.” It is important to remember that you are in control of your choices and the cost of your college experience, and you do not have to be a sad statistic.
Education Choices
Education is vital to living. Education starts at the beginning of our life, and as we grow, we learn language, sharing, and to look both ways before crossing the street. We also generally pursue a secular or public education that often ends at high school graduation. After that, we have many choices, including getting a job and stopping our education, working at a trade or business started by our parents and bypassing additional schooling, earning a certificate from a community college or four-year college or university, earning a two-year or associate degree from one of the same schools, and completing a bachelor’s or advanced degree at a college or university. We can choose to attend a public or private school. We can live at home or on a campus.
Each of these choices impacts our debt, happiness, and earning power. The average income goes up with an increase in education, but that is not an absolute rule. The New York Federal Reserve Bank reported in 2017 that approximately 34 percent of college graduates worked in a job that did not require a college degree, 14 and in 2013, CNN Money reported on a study from Georgetown University’s Center on Education and the Workforce showing that nearly 30 percent of Americans with two-year degrees are now earning more than graduates with bachelor’s degrees. 15 Of course, many well-paying occupations do require a bachelor’s or master’s degree. You have started on a path that may be perfect for you, but you may also choose to make adjustments.
College success from a financial perspective means that you must:
- Know the total cost of the education
- Consider job market trends
- Work hard at school during the education
- Pursue ways to reduce costs
Most importantly: Buy only the amount of education that returns more than you invest.
According to US News & World Report , the average cost of college (including university) tuition and fees varies widely. In-state colleges average $9,716 while out-of-state students pay $21,629 for the same state college. Private colleges average $35,676. The local community college averages approximately $3,726. On-campus housing and meals, if available, can add approximately $10,000 per year. 16 See the table below, and create your own chart after you research.
| Sample College Costs | |||
|---|---|---|---|
| Type of School | Annual Tuition without Housing | Tuition If Living on Campus | Total Cost at Planned Completion |
| Community College (2 yr.) | $3,726 | Live at Home | $7,452 |
| Public University, In State (4 yr.) | $9,716 | Live at Home | $38,864 |
| Public University, In State (4 yr.) | $19,716 | $78,864 | |
| Public University, Out of State (4 yr.) | $21,629 | $31,629 | $126,516 |
| Private College (4 yr.) | $35,676 | $45,676 | $182,704 |
You may need to adjust your college plan as circumstances change for you and in the job market. You can modify plans based on funding opportunities available to you (see next sections) and your location. You may prefer a community-college-only education, or you may complete two years at a community college and then transfer to a university to complete a bachelor’s degree. Living at home for the first two years or all of your college education will save a lot of money if your circumstances allow. Be creative!
Key to Success: Matching Student Debt to Postgraduation Income
Students and parents often ask, “How much debt should I have?” The problem is that the correct answer depends on your personal situation. A big-firm attorney in a major city might make $120,000 in their first year as a lawyer. Having $100,00 or even $200,000 in student debt in this situation may be reasonable. But a high school teacher making $40,000 in their first year would never be able to pay off the debt.
The amount of debt you take on should be tied to the income you expect.
 Most students expect to make significantly more than they will actually make. 17 As a result, your salary expectations are likely much higher than reality. Ask professors at your college what is typical for a recent graduate in your field, or do informational interviews with human resource managers at local companies. Explore the US Bureau of Labor Statistics’ Occupational Outlook Handbook . PayScale also has a handy tool for getting general information based on your personal experience and location. Search websites and talk to employees of companies that interest you for future employment to identify real starting salaries.
Undergraduate Degree: 1 x Annual Salary
For students working toward a bachelor’s or associate degree, both forms of undergraduate degrees, you should try to keep your student loans equal to or less than your expected first year’s salary. So if, based on research, you expect to make $40,000 in your first year out of college, then $33,000 in student loans would be a reasonable amount for you to pay out of a monthly budget with some sacrifice.
Advanced Degrees: 1–2 x Annual Salary
Once you’ve graduated with your bachelor’s degree, you may want to get an advanced degree such as a master’s degree, a law degree, a medical degree, or a doctorate. While these degrees can greatly increase your income, you still need to match your student debt to your expected income. Advanced degrees can often double your expected annual salary, meaning your total debt for all your degrees should be equal to or less than twice your expected first job income. A lower number for the debt portion of your education would be more manageable.
Your goal should be to pay for college using multiple methods so your student loan debt can be as small as possible, rather than just making low monthly payments on a large loan that will lead to a higher overall cost.
Types of Financial Aid: How to Pay for College
The true cost of college may be more than you expected, but you can make an effort to make the cost less than many might think. While the price tag for a school might say $40,000, the net cost of college may be significantly less. The net price for a college is the true cost a family will pay when grants, scholarships, and education tax benefits are factored in. The net cost for the average family at a public in-state school is only $3,980. And for a private school, free financial aid money reduces the cost to the average family from $32,410 per year to just $14,890.
If you haven’t visited your college’s financial aid office recently, it’s probably worth it to talk with them. You must seek out opportunities, complete paperwork, and learn and meet criteria, but it can save you thousands of dollars.
| Type of College | Average Published Yearly Tuition and Fees |
|---|---|
| Public Two-Year College (in-district students) | $3,440 |
| Public Four-Year College (in-state students) | $9,410 |
| Public Two-Year College (out-of-state students) | $23,890 |
| Private Four-Year College | $32,410 |
Grants and Scholarships
Grants and scholarships are free money you can use to pay for college. Unlike loans, you never have to pay back a grant or a scholarship.  And you don’t have to be a straight-A student to get grants and scholarships. There is so much free money, in fact, that billions of dollars go unclaimed every year. 18
While some grants and scholarships are based on a student’s academic record, many are given to average students based on their major, ethnic background, gender, religion, or other factors. There are likely dozens or hundreds of scholarships and grants available to you personally if you look for them.
Federal Grants
Federal Pell Grants are awarded to students based on financial need, although there is no income or wealth limit on the grant program. The Pell Grant can give you more than $6,000 per year in free money toward tuition, fees, and living expenses. 19 If you qualify for a Pell Grant based on your financial need, you will automatically get the money.
Federal Supplemental Educational Opportunity Grants (FSEOGs) are additional free money available to students with financial need. Through the FSEOG program, you can receive up to an additional $4,000 in free money. These grants are distributed through your school’s financial aid department on a first-come, first-served basis, so pay close attention to deadlines.
Teacher Education Assistance for College and Higher Education (TEACH) Grants are designed to help students who plan to go into the teaching profession. You can receive up to $4,000 per year through the TEACH Grant. To be eligible for a TEACH Grant, you must take specific classes and majors and must hold a qualifying teaching job for at least four years after graduation. If you do not fulfill these obligations, your TEACH Grant will be converted to a loan, which you will have to pay back with both interest and back interest.
There are numerous other grants available through individual states, employers, colleges, and private organizations.
State Grants
Most states also have grant programs for their residents, often based on financial need. Eleven states have even implemented free college tuition programs for residents who plan to continue to live in the state. Even some medical schools are beginning to be tuition free. Check your school’s financial aid office and your state’s department of education for details.
College/University Grants and Scholarships
Most colleges and universities have their own scholarships and grants. These are distributed through a wide variety of sources, including the school’s financial aid office, the school’s endowment fund, individual departments, and clubs on campus.
Private Organization Grants and Scholarships
A wide variety of grants and scholarships and are awarded by foundations, civic groups, companies, religious groups, professional organizations, and charities. Most are small awards under $4,000, but multiple awards can add up to large amounts of money each year. Your financial aid office can help you find these opportunities.
Employer Grants and Scholarships
Many employers also offer free money to help employees go to school. A common work benefit is a tuition reimbursement program, where employers will pay students extra money to cover the cost of tuition once they’ve earned a passing grade in a college class. And some companies are going even further, offering to pay 100 percent of college costs for employees. Check to see whether your employer offers any kind of educational support.
Additional Federal Support
The federal government offers a handful of additional options for college students to find financial support.
Education Tax Credits
The IRS gives out free money to students and their parents through two tax credits, although you will have to choose between them. The American opportunity tax credit (AOTC) will refund up to $2,500 of qualifying education expenses per eligible student, while the lifetime learning credit (LLC) refunds up to $2,000 per year regardless of the number of qualifying students.
While the AOTC may be a better tax credit to choose for some, it can only be claimed for four years for each student, and it has other limitations. The LLC has fewer limitations, and there is no limit on the number of years you can claim it. Lifetime learners and nontraditional students may consider the LLC a better choice. Calculate the benefits for your situation.
The IRS warns taxpayers to be careful when claiming the credits. There are potential penalties for incorrectly claiming the credits, and you or your family should consult a tax professional or financial adviser when claiming these credits.
Federal Work-Study Program
The Federal Work-Study Program provides part-time jobs through colleges and universities to students who are enrolled in the school. The program offers students the opportunity to work in their field, for their school, or for a nonprofit or civic organization to help pay for the cost of college. If your school participates in the program, it will be offered through your school’s financial aid office.
Student Loans
Federal student loans are offered through the US Department of Education and are designed to give easy and inexpensive access to loans for school. You don’t have to make payments on the loans while you are in school, and the interest on the loans is tax deductible for most people. Direct Loans, also called Federal Stafford Loans, have a competitive fixed interest rate and don’t require a credit check or cosigner.
Direct Subsidized Loans
Direct Subsidized Loans are federal student loans on which the government pays the interest while you are in school. Direct Subsidized Loans are made based on financial need as calculated from the information you provide in your application. Qualifying students can get up to $3,500 in subsidized loans in their first year, $4,500 in their second year, and $5,500 in later years of their college education.
 If you don’t make interest payments while in school, the interest will be added to the loan amount each year and will result in a larger student loan balance when you graduate. The amount you can borrow each year depends on numerous factors, with a maximum of $12,500 annually for undergraduates and $20,500 annually for professional or graduate students.
There are also aggregate loan limits that apply to put a maximum cap on the total amount you can borrow for student loans.
Direct PLUS Loans
Direct PLUS Loans are additional loans a parent, grandparent, or graduate student can take out to help pay for additional costs of college. PLUS loans require a credit check and have higher interest rates, but the interest is still tax deductible. The maximum PLUS loan you can receive is the remaining cost of attending the school.
Parents and other family members should be careful when taking out PLUS loans on behalf of a child. Whoever is on the loan is responsible for the loan forever, and the loan generally cannot be forgiven in bankruptcy. The government can also take Social Security benefits should the loan not be repaid.
Private Loans
Private loans are also available for students who need them from banks, credit unions, private investors, and even predatory lenders. But with all the other resources for paying for college, a private loan is generally unnecessary and unwise. Private loans will require a credit check and potentially a cosigner, they will likely have higher interest rates, and the interest is not tax deductible. As a general rule, you should be wary of private student loans or avoid them altogether.
Repayment Strategies
Payments on student loans will begin shortly after you graduate. While many websites, financial “gurus,” and talking heads in the media will encourage you to pay off your student loans as quickly as possible, you should give careful consideration to your repayment options and how they may impact your financial plans. Quickly paying off your student loans or refinancing your student loans into a private loan may be the worst option available to you.
Payment Plans
The federal government has eight separate loan repayment programs, each with their own way of calculating the payment you owe. Five of the programs tie loan payments to your income, which can make it easier to afford your student loans when you are just starting off in your career. The programs are described briefly below, but you should seek the help of a licensed fiduciary financial adviser familiar with student loans when making decisions related to student loan payment plans.
The standard repayment plan sets a consistent monthly payment to pay off your loan within 10 years (or up to 30 years for consolidated loans). You can also choose a graduated repayment plan, which will begin with lower payments and then increase the payment every two years. The graduated plan is also designed to pay off your student loans in 10 years (or up to 30 years for consolidated loans). A third option is the extended repayment plan, which provides a fixed or graduated payment for up to 25 years. However, none of these programs are ideal for individuals planning to seek loan forgiveness options, which are discussed below.
Beyond the “normal” repayment options, the government offers five income-based repayment options: (1) the Pay As You Earn (PAYE) repayment plan, (2) the Revised Pay As You Earn (REPAYE) repayment plan, (3) the Income-Based Repayment (IBR) plan, (4) the Income-Contingent Repayment (ICR) plan, and (5) the Income-Sensitive Repayment (ISR) plan. Each program has its method of calculating payments, along with specific requirements for eligibility and rules for staying eligible in the program. Many income-based repayment plans are also eligible for loan forgiveness after a set period of time, assuming you follow all the rules and remain eligible.
Loan Forgiveness Programs
Many income-based repayment options also have a loan forgiveness feature built into the repayment plan. If you make 100 percent of your payments on time and follow all the other plan rules, any remaining loan balance at the end of the plan repayment term (typically 20 to 30 years) will be forgiven. This means you will not have to pay the remainder on your student loans.
This loan forgiveness, however, comes with a catch: taxes. Any forgiven balance will be counted and taxed as income during that year. So if you have a $100,000 loan forgiven, you could be looking at an additional $20,000 tax bill that year (assuming you were in the 20 percent marginal tax rate).
Another option is the Public Service Loan Forgiveness (PSLF) program for students who go on to work for a nonprofit or government organization. If eligible, you can have your loans forgiven after working for 10 years in a qualifying public service job and making 120 on-time payments on your loans. A major advantage of PSLF is that the loan forgiveness may not be taxed as income in the year the loan is forgiven.
Consider Professional Advice
The complexity of the payment and forgiveness programs makes it difficult for nonexperts to choose the best strategy to minimize costs. Additionally, the strict rules and potential tax implications create a minefield of potential financial problems. In 2017, the first year graduates were eligible for the PSLF program, 99 percent of applicants were denied due to misunderstanding the programs or having broken one of the many requirements for eligibility. 20
Your Rights as a Loan Recipient
As a recipient of a federal student loan, you have the same rights and protections as you would for any other loan. This includes the right to know the terms and conditions for any loan before signing the paperwork. You also have the right to know information on your credit report and to dispute any loan or information on your credit file.
If you end up in collections, you also have several rights, even though you have missed loan payments. Debt collectors can only call you between 8 a.m. and 9 p.m.  The United States doesn’t have debtors’ prisons, so anyone threatening you with arrest or jail time is automatically breaking the law.
Federal student loans also come with many other rights, including the right to put your loan in deferment or forbearance (pushing pause on making payments) under qualifying circumstances. Deferment or forbearance can be granted if you lose your job, go back to school, or have an economic hardship. If you have a life event that makes it difficult to make your payments, immediately contact the student loan servicing company on your loan statements to see if you can pause your student loan payments.
The Consumer Financial Protection Bureau (CFPB) has created a series of sample letters you can use to respond to a debt collector. You can also file a complaint with the CFPB if you believe your rights have been violated.
Applying for Financial Aid, FAFSA, and Everything Else
Take this first step—you will need to do it. The federal government offers a standard form called the Free Application for Federal Student Aid (FAFSA), which qualifies you for federal financial aid and also opens the door for nearly all other financial aid. Most grants and scholarships require you to fill out the FAFSA, and they base their decisions on the information in the application.
The FAFSA only requests financial aid for the specific year you file your application. This means you will need to file a FAFSA for each year you are in college. Since your financial needs will change over time, you may qualify for financial aid even if you did not qualify before.
You can apply for the FAFSA through your college’s financial aid office or at studentaid.gov if you don’t have access to a financial aid office. Once you file a FAFSA, any college can gain access to the information (with your approval), so you can shop around for financial aid offers from colleges.
Maintaining Financial Aid
To maintain your financial aid throughout your college, you need to make sure you meet the eligibility requirements for each year you are in school, not just the year of your initial application. The basic requirements include being a US citizen or eligible noncitizen, having a valid Social Security number, and registering for selective service if required. Undocumented residents may receive financial aid as well and should check with their school’s financial aid office.
You also must make satisfactory academic progress, including meeting a minimum grade-point average, taking and completing a minimum number of classes, and making progress toward graduation or a certificate. Your school will have a policy for satisfactory academic progress, which you can get from the financial aid office.
What to Do with Extra Financial Aid Money
One expensive mistake that students make with financial aid money is spending the money on noneducation expenses. Students often use financial aid, including student loans, to purchase clothing, take vacations, or dine out at restaurants. Nearly 3 percent spend student loan money on alcohol and drugs. 21 While this seems like fun now, these noneducation expenses are major contributors to student loan debt, which will make it harder for you to afford a home, take vacations, or save for your retirement after you graduate.
When you have extra student loan money, consider saving it for future education expenses. Just like you will need an emergency fund all your adult life, you will want an emergency fund for college when expensive books or travel abroad programs present unexpected costs. If you make it through your college years with extra money in your savings, you can use the money to help pay down debt.
ANALYSIS QUESTION
A closer look: How much student loan debt do you currently have, and how much do you think you’ll have by the end of college? How could this debt impact your future?
Footnotes
- https://www.forbes.com/sites/preston.../#55be172f40d8
- https://www.communitycollegereview.c...r-year-schools
- www.usnews.com/education/bes...-tuition-costs
- Hess, Abigail. “College Grades expect to earn $60,000.” 2019. CNBC. https://www.cnbc.com/2019/02/15/coll...---few-do.html
- https://www.usatoday.com/story/colle...year/37399897/
- studentaid.ed.gov/sa/types/g...olarships/pell
- https://www.forbes.com/sites/zackfri...n-forgiveness/
- https://studentloanhero.com/featured...oves-students/


10.1: Chronology
18,000 – 15,000 BCE First humans migrate to the Americas
c. 13,000 BCE Big game hunters inhabit the Great Plains
c. 10,000 BCE Mesoamericans begin to cultivate squash
10,000 – 3,500 BCE Paleo-Indian Period
5600 – 3000 BCE Early Plains Archaic Period
2000 BCE – 250 CE Preclassic or Formative period in Mesoamerica
c. 1900 BCE Mesoamericans begin to make pottery
1800 – 800 BCE Late Initial Period in Peru
1500 – 400 BCE Middle Formative Period in Mesoamerica. Peak of Olmec statue carving
c. 1000 BCE Maize becomes widespread in North America
400 BCE – 100 CE Late Formative Period in Mesoamerica
200 BCE The Moche begin their conquest of Peru’s north coast
200 BCE – 400 CE The Hopewell culture flourishes in North America
100 BCE – 600 CE The Nazca culture flourishes in Peru
400s CE Tiwankau founded
550 CE Teotihuacán reaches 125,000 residents
700 CE The Huari Empire reaches its height
700 – 1400 CE Cahokia
750 CE Tikal reaches 80,000 residents
800 CE The Toltec city of Tula reaches a population of 35,000
1000 CE The Chimu establish the capital city of Chan Chan
1050 CE The population of Chaco Canyon’s five great pueblos reaches 5,000 inhabitants
1325 CE Tenochtitlán founded
1471 CE Death of Inca Pachacuti


20.11: Exercises
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1. Write the following isotopes in hyphenated form (e.g., “carbon-14”)
(a) \({ }_{11}^{24} \mathrm{Na}\)
(b) \({ }_{13}^{29} \mathrm{Al}\)
(c) \({ }_{36}^{73} \mathrm{Kr}\)
(d) \({ }_{77}^{194} \mathrm{Ir}\)
2. Write the following isotopes in nuclide notation (e.g.,
- oxygen-14
- copper-70
- tantalum-175
- francium-217
3. For the following isotopes that have missing information, fill in the missing information to complete the notation
(a) \({ }_{14}^{34} \mathrm{X}\)
(b) \({ }_{\mathrm{X}}^{36} \mathrm{P}\)
(c) \({ }_{\mathrm{X}}^{57} \mathrm{Mn}\)
(d) \({ }_{56}^{121} \mathrm{X}\)
4. For each of the isotopes in Exercise 21.1, determine the numbers of protons, neutrons, and electrons in a neutral atom of the isotope.
5. Write the nuclide notation, including charge if applicable, for atoms with the following characteristics:
- 25 protons, 20 neutrons, 24 electrons
- 45 protons, 24 neutrons, 43 electrons
- 53 protons, 89 neutrons, 54 electrons
- 97 protons, 146 neutrons, 97 electrons
6. Calculate the density of the nucleus in g/mL, assuming that it has the typical nuclear diameter of 1 10 –13 cm and is spherical in shape. 7. What are the two principal differences between nuclear reactions and ordinary chemical changes?
8. The mass of the atom is 22.9898 amu.
- Calculate its binding energy per atom in millions of electron volts.
- Calculate its binding energy per nucleon.
9. Which of the following nuclei lie within the band of stability shown in Figure 21.2?
- chlorine-37
- calcium-40
- 204 Bi
- 56 Fe
- 206 Pb
- 211 Pb
- 222 Rn
- carbon-14
10. Which of the following nuclei lie within the band of stability shown in Figure 21.2?
- argon-40
- oxygen-16
- 122 Ba
- 58 Ni
- 205 Tl
- 210 Tl
- 226 Ra
- magnesium-24
11. Write a brief description or definition of each of the following:
- nucleon
- α particle
- β particle
- positron
- γ ray
- nuclide
- mass number
- atomic number
12. Which of the various particles (α particles, β particles, and so on) that may be produced in a nuclear reaction are actually nuclei?
13. Complete each of the following equations by adding the missing species:
(a) \({ }_{13}^{27} \mathrm{Al}+{ }_2^4 \mathrm{He} \rightarrow\) ? \(+{ }_0^1 \mathrm{n}\)
(b) \({ }_{94}^{239} \mathrm{Pu}+? \rightarrow{ }_{96}^{242} \mathrm{Cm}+{ }_0^1 \mathrm{n}\)
(c) \({ }_7^{14} \mathrm{~N}+{ }_2^4 \mathrm{He} \rightarrow ?+{ }_1^1 \mathrm{H}\)
(d) \({ }_{92}^{235} \mathrm{U} \rightarrow ?+{ }_{55}^{135} \mathrm{Cs}+4{ }_0^1 \mathrm{n}\)
14. Complete each of the following equations:
(a) \({ }_3^7 \mathrm{Li}+? \rightarrow 2{ }_2^4 \mathrm{He}\)
(b) \({ }_6^{14} \mathrm{C} \rightarrow{ }_7^{14} \mathrm{~N}+\) ?
(c) \({ }_{13}^{27} \mathrm{Al}+{ }_2^4 \mathrm{He} \rightarrow ?+{ }_0^1 \mathrm{n}\)
(d) \({ }_{96}^{250} \mathrm{Cm} \rightarrow ?+{ }_{38}^{98} \mathrm{Sr}+4{ }_0^1 \mathrm{n}\)
15. Write a balanced equation for each of the following nuclear reactions:
- the production of 17 O from 14 N by α particle bombardment
- the production of 14 C from 14 N by neutron bombardment
- the production of 233 Th from 232 Th by neutron bombardment
- the production of 239 U from 238 U by bombardment
16. Technetium-99 is prepared from 98 Mo. Molybdenum-98 combines with a neutron to give molybdenum-99, an unstable isotope that emits a β particle to yield an excited form of technetium-99, represented as 99 Tc * . This excited nucleus relaxes to the ground state, represented as 99 Tc, by emitting a γ ray. The ground state of 99 Tc then emits a β particle. Write the equations for each of these nuclear reactions.
17. The mass of the atom is 18.99840 amu.
- Calculate its binding energy per atom in millions of electron volts.
- Calculate its binding energy per nucleon.
18. For the reaction if 100.0 g of carbon reacts, what volume of nitrogen gas (N 2 ) is produced at 273K and 1 atm?
19. What are the types of radiation emitted by the nuclei of radioactive elements?
20. What changes occur to the atomic number and mass of a nucleus during each of the following decay scenarios?
- an α particle is emitted
- a β particle is emitted
- γ radiation is emitted
- a positron is emitted
- an electron is captured
21. What is the change in the nucleus that results from the following decay scenarios?
- emission of a β particle
- emission of a β + particle
- capture of an electron
22. Many nuclides with atomic numbers greater than 83 decay by processes such as electron emission. Explain the observation that the emissions from these unstable nuclides also normally include α particles.
23. Why is electron capture accompanied by the emission of an X-ray?
24. Explain, in terms of Figure 21.2, how unstable heavy nuclides (atomic number > 83) may decompose to form nuclides of greater stability (a) if they are below the band of stability and (b) if they are above the band of stability.
25. Which of the following nuclei is most likely to decay by positron emission? Explain your choice.
- chromium-53
- manganese-51
- iron-59
26. The following nuclei do not lie in the band of stability. How would they be expected to decay? Explain your answer.
(a) \({ }_{15}^{34} \mathrm{P}\)
(b) \({ }_{92}^{239} U\)
(c) \({ }_{20}^{38} \mathrm{Ca}\)
(d) \({ }_1^3 \mathrm{H}\)
(e) \({ }_{94}^{245} \mathrm{Pu}\)
27. The following nuclei do not lie in the band of stability. How would they be expected to decay?
(a) \({ }_{15}^{28} \mathrm{P}\)
(b) \({ }_{92}^{235} \mathrm{U}\)
(c) \({ }_{20}^{37} \mathrm{Ca}\)
(d) \({ }_3^9 \mathrm{Li}\)
(e) \({ }_{96}^{245} \mathrm{Cm}\)
28. Predict by what mode(s) of spontaneous radioactive decay each of the following unstable isotopes might proceed:
(a) \({ }_2^6 \mathrm{He}\)
(b) \({ }_{30}^{60} \mathrm{Zn}\)
(c) \({ }_{91}^{235} \mathrm{~Pa}\)
(d) \({ }_{94}^{241} \mathrm{~Np}\)
(e) \({ }^{18} \mathrm{~F}\)
(f) \({ }^{129} \mathrm{Ba}\)
(g) \({ }^{237} \mathrm{Pu}\)
29. Write a nuclear reaction for each step in the formation of from which proceeds by a series of decay reactions involving the step-wise emission of α, β, β, α, α, α particles, in that order.
30. Write a nuclear reaction for each step in the formation of from which proceeds by a series of decay reactions involving the step-wise emission of α, α, α, α, β, β, α particles, in that order.
31. Define the term half-life and illustrate it with an example.
32. A 1.00 10 –6 -g sample of nobelium, has a half-life of 55 seconds after it is formed. What is the percentage of remaining at the following times?
- 5.0 min after it forms
- 1.0 h after it forms
33. 239 Pu is a nuclear waste byproduct with a half-life of 24,000 y. What fraction of the 239 Pu present today will be present in 1000 y?
34. The isotope 208 Tl undergoes β decay with a half-life of 3.1 min.
- What isotope is produced by the decay?
- How long will it take for 99.0% of a sample of pure 208 Tl to decay?
- What percentage of a sample of pure 208 Tl remains un-decayed after 1.0 h?
35. If 1.000 g of produces 0.0001 mL of the gas at STP (standard temperature and pressure) in 24 h, what is the half-life of 226 Ra in years?
36. The isotope is one of the extremely hazardous species in the residues from nuclear power generation. The strontium in a 0.500-g sample diminishes to 0.393 g in 10.0 y. Calculate the half-life.
37. Technetium-99 is often used for assessing heart, liver, and lung damage because certain technetium compounds are absorbed by damaged tissues. It has a half-life of 6.0 h. Calculate the rate constant for the decay of
38. What is the age of mummified primate skin that contains 8.25% of the original quantity of 14 C?
39. A sample of rock was found to contain 8.23 mg of rubidium-87 and 0.47 mg of strontium-87.
- Calculate the age of the rock if the half-life of the decay of rubidium by β emission is 4.7 10 10 y.
- If some was initially present in the rock, would the rock be younger, older, or the same age as the age calculated in (a)? Explain your answer.
40. A laboratory investigation shows that a sample of uranium ore contains 5.37 mg of and 2.52 mg of Calculate the age of the ore. The half-life of is 4.5 10 9 yr.
41. Plutonium was detected in trace amounts in natural uranium deposits by Glenn Seaborg and his associates in 1941. They proposed that the source of this 239 Pu was the capture of neutrons by 238 U nuclei. Why is this plutonium not likely to have been trapped at the time the solar system formed 4.7 10 9 years ago?
42. A atom (mass = 7.0169 amu) decays into a atom (mass = 7.0160 amu) by electron capture. How much energy (in millions of electron volts, MeV) is produced by this reaction?
43. A atom (mass = 8.0246 amu) decays into a atom (mass = 8.0053 amu) by loss of a β + particle (mass = 0.00055 amu) or by electron capture. How much energy (in millions of electron volts) is produced by this reaction?
44. Isotopes such as 26 Al (half-life: 7.2 10 5 years) are believed to have been present in our solar system as it formed, but have since decayed and are now called extinct nuclides.
- 26 Al decays by β + emission or electron capture. Write the equations for these two nuclear transformations.
- The earth was formed about 4.7 10 9 (4.7 billion) years ago. How old was the earth when 99.999999% of the 26 Al originally present had decayed?
45. Write a balanced equation for each of the following nuclear reactions:
- bismuth-212 decays into polonium-212
- beryllium-8 and a positron are produced by the decay of an unstable nucleus
- neptunium-239 forms from the reaction of uranium-238 with a neutron and then spontaneously converts into plutonium-239
- strontium-90 decays into yttrium-90
46. Write a balanced equation for each of the following nuclear reactions:
- mercury-180 decays into platinum-176
- zirconium-90 and an electron are produced by the decay of an unstable nucleus
- thorium-232 decays and produces an alpha particle and a radium-228 nucleus, which decays into actinium-228 by beta decay
- neon-19 decays into fluorine-19
47. Write the balanced nuclear equation for the production of the following transuranium elements:
- berkelium-244, made by the reaction of Am-241 and He-4
- fermium-254, made by the reaction of Pu-239 with a large number of neutrons
- lawrencium-257, made by the reaction of Cf-250 and B-11
- dubnium-260, made by the reaction of Cf-249 and N-15
48. How does nuclear fission differ from nuclear fusion? Why are both of these processes exothermic?
49. Both fusion and fission are nuclear reactions. Why is a very high temperature required for fusion, but not for fission?
50.Cite the conditions necessary for a nuclear chain reaction to take place. Explain how it can be controlled to produce energy, but not produce an explosion.
51. Describe the components of a nuclear reactor.
52. In usual practice, both a moderator and control rods are necessary to operate a nuclear chain reaction safely for the purpose of energy production. Cite the function of each and explain why both are necessary.
53. Describe how the potential energy of uranium is converted into electrical energy in a nuclear power plant.
54. The mass of a hydrogen atom is 1.007825 amu; that of a tritium atom is 3.01605 amu; and that of an α particle is 4.00150 amu. How much energy in kilojoules per mole of produced is released by the following fusion reaction:
55. How can a radioactive nuclide be used to show that the equilibrium is a dynamic equilibrium?
56. Technetium-99m has a half-life of 6.01 hours. If a patient injected with technetium-99m is safe to leave the hospital once 75% of the dose has decayed, when is the patient allowed to leave?
57. Iodine that enters the body is stored in the thyroid gland from which it is released to control growth and metabolism. The thyroid can be imaged if iodine-131 is injected into the body. In larger doses, I-133 is also used as a means of treating cancer of the thyroid. I-131 has a half-life of 8.70 days and decays by β − emission.
- Write an equation for the decay.
- How long will it take for 95.0% of a dose of I-131 to decay?
58. If a hospital were storing radioisotopes, what is the minimum containment needed to protect against:
- cobalt-60 (a strong γ emitter used for irradiation)
- molybdenum-99 (a beta emitter used to produce technetium-99 for imaging)
59. Based on what is known about Radon-222’s primary decay method, why is inhalation so dangerous?
60. Given specimens uranium-232 ( t 1/2 = 68.9 y) and uranium-233 ( t 1/2 = 159,200 y) of equal mass, which one would have greater activity and why?
61. A scientist is studying a 2.234 g sample of thorium-229 ( t 1/2 = 7340 y) in a laboratory.
- What is its activity in Bq?
- What is its activity in Ci?
62. Given specimens neon-24 ( t 1/2 = 3.38 min) and bismuth-211 ( t 1/2 = 2.14 min) of equal mass, which one would have greater activity and why?


2.1: What Business Are You Even In?
In this section, we’ll dive into the art of crafting a rock-solid social media strategy that’s custom-made for your business. But first things first: let’s nail down what type of business you’re running. Are you in the apparel game, living that lifestyle brand life, flexing your creative muscles, schooling folks in education, or hustling in the service or B2B/B2C support industry? Pinpointing your business type is crucial, but remember, you’re not boxed in – think of how Amazon evolved from a mere online bookstore to an everything store.
Now, before we go all-in on marketing strategies (be it traditional or social media), we need to make sure we can explain our business in a way that’s snappy and easy to grasp. No, you don’t have to cram it all into a 30-second elevator pitch, but you should be able to convey your message clearly and efficiently across different organizations and industries. Your message should scream, “This is who we are, and this is why we do what we do!” Easy peasy.
So, what’s next? Let’s dig deeper into the components of a killer social media strategy. We’ll talk about how to align your social media goals with your overall business objectives, define your target audience, and create content that speaks their language. We’ll also explore the importance of analyzing your competition and staying on top of trends that are relevant to your industry.
And, of course, we’ll dive into the nitty-gritty of choosing the right social media platforms and tools that suit your unique business needs. With the right blend of creativity, analytics, and good ol’ fashioned hustle, you’ll be well on your way to crafting a social media strategy that gets your message out there, engages your audience, and drives results for your business.
It’s time to make your social media game unstoppable with a strategy that’s tailored to your one-of-a-kind business. Let’s do this!


9.5: When to Quote and When to Paraphrase
“When to Quote and When to Paraphrase” was written by Brianna Jerman
Academic writing requires authors to connect information from outside sources to their own ideas in order to establish credibility and produce an effective argument.
Sometimes, the rules surrounding source integration and plagiarism may seem confusing, so many new writers err on the side of caution by using the simplest form of integration: direct quotation. However, using direct quotes is not always the best way to use a source. Paraphrasing or summarizing a text is sometimes a more effective means of supporting a writer’s argument than directly quoting. Taking into consideration the purpose of their own writing and the purpose of utilizing the outside source, authors should seek to vary the ways in which they work sources into their own writing.
Paraphrasing and quoting are two of the three ways an author can integrate sources. The two methods are closely related, and therefore, can sometimes be confused with one another. Quoting borrows the exact wording used in a source and is indicated by placing quotes around the borrowed material. Paraphrasing, on the other hand, borrows an idea found in a shorter passage but communicates this idea using different words and word order. While it is acceptable to loosely follow a similar structure, paraphrasing requires more than simply changing a few of the original words to synonyms. Both paraphrasing and directly quoting have their merit, but they should be used at different times for different purposes. An author chooses to use one of these strategies depending on why the source is being used and what information the source provides.
When to Paraphrase
Paraphrasing provides an author the opportunity to tailor the passage for the purpose of his or her own essay, which cannot always be done when using a direct quote. Paraphrasing should be used to
- Further explain or simplify a passage that may be difficult to understand. It could be that the topic, such as the process of extracting stem cells, is particularly difficult to follow, or that the author has used language that further complicates the topic. In such situations, paraphrasing allows an author to clarify or simplify a passage so the audience can better understand the idea.
- Establish the credibility of the author. In connection to the above point, paraphrasing a complicated passage can help the author establish trust with his or her audience. If an author directly quotes a difficult passage without analysis or further explanation, it may appear that he or she does not understand the idea. Paraphrasing not only clarifies the idea in the passage but also illustrates that the writer, since he or she can articulate this difficult message to the reader, is knowledgeable about the topic and should be trusted.
- Maintain the flow of the writing. Each author has a unique voice, and using direct quotes can interrupt this voice. Too many quotes can make an essay sound choppy and difficult to follow. Paraphrasing can help communicate an important idea in a passage or source without interrupting the flow of the essay.
- Eliminate less relevant information. Since paraphrasing is written using the author’s own words, he or she can be more selective in what information from a passage should be included or omitted. While an author should not manipulate a passage unnecessarily, paraphrasing allows an author to leave out unrelated details that would have been part of a direct quote.
- Communicate relevant statistics and numerical data. A lot of times, sources offer statistical information about a topic that an author may find necessary to developing his or her own argument. For example, statistics about the percentage of mothers who work more than one job may be useful to explaining how the economy has affected children rearing practices. Directly quoting statistics such as this should be avoided.
When to Quote
Direct quotes should be used sparingly, but when they are used, they can be a powerful rhetorical tool. As a rule, avoid using long quotes when possible, especially those longer than three lines. When quotes are employed, they should be used to
- Provide indisputable evidence of an incredible claim. Directly quoting a source can show the audience exactly what the source says so there is not suspicion of misinterpretation on the author’s part.
- Communicate an idea that is stated in a particularly striking or unique way. A passage should be quoted if the source explains an idea in the best way possible or in a way that cannot be reworded. Additionally, quoting should be used when the original passage is particularly moving or striking.
- Serve as a passage for analysis. If an author is going to analyze the quote or passage, the exact words should be included in the essay either before or following the author’s analysis.
- Provide direct evidence for or proof of an author’s own claim. An author can use a direct quote as evidence for a claim he or she makes. The direct quote should follow the author’s claim and a colon, which indicates that the following passage is evidence of the statement that precedes it.
- Support or clarify information you’ve already reported from a source. Similar to the above principle, an author can use a direct quote as further evidence or to emphasize a claim found in the source. This strategy should be used when an idea from a source is particularly important to an author’s own work.
- Provide a definition of a new or unfamiliar term or phrase. When using a term that is used or coined by the source’s author or that is unfamiliar to most people, use direct quotes to show the exact meaning of the phrase or word according to the original source.