2.2: The Unit Circle
- What is the radian measure of an angle?
- Are there natural special points on the unit circle whose coordinates we can identify exactly?
- How can we determine arc length and the location of special points in circles other than the unit circle?
As demonstrated by several different examples in Section 2.1 , certain periodic phenomena are closely linked to circles and circular motion. Rather than regularly work with circles of different center and radius, it turns out to be ideal to work with one standard circle and build all circular functions from it. The unit circle is the circle of radius \(1\) that is centered at the origin, \((0,0)\text{.}\)
If we pick any point \((x,y)\) that lies on the unit circle, the point is associated with a right triangle whose horizontal leg has length \(|x|\) and whose vertical leg has length \(|y|\text{,}\) as seen in Figure \(\PageIndex{1}\) By the Pythagorean Theorem, it follows that
and this is the equation of the unit circle: a point \((x,y)\) lies on the unit circle if and only if \(x^2 + y^2 = 1\text{.}\)
To study the circular functions generated by the unit circle, we will also animate a point and let it traverse the circle. Starting at \((1,0)\) indicated by \(t_0\) in Figure \(\PageIndex{2}\) , we see a sequence of points that result from traveling a distance along the circle that is \(1/24\) the circumference of the unit circle. Since the unit circle's circumference is \(C = 2\pi r = 2\pi\text{,}\) it follows that the distance from \(t_0\) to \(t_1\) is
As we work to better understand the unit circle, we will commonly use fractional multiples of \(\pi\) as these result in natural distances traveled along the unit circle.
In Figure \(\PageIndex{3}\) there are 24 equally spaced points on the unit circle. Since the circumference of the unit circle is \(2\pi\text{,}\) each of the points is \(\frac{1}{24} \cdot 2\pi = \frac{\pi}{12}\) units apart (traveled along the circle). Thus, the first point counterclockwise from \((1,0)\) corresponds to the distance \(t = \frac{\pi}{12}\) traveled along the unit circle. The second point is twice as far, and thus \(t = 2 \cdot \frac{\pi}{12} = \frac{\pi}{6}\) units along the circle away from \((1,0)\text{.}\)
- Label each of the subsequent points on the unit circle with the exact distance they lie counter-clockwise away from \((1,0)\text{;}\) write each fraction in lowest terms.
- Which distance along the unit circle corresponds to \(\frac{1}{4}\) of a full rotation around? to \(\frac{5}{8}\) of a full rotation?
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One way to measure angles is connected to the arc length along a circle. For an angle whose vertex is at \((0,0)\) in the unit circle, we say the angle's measure is \(1\) radian provided that the angle intercepts an arc of the circle that is \(1\) unit in length, as pictured in Figure \(\PageIndex{4}\) Note particularly that an angle measuring \(1\) radian intercepts an arc of the same length as the circle's radius.
Suppose that \(\alpha\) and \(\beta\) are each central angles and that their respective radian measures are \(\alpha = \frac{\pi}{3}\) and \(\beta = \frac{3\pi}{4}\text{.}\) Sketch the angles \(\alpha\) and \(\beta\) on the unit circle in Figure 2.2.3.
- What is the radian measure that corresonds to a \(90^\circ\) angle?
Radians and degrees
In Preview Activity \(\PageIndex{1}\) , we introduced the idea of radian measure of an angle. Here we state the formal definition of this term.
An angle whose vertex is at the center of a circle 1 measures \(1\) radian provided that the arc the angle intercepts on the circle equals the radius of the circle.
As seen in Figure \(\PageIndex{4}\) , in the unit circle this means that a central angle has measure \(1\) radian whenever it intercepts an arc of length \(1\) unit along the circumference. Because of this important correspondence between the unit circle and radian measure (one unit of arc length on the unit circle corresponds to \(1\) radian), we focus our discussion of radian measure within the unit circle.
Since there are \(2\pi\) units of length along the unit circle's circumference it follows there are \(\frac{1}{4} \cdot 2\pi = \frac{\pi}{2}\) units of length in \(\frac{1}{4}\) of a revolution. We also know that \(\frac{1}{4}\) of a revolution corresponds to a central angle that is a right angle, whose familiar degree measure is \(90^\circ\text{.}\) If we extend to a central angle that intercepts half the circle, we see similarly that \(\pi\) radians corresponds to \(180^\circ\text{;}\) this relationship enables us to convert angle measures from radians to degrees and vice versa.
An angle whose radian measure is \(1\) radian has degree measure \(\frac{180}{\pi} ^\circ\text{.}\) An angle whose degree measure is \(1^\circ\) has radian measure \(\frac{\pi}{180}\text{.}\)
Convert each of the following quantities to the alternative measure: degrees to radians or radians to degrees.
- \(\displaystyle 30^\circ\)
- \(\frac{2\pi}{3}\) radians
- \(\frac{5\pi}{4}\) radians
- \(\displaystyle 240^\circ\)
- \(\displaystyle 17^\circ\)
- \(2\) radians
Note that in Figure \(\PageIndex{3}\) in the Preview Activity, we labeled \(24\) equally spaced points with their respective distances around the unit circle counterclockwise from \((1,0)\text{.}\) Because these distances are on the unit circle, they also correspond to the radian measure of the central angles that intercept them. In particular, each central angle with one of its sides on the positive \(x\)-axis generates a unique point on the unit circle, and with it, an associated length intercepted along the circumference of the circle. A good exercise at this point is to return to Figure \(\PageIndex{3}\) and label each of the noted points with the degree measure that is intercepted by a central angle with one side on the positive \(x\)-axis, in addition to the arc lengths (radian measures) already identified.
Special points on the unit circle
Our in-depth study of the unit circle is motivated by our desire to better understand the behavior of circular functions. Recall that as we traverse a circle, the height of the point moving along the circle generates a function that depends on distance traveled along the circle. Wherever possible, we'd like to be able to identify the exact height of a given point on the unit circle. Two special right triangles enable us to locate exactly an important collection of points on the unit circle.
Activity \(\PageIndex{3}\)
In what follows, we work to understand key relationships in \(45^\circ\)-\(45^\circ\)-\(90^\circ\) and \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangles.
- For the \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle with legs \(x\) and \(y\) and hypotenuse \(1\text{,}\) what does the fact that the triangle is isosceles tell us about the relationship between \(x\) and \(y\text{?}\) What are their exact values?
- Now consider the \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle with hypotenuse \(1\) and the longer leg lying along the positive \(x\)-axis. What special kind of triangle is formed when we reflect this triangle across the \(x\)-axis? How can we use this perspective to determine the exact values of \(x\) and \(y\text{?}\)
- Suppose we consider the related \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle with hypotenuse \(1\) and the shorter leg lying along the positive \(x\)-axis. What are the exact values of \(x\) and \(y\) in this triangle?
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We know from the conversion factor from degrees to radians that an angle of \(30^\circ\) corresponds to an angle measuring \(\frac{\pi}{6}\) radians, an angle of \(45^\circ\) corresponds to \(\frac{\pi}{4}\) radians, and \(60^\circ\) corresponds to \(\frac{\pi}{3}\) radians.
Use your work in (a), (b), and (c) to label the noted point in each of Figure \(\PageIndex{8}\) , Figure \(\PageIndex{9}\) , and Figure \(\PageIndex{10}\) , respectively, with its exact coordinates.
Our work in Activity \(\PageIndex{3}\) enables us to identify exactly the location of \(12\) special points on the unit circle. In part (d) of the activity, we located the three noted points in Figure \(\PageIndex{11}\) along with their respective radian measures. By symmetry across the coordinate axes and thinking about the signs of coordinates in the other three quadrants, we can now identify all of the coordinates of the remaining \(9\) points.
In addition, we note that there are four additional points on the circle that we can locate exactly: the four points that correspond to angle measures of \(0\text{,}\) \(\frac{\pi}{2}\text{,}\) \(\pi\text{,}\) and \(\frac{3\pi}{2}\) radians, which lie where the coordinate axes intersect the circle. Each such point has \(0\) for one coordinate and \(\pm 1\) for the other. Labeling all of the remaining points in Figure 2.2.11 is an important exercise that you should do on your own.
Finally, we note that we can identify any point on the unit circle exactly simply by choosing one of its coordinates. Since every point \((x,y)\) on the unit circle satisfies the equation \(x^2 + y^2 = 1\text{,}\) if we know the value of \(x\) or \(y\) and the quadrant in which the point lies, we can determine the other coordinate exactly.
Special points and arc length in non-unit circles
All of our work with the unit circle can be extended to circles centered at the origin with different radii, since a circle with a larger or smaller radius is a scaled version of the unit circle. For instance, if we instead consider a circle of radius \(7\text{,}\) the coordinates of every point on the unit circle are magnified by a factor of \(7\text{,}\) so the point that corresponds to an angle such as \(\theta = \frac{2\pi}{3}\) has coordinates \(\left( -\frac{7}{2}, \frac{7\sqrt{3}}{2} \right)\text{.}\) Distance along the circle is magnified by the same factor: the arc length along the unit circle from \((0,0)\) to \(\left( -\frac{7}{2}, \frac{7\sqrt{3}}{2} \right)\) is \(7 \cdot \frac{2\pi}{3}\text{,}\) since the arc length along the unit circle for this angle is \(\frac{2\pi}{3}\text{.}\)
If we think more generally about a circle of radius \(r\) with a central angle \(\theta\) that intercepts an arc of length \(s\text{,}\) we see how the magnification factor \(r\) (in comparison to the unit circle) connects arc length and the central angle according to the following principle.
If a central angle measuring \(\theta\) radians intercepts an arc of length \(s\) in a circle of radius \(r\text{,}\) then
In the unit circle, where \(r = 1\text{,}\) the equation \(s = r\theta\) demonstrates the familiar fact that arc length matches the radian measure of the central angle. Moreover, we also see how this formula aligns with the definition of radian measure: if the arc length and radius are equal, then the angle measures \(1\) radian.
Connecting arc length and angles in non-unit circles.
If a central angle measuring \(\theta\) radians intercepts an arc of length \(s\) in a circle of radius \(r\text{,}\) then
In the unit circle, where \(r = 1\text{,}\) the equation \(s = r\theta\) demonstrates the familiar fact that arc length matches the radian measure of the central angle. Moreover, we also see how this formula aligns with the definition of radian measure: if the arc length and radius are equal, then the angle measures \(1\) radian.
Determine each of the following values or points exactly.
- In a circle of radius \(11\text{,}\) the arc length intercepted by a central angle of \(\frac{5\pi}{3}\text{.}\)
- In a circle of radius \(3\text{,}\) the central angle measure that intercepts an arc of length \(\frac{\pi}{4}\text{.}\)
- The radius of the circle in which an angle of \(\frac{7\pi}{6}\) intercepts an arc of length \(\frac{\pi}{2}\text{.}\)
- The exact coordinates of the point on the circle of radius \(5\) that lies \(\frac{25\pi}{6}\) units counterclockwise along the circle from \((5,0)\text{.}\)
Summary
- The radian measure of an angle connects the measure of a central angle in a circle to the radius of the circle. A central angle has radian measure \(1\) provided that it intercepts an arc of length equal to the circle's radius. In the unit circle, a central angle's radian measure is precisely the same numerical value as the length of the arc it intercepts along the circle.
- If we begin at the point \((1,0)\) and move counterclockwise along the unit circle, there are natural special points on the unit circle that correspond to angles of measure \(30^\circ\text{,}\) \(45^\circ\text{,}\) \(60^\circ\text{,}\) and their multiples. We can count in \(30^\circ\) increments and identify special points that correspond to angles of measure \(30^\circ\text{,}\) \(60^\circ\text{,}\) \(90^\circ\text{,}\) \(120^\circ\text{,}\) and so on; doing likewise with \(45^\circ\text{,}\) these correspond to angles of \(45^\circ\text{,}\) \(90^\circ\text{,}\) \(135^\circ\text{,}\) etc. In radian measure, these sequences together give us the important angles \(\frac{\pi}{6}\text{,}\) \(\frac{\pi}{4}\text{,}\) \(\frac{\pi}{3}\text{,}\) \(\frac{\pi}{2}\text{,}\) \(\frac{2\pi}{3}\text{,}\) \(\frac{3\pi}{4}\text{,}\) and so on. Together with our work involving \(45^\circ\)-\(45^\circ\)-\(90^\circ\) and \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangles in Activity \(\PageIndex{3}\) , we are able to identify the exact locations of all of the points in Figure \(\PageIndex{11}\) .
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In any circle of radius \(r\text{,}\) if a central angle of measure \(\theta\) radians intercepts an arc of length \(s\text{,}\) then it follows that
\[ s = r\theta\text{.} \nonumber \]
This shows that arc length, \(s\text{,}\) is magnified along with the size of the radius, \(r\text{,}\) of the circle.
Exercises
Let \((x,y)\) by a point on the unit circle. In each of the following situations, determine requested value exactly.
- Suppose that \(x = -0.3\) and \(y\) is negative. Find the value of \(y\text{.}\)
- Suppose that \((x,y)\) lies in Quadrant II and \(x = -2y\text{.}\) Find the values of \(x\) and \(y\text{.}\)
- Suppose that \((x,y)\) lies a distance of \(\frac{29\pi}{6}\) units clockwise around the circle from \((1,0)\text{.}\) Find the values of \(x\) and \(y\text{.}\)
- At what exact point(s) does the line \(y = \frac{1}{2}x + \frac{1}{2}\) intersect the unit circle?
The unit circle is centered at \((0,0)\) and radius \(r = 1\text{,}\) from which the Pythagorean Theorem tells us that any point \((x,y)\) on the unit circle satisfies the equation \(x^2 + y^2 = 1\text{.}\)
- Explain why any point \((x,y)\) on a circle of radius \(r\) centered at \((h,k)\) satisfies the equation \((x-h)^2 + (y-k)^2 = r^2\text{.}\)
- Determine the equation of a circle centered at \((-3,5)\) with radius \(r = 2\text{.}\)
- Suppose that the unit circle is magnified by a factor of \(5\) and then shifted \(4\) units right and \(7\) units down. What is the equation of the resulting circle?
- What is the length of the arc intercepted by a central angle of \(\frac{2\pi}{3}\) radians in the circle \((x-1)^2 + (y-3)^2 = 16\text{?}\)
- Suppose that the line segment from \((-2,-1)\) to \((4,2)\) is a diameter of a circle. What is the circle's center, radius, and equation?
Consider the circle whose center is \((0,0)\) and whose radius is \(r = 5\text{.}\) Let a point \((x,y)\) traverse the circle counterclockwise from \((5,0)\text{,}\) and say the distance along the circle from \((5,0)\) is represented by \(d\text{.}\)
- Consider the point \((a,b)\) that is generated by the central angle \(\theta\) with vertices \((5,0)\text{,}\) \((0,0)\text{,}\) and \((a,b)\text{.}\) If \(\theta = \frac{\pi}{6}\text{,}\) what are the exact values of \(a\) and \(b\text{?}\)
- Answer the same question as in (a) except with \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{\pi}{3}\text{.}\)
- How far has the point \((x,y)\) traveled after it has traversed the circle one full revolution?
- Let \(h = f(d)\) be the circular function that tracks the height of the point \((x,y)\) as a function of distance, \(d\text{,}\) traversed counterclockwise from \((5,0)\text{.}\) Sketch an accurate graph of \(f\) through two full periods, labeling several special points on the graph as well as the horizontal and vertical scale of the axes.
<1> We often call such an angle a central angle.


6.13: Forces and constaints
In this section we discuss the properties of some of the most relevant forces and constraints that can occur in dynamics, as have been experimentally approximately determined using Newton’s laws.
6.13.1 Gravitational force
The gravitational force is important, because it acts on every point mass, and because it is the most prominent force that acts over a distance. In this textbook we focus on the dynamics of objects on or near the surface of the earth, and therefore we can often assume a constant distance to the earth’s centre. In this case the gravitational force vector on a mass \(m_{i}\) can be described by
\[\overrightarrow{\boldsymbol{F}}_{i, g}=m_{i} \overrightarrow{\boldsymbol{g}} \tag{6.27} \label{6.27}\]
where \(\overrightarrow{\boldsymbol{g}}\) is the gravitational acceleration vector with magnitude \(g=|\overrightarrow{\boldsymbol{g}}| \approx\) \(9.8 \mathrm{~m} / \mathrm{s}^{2}\) that points downward in the direction of the centre of the earth.
6.13.2 Contact force
At a point \(C\) where the surfaces of two objects touch, the atoms in the objects can generate a contact force vector \(\overrightarrow{\boldsymbol{F}}_{C}\). One can define a contact plane through \(C\) that is tangential to the surface of the objects. Then, like shown in Figure 6.3, the component of the contact force \(\overrightarrow{\boldsymbol{F}}_{C}\) perpendicular to this plane is called the normal force \(\overrightarrow{\boldsymbol{F}}_{N}\), and the component parallel to the contact plane is the called the friction force \(\overrightarrow{\boldsymbol{F}}_{f}\). So, it always holds that \(\overrightarrow{\boldsymbol{F}}_{C}=\overrightarrow{\boldsymbol{F}}_{N}+\overrightarrow{\boldsymbol{F}}_{f}\), and it is allowed to draw and treat the contact force as a single vector \(\overrightarrow{\boldsymbol{F}}_{C}\). However, since the expressions for \(\overrightarrow{\boldsymbol{F}}_{N}\) and \(\overrightarrow{\boldsymbol{F}}_{f}\) are different, it can be useful to split them. Let us now discuss the normal forces and friction forces separately.
6.13.3 Normal force
Normal forces are the result of repulsive Coulomb forces and quantum mechanical effects. By definition, the normal force vector \(\overrightarrow{\boldsymbol{F}}_{N}\) points perpendicular to the contact plane. Normal forces can be calculated by combining Newton’s laws with constraint equations. If the contact surface is flat, the constraint equations for the motion components normal to the surface simplify to \(v_{n}=0\) and \(a_{n}=0\), otherwise one needs to use Equation 5.86. An example of how to determine the normal force \(F_{N}\) from the EoM was given in Equation 6.10 and Equation 6.17, where the movement of block \(A\) was constrained by the slope.
6.13.4 Friction force
The friction force is the component of the contact force tangential to the contact plane. We distinguish static or kinetic friction forces, which are characterised by a static \(\mu_{s}\) and kinetic \(\mu_{k}\) friction coefficient with \(\mu_{s} \geq \mu_{k}>0\). We now consider the friction force between two points \(A\) and \(B\) on different objects that are in contact.
Static friction
The static friction acting on point \(A\) is best described by this constraint equation:
\[\left\{\begin{array}{cl} a_{A / B, t}=0 & \text { if }\left|\overrightarrow{\boldsymbol{F}}_{A, f s}\right| \leq \mu_{s}\left|F_{A, N}\right| \text { and } \overrightarrow{\boldsymbol{v}}_{A / B}=\overrightarrow{\mathbf{0}} \tag{6.28} \label{6.28}\\[4pt] & \text { otherwise use kinetic friction } \end{array}\right.\]
The first condition for static friction is that the points \(A\) and \(B\) are not moving relative to each other \(\overrightarrow{\boldsymbol{v}}_{A / B}=0\). In that case Equation 6.28 states that the tangential component of acceleration of point \(A\) is zero \(\left(a_{A / B, t}=0\right)\) as long as the friction force \(\left|\overrightarrow{\boldsymbol{F}}_{A, f_{s}}\right|\) needed to satisfy that constraint equation does not exceed the maximum static friction force \(F_{A, f s, \max }=\mu_{s}\left|F_{A, N}\right|\), where \(F_{A, N}\) is the normal force on \(A\). In practice the EoM in the presence of the constraint \(a_{A / B, t}=0\) needs to be solved to determine the friction force \(\overrightarrow{\boldsymbol{F}}_{A, f s}\) after which it is checked if the static friction condition holds. If it does not hold, point \(A\) will accelerate and kinetic friction can be used.
Kinetic friction
When the points \(A\) and \(B\) slide or slip along each other, a kinetic friction force acts on \(A\). The kinetic friction force vector \(\overrightarrow{\boldsymbol{F}}_{A, f k}\) is tangential to the plane of contact and points opposite to the direction of the relative velocity vector:
\[\overrightarrow{\boldsymbol{F}}_{A, f k}= \begin{cases}-\mu_{k}\left|\overrightarrow{\boldsymbol{F}}_{A, N}\right| \hat{\boldsymbol{v}}_{A / B} & \text { if } \overrightarrow{\boldsymbol{v}}_{A / B} \neq 0 \text { or }\left|\overrightarrow{\boldsymbol{F}}_{A, f s}\right|>\mu_{s}\left|F_{A, N}\right| \tag{6.29} \label{6.29}\\[4pt] & \text { otherwise use static friction }\end{cases}\]
Solving problems with friction
If it is not known whether one should deal with static or kinetic friction, and/or if the direction of the velocity is not known in a kinetic friction situation, the following procedure can be used to solve the problem step by step:
- Determine the normal contact force \(F_{N}\) using constraint equations and the EoMs, like in Equation 6.17.
- Assume that the relative velocity is zero \(\overrightarrow{\boldsymbol{v}}_{A / B}=\overrightarrow{\mathbf{0}}\) and use the constraint equation \(a_{A / B, t}=0\) to determine the static friction force \(\overrightarrow{\boldsymbol{F}}_{A, f_{s}}\) from the EoMs.
- Check if \(\left|\overrightarrow{\boldsymbol{F}}_{A, f_{s}}\right| \leq \mu_{s} F_{N}\). If that is true, the static friction assumption is correct.
- Otherwise use kinetic friction. Guess a direction for the velocity \(\hat{\boldsymbol{v}}_{A / B}\) and use it to determine \(\overrightarrow{\boldsymbol{F}}_{A, f k}\) using Equation 6.29.
- Then solve the EoMs to determine if the direction of the velocity \(\overrightarrow{\boldsymbol{v}}_{A / B}\) matches the guessed direction of \(\hat{\boldsymbol{v}}_{A / B}\).
- If not, repeat from step 4 for the other direction of \(\hat{\boldsymbol{v}}_{A / B}\).
For friction of a rolling wheel with the ground two conditions can occur. If there is no slip, only static friction is present and \(a_{A / B, t}=0\). While in the case of slip or sliding, one deals with kinetic friction forces. Gearwheels always behave according to the no slip condition, since their teeth prevent slipping. Finally, a special situation are so-called frictionless contact problems, which have \(\mu_{s}=\mu_{k}=0\). For these cases the friction force is always zero \(\overrightarrow{\boldsymbol{F}}_{f}=\overrightarrow{\mathbf{0}}\), and the contact force is always normal to the contact plane.
6.13.5 Constraint equations on points
Some examples of common constraint equations that limit the motion of a point \(A\) in 3,2 or 1 dimensions are:
- A joint that fixes a point \(A\) to a certain other point \(P\), fixing it along all 3 coordinate directions:
\(\overrightarrow{\boldsymbol{r}}_{A}=\overrightarrow{\boldsymbol{r}}_{P}\)
- A point \(A\) is constrained to move along a slider joint, collar or rail with path curve \(\overrightarrow{\boldsymbol{r}}_{s}(s)\), constraining it along 2 coordinate axes:
\(\overrightarrow{\boldsymbol{r}}_{A}=\overrightarrow{\boldsymbol{r}}_{s}(s)\).
- Motion over a surface, for instance because the gravitational force ensures object \(A\) does not move in the \(z\) direction, constraining motion along 1 coordinate direction:
\(z_{A}(t)=0\)
- Angular constraints can fix certain rotation angles, e.g. \(\phi(t)=\phi_{0}\), they can result in moments on the constrained object.
The contact forces and moments resulting from these constraint equations can be analyzed by combining them with the EoMs.
6.13.6 Dynamics of massless mechanisms
When a mechanism \(m\) has zero mass, so \(m_{m}=0\), application of Newton’s second law tells us that the sum of external forces that act on the mechanism is zero, since otherwise its acceleration would be infinite:
\[\sum \overrightarrow{\boldsymbol{F}}_{\mathrm{ext}}=m_{m} \overrightarrow{\boldsymbol{a}}_{m}=\overrightarrow{\mathbf{0}} \tag{6.30} \label{6.30}\]
This means that the sum of the forces on the mechanism is always zero, just like in statics. Therefore you can use all methods you have learned in statics courses to determine the positions and forces of the massless elements in the mechanism. Unless it is explicitly mentioned they have mass, you may assume all mechanisms in this textbook to be massless.
6.13.7 Two-force members
A massless mechanism on which only two external forces \(\overrightarrow{\boldsymbol{F}}_{A}\) and \(\overrightarrow{\boldsymbol{F}}_{B}\) are acting is called a two-force member. Then we get from Equation 6.30 that \(\overrightarrow{\boldsymbol{F}}_{A}+\overrightarrow{\boldsymbol{F}}_{B}=\overrightarrow{\boldsymbol{0}}\) and:
\[\overrightarrow{\boldsymbol{F}}_{A}=-\overrightarrow{\boldsymbol{F}}_{B} \tag{6.31} \label{6.31}\]
We see from Equation 6.30 that because the mass of the mechanism is zero \(\left(m_{m}=0\right)\), its acceleration \(\overrightarrow{\boldsymbol{a}}_{m}\) can take any value, which means that the mechanism can translate extremely fast, such that it will instantaneously assume a position that ensures \(\sum \overrightarrow{\boldsymbol{F}}=\overrightarrow{\mathbf{0}}\). Note that, as will become clearer later, the same argumentation holds for rotations, such that the mechanism also instantaneously assumes a state where the sum of external moments is zero \(\sum \overrightarrow{\boldsymbol{M}}=\overrightarrow{\mathbf{0}}\), which is only possible if the two forces \(\overrightarrow{\boldsymbol{F}}_{A}\) and \(\overrightarrow{\boldsymbol{F}}_{B}\) are collinear. We will now discuss a few important massless mechanisms, show in Figure 6.7.
6.13.8 Rods and ropes
Probably the simplest two-force members are rods and ropes. Since these mechanisms have a fixed length \(L\) their operation is governed by this constraint equation:
\[\left|\overrightarrow{\boldsymbol{r}}_{A}-\overrightarrow{\boldsymbol{r}}_{B}\right|=L \tag{6.32} \label{6.32}\]
Ropes are flexible, therefore they can only handle tensile (’pulling’) forces that are parallel to the rope. The absolute value of those force vectors is called the tension in the rope and Equation 6.32 only holds if the rope is experiencing tensile force. If ropes are bent (e.g. by pulleys) they can be dealt with as discussed in 5.2.4. Since rods are rigid, they can handle also compressive forces and can transfer moments.
As shown in second FBD in Figure 6.7 the sum of the forces on a rope is not zero anymore \(\left(\overrightarrow{\boldsymbol{F}}_{A}+\overrightarrow{\boldsymbol{F}}_{B} \neq 0\right)\) if there is mass connected to the rope, because part of the force \(\overrightarrow{\boldsymbol{F}}_{A}\) is needed to accelerate the mass. So, be careful, \(\overrightarrow{\boldsymbol{F}}_{A}=-\overrightarrow{\boldsymbol{F}}_{B}\) can only be applied when no masses are accelerated by the forces.
6.13.9 Spring
A linear spring is a mechanism whose length extends by a distance \(\Delta L\) that is proportional to the external pulling force (see Figure 6.7). When there is no external force \(\left(\overrightarrow{\boldsymbol{F}}_{A}=\overrightarrow{\boldsymbol{F}}_{B}=\overrightarrow{\mathbf{0}}\right)\), it has a relaxed length \(L=L_{0}\), and when a force is applied its length increases by a distance \(\Delta L=L-L_{0}\) :
\[\overrightarrow{\boldsymbol{F}}_{k}=k \Delta L \hat{\boldsymbol{s}} \tag{6.33} \label{6.33}\]
Here \(k\) is the stiffness or spring constant of the spring, with unit \(\mathrm{N} / \mathrm{m}\), which normally has a positive value. The direction of the force can be indicated using a unit vector \(\hat{\boldsymbol{s}}\) that points outward parallel to the spring at point \(A\).
6.13.10 Damper
A linear damper is a mechanism whose length changes at a rate that is proportional to the externally applied pulling force (see Figure 6.7).
\[\overrightarrow{\boldsymbol{F}}_{c}=c \frac{\mathrm{d} L}{\mathrm{~d} t} \hat{\boldsymbol{s}} \tag{6.34} \label{6.34}\]
where \(c\) is the damping constant (unit \(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}\) ), which is always positive. If point \(B\) is fixed, the velocity of point \(A\) is in the same direction as the applied force vector.
It is important to note that besides extending, springs and dampers can also be compressed, such that the distance \(\Delta L\) and time derivative \(\dot{L}\) become negative and the forces change sign. Secondly we note that we have discussed in Figure 6.7 the effect of external forces \(\overrightarrow{\boldsymbol{F}}_{A}\) and \(\overrightarrow{\boldsymbol{F}}_{B}\) generated on the mechanisms. But if one e.g. connects a point mass \(m\) at point \(A\) to a spring, the force generated by the mechanism is of opposite sign \(\overrightarrow{\boldsymbol{F}}_{m, k}=-k \Delta L \hat{\boldsymbol{s}}\), where the minus sign arises from Newton’s third law.
6.13.11 Other forces
There are many other types of forces, like thermodynamic, aerodynamic, fluid dynamic, electromagnetic, electrostatic and material forces. It carries too far here to discuss them all, and the methodology discussed in this chapter can be applied for all types of forces once the functional form is known.


1.4 Corpus data in other sub-disciplines of linguistics
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Before we conclude our discussion of the supposed weaknesses of corpus data and the supposed strengths of intuited judgments, it should be pointed out that this discussion is limited largely to the field of grammatical theory. This in itself would be surprising if intuited judgments were indeed superior to corpus evidence: after all, the distinction between linguistic behavior and linguistic knowledge is potentially relevant in other areas of linguistic inquiry, too. Yet, no other sub-discipline of linguistics has attempted to make a strong case against observation and for intuited “data”.
In some cases, we could argue that this is due to the fact that intuited judgments are simply not available. In language acquisition or in historical linguistics, for example, researchers could not use their intuition even if they wanted to, since not even the most fervent defendants of intuited judgments would want to argue that speakers have meaningful intuitions about earlier stages of their own linguistic competence or their native language as a whole. For language acquisition research, corpus data and, to a certain extent, psycholinguistic experiments are the only sources of data available, and historical linguists must rely completely on textual evidence.
In dialectology and sociolinguistics, however, the situation is slightly different: at least those researchers whose linguistic repertoire encompasses more than one dialect or sociolect (which is not at all unusual), could, in principle, attempt to use intuition data to investigate regional or social variation. To my knowledge, however, nobody has attempted to do this. There are, of course, descriptions of individual dialects that are based on introspective data – the description of the grammar of African-American English in Green (2002) is an impressive example. But in the study of actual variation , systematically collected survey data (e.g. Labov et al. 2006) and corpus data in conjunction with multivariate statistics (e.g. Tagliamonte 2006) were considered the natural choice of data long before their potential was recognized in other areas of linguistics.
The same is true of conversation and discourse analysis. One could theoretically argue that our knowledge of our native language encompasses knowledge about the structure of discourse and that this knowledge should be accessible to introspection in the same way as our knowledge of grammar. However, again, no conversation or discourse analyst has ever actually taken this line of argumentation, relying instead on authentic usage data. 6
Even lexicographers, who could theoretically base their descriptions of the meaning and grammatical behavior of words entirely on the introspectively accessed knowledge of their native language have not generally done so. Beginning with the Oxford English Dictionary (OED), dictionary entries have been based at least in part on citations – authentic usage examples of the word in question (see Chapter 2).
If the incompleteness of linguistic corpora or the fact that corpus data have to be interpreted were serious arguments against their use, these sub-disciplines of linguistics should not exist, or at least, they should not have yielded any useful insights into the nature of language change, language acquisition, language variation, the structure of linguistic interactions or the lexicon. Yet all of these disciplines have, in fact, yielded insightful descriptive and explanatory models of their respective research objects.
The question remains, then, why grammatical theory is the only sub-discipline of linguistics whose practitioners have rejected the common practice of building models of underlying principles on careful analysis of observable phenomena. If I were willing to speculate, I would consider the possibility that the rejection of corpora and corpus-linguistic methods in (some schools of) grammatical theorizing are based mostly on a desire to avoid having to deal with actual data, which are messy, incomplete and often frustrating, and that the arguments against the use of such data are, essentially, post-hoc rationalizations. But whatever the case may be, we will, at this point, simply stop worrying about the wholesale rejection of corpus linguistics by some researchers until the time that they come up with a convincing argument for this rejection, and turn to a question more pertinent to this book: what exactly constitutes corpus linguistics?
6 Perhaps Speech Act Theory could be seen as an attempt at discourse analysis on the basis of intuition data: its claims are often based on short snippets of invented conversations. The difference between intuition data and authentic usage data is nicely demonstrated by the contrast between the relatively broad but superficial view of linguistic interaction found in philosophical pragmatics and the rich and detailed view of linguistic interaction found in Conversation Analysis (e.g. Sacks et al. 1974, Sacks 1992) and other discourse-analytic traditions.


3.7: Common Sources of Hazard
The easiest way to decrease skin resistance is to get it wet. Therefore, touching electrical devices with wet hands, wet feet, or especially in a sweaty condition (salt water is a much better conductor of electricity than fresh water) is dangerous. In the household, the bathroom is one of the more likely places where wet people may contact electrical appliances, and so shock hazard is a definite threat there. Good bathroom design will locate power receptacles away from bathtubs, showers, and sinks to discourage the use of appliances nearby. Telephones that plug into a wall socket are also sources of hazardous voltage (the open circuit voltage is 48 volts DC, and the ringing signal is 150 volts AC—remember that any voltage over 30 is considered potentially dangerous!). Appliances such as telephones and radios should never, ever be used while sitting in a bathtub. Even battery-powered devices should be avoided. Some battery-operated devices employ voltage-increasing circuitry capable of generating lethal potentials.
Swimming pools are another source of trouble, since people often operate radios and other powered appliances nearby. The National Electrical Code requires that special shock-detecting receptacles called Ground-Fault Current Interrupting (GFI or GFCI) be installed in wet and outdoor areas to help prevent shock incidents. More on these devices in a later section of this chapter. These special devices have no doubt saved many lives, but they can be no substitute for common sense and diligent precaution. As with firearms, the best “safety” is an informed and conscientious operator.
Extension cords, so commonly used at home and in industry, are also sources of potential hazard. All cords should be regularly inspected for abrasion or cracking of insulation, and repaired immediately. One sure method of removing a damaged cord from service is to unplug it from the receptacle, then cut off that plug (the “male” plug) with a pair of side-cutting pliers to ensure that no one can use it until it is fixed. This is important on jobsites, where many people share the same equipment, and not all people there may be aware of the hazards.
Any power tool showing evidence of electrical problems should be immediately serviced as well. I’ve heard several horror stories of people who continue to work with hand tools that periodically shock them.  Like extension cords, a bad power tool can be removed from service by unplugging it and cutting off the plug at the end of the cord.
Downed power lines are an obvious source of electric shock hazard and should be avoided at all costs. The voltages present between power lines or between a power line and earth ground are typically very high (2400 volts being one of the lowest voltages used in residential distribution systems). If a power line is broken and the metal conductor falls to the ground, the immediate result will usually be a tremendous amount of arcing (sparks produced), often enough to dislodge chunks of concrete or asphalt from the road surface, and reports rivaling that of a rifle or shotgun. To come into direct contact with a downed power line is almost sure to cause death, but other hazards exist which are not so obvious.
When a line touches the ground, current travels between that downed conductor and the nearest grounding point in the system, thus establishing a circuit:
The earth, being a conductor (if only a poor one), will conduct current between the downed line and the nearest system ground point, which will be some kind of conductor buried in the ground for good contact. Being that the earth is a much poorer conductor of electricity than the metal cables strung along the power poles, there will be substantial voltage dropped between the point of cable contact with the ground and the grounding conductor, and little voltage dropped along the length of the cabling (the following figures are very approximate):
If the distance between the two ground contact points (the downed cable and the system ground) is small, there will be substantial voltage dropped along short distances between the two points. Therefore, a person standing on the ground between those two points will be in danger of receiving an electric shock by intercepting a voltage between their two feet!
Again, these voltage figures are very approximate, but they serve to illustrate a potential hazard: that a person can become a victim of electric shock from a downed power line without even coming into contact with that line!
One practical precaution a person could take if they see a power line falling towards the ground is to only contact the ground at one point, either by running away (when you run, only one foot contacts the ground at any given time), or if there’s nowhere to run, by standing on one foot. Obviously, if there’s somewhere safer to run, running is the best option. By eliminating two points of contact with the ground, there will be no chance of applying deadly voltage across the body through both legs.
Review
- Wet conditions increase risk of electric shock by lowering skin resistance.
- Immediately replace worn or damaged extension cords and power tools. You can prevent innocent use of a bad cord or tool by cutting the male plug off the cord (while its unplugged from the receptacle, of course).
- Power lines are very dangerous and should be avoided at all costs. If you see a line about to hit the ground, stand on one foot or run (only one foot contacting the ground) to prevent shock from voltage dropped across the ground between the line and the system ground point.


21.2: Anatomy of the Lymphatic and Immune Systems
By the end of this section, you will be able to:
- Describe the structure and function of the lymphatic tissue (lymph fluid, vessels, ducts, and organs)
- Describe the structure and function of the primary and secondary lymphatic organs
- Discuss the cells of the immune system, how they function, and their relationship with the lymphatic system
The immune system is the complex collection of cells and organs that destroys or neutralizes pathogens that would otherwise cause disease or death. The lymphatic system, for most people, is associated with the immune system to such a degree that the two systems are virtually indistinguishable. The lymphatic system is the system of vessels, cells, and organs that carries excess fluids to the bloodstream and filters pathogens from the blood. The swelling of lymph nodes during an infection and the transport of lymphocytes via the lymphatic vessels are but two examples of the many connections between these critical organ systems.
Functions of the Lymphatic System
A major function of the lymphatic system is to drain body fluids and return them to the bloodstream. Blood pressure causes leakage of fluid from the capillaries, resulting in the accumulation of fluid in the interstitial space—that is, spaces between individual cells in the tissues. In humans, 20 liters of plasma is released into the interstitial space of the tissues each day due to capillary filtration. Once this filtrate is out of the bloodstream and in the tissue spaces, it is referred to as interstitial fluid. Of this, 17 liters is reabsorbed directly by the blood vessels. But what happens to the remaining three liters? This is where the lymphatic system comes into play. It drains the excess fluid and empties it back into the bloodstream via a series of vessels, trunks, and ducts. Lymph is the term used to describe interstitial fluid once it has entered the lymphatic system. When the lymphatic system is damaged in some way, such as by being blocked by cancer cells or destroyed by injury, protein-rich interstitial fluid accumulates (sometimes “backs up” from the lymph vessels) in the tissue spaces. This inappropriate accumulation of fluid referred to as lymphedema may lead to serious medical consequences.
As the vertebrate immune system evolved, the network of lymphatic vessels became convenient avenues for transporting the cells of the immune system. Additionally, the transport of dietary lipids and fat-soluble vitamins absorbed in the gut uses this system.
Cells of the immune system not only use lymphatic vessels to make their way from interstitial spaces back into the circulation, but they also use lymph nodes as major staging areas for the development of critical immune responses. A lymph node is one of the small, bean-shaped organs located throughout the lymphatic system.
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Structure of the Lymphatic System
The lymphatic vessels begin as a blind ending, or closed at one end, capillaries, which feed into larger and larger lymphatic vessels, and eventually empty into the bloodstream by a series of ducts. Along the way, the lymph travels through the lymph nodes, which are commonly found near the groin, armpits, neck, chest, and abdomen. Humans have about 500–600 lymph nodes throughout the body (Figure \(\PageIndex{1}\)).
A major distinction between the lymphatic and cardiovascular systems in humans is that lymph is not actively pumped by the heart, but is forced through the vessels by the movements of the body, the contraction of skeletal muscles during body movements, and breathing. One-way valves (semi-lunar valves) in lymphatic vessels keep the lymph moving toward the heart. Lymph flows from the lymphatic capillaries, through lymphatic vessels, and then is dumped into the circulatory system via the lymphatic ducts located at the junction of the jugular and subclavian veins in the neck.
Lymphatic Capillaries
Lymphatic capillaries , also called the terminal lymphatics, are vessels where interstitial fluid enters the lymphatic system to become lymph fluid. Located in almost every tissue in the body, these vessels are interlaced among the arterioles and venules of the circulatory system in the soft connective tissues of the body (Figure \(\PageIndex{2}\)). Exceptions are the central nervous system, bone marrow, bones, teeth, and the cornea of the eye, which do not contain lymph vessels.
Lymphatic capillaries are formed by a one cell-thick layer of endothelial cells and represent the open end of the system, allowing interstitial fluid to flow into them via overlapping cells (see Figure \(\PageIndex{2}\)). When interstitial pressure is low, the endothelial flaps close to prevent “backflow.” As interstitial pressure increases, the spaces between the cells open up, allowing the fluid to enter. Entry of fluid into lymphatic capillaries is also enabled by the collagen filaments that anchor the capillaries to surrounding structures. As interstitial pressure increases, the filaments pull on the endothelial cell flaps, opening up them even further to allow easy entry of fluid.
In the small intestine, lymphatic capillaries called lacteals are critical for the transport of dietary lipids and lipid-soluble vitamins to the bloodstream. In the small intestine, dietary triglycerides combine with other lipids and proteins, and enter the lacteals to form a milky fluid called chyle . The chyle then travels through the lymphatic system, eventually entering the bloodstream.
Larger Lymphatic Vessels, Trunks, and Ducts
The lymphatic capillaries empty into larger lymphatic vessels, which are similar to veins in terms of their three-tunic structure and the presence of valves. These one-way valves are located fairly close to one another, and each one causes a bulge in the lymphatic vessel, giving the vessels a beaded appearance (see Figure \(\PageIndex{2}\)).
The superficial and deep lymphatics eventually merge to form larger lymphatic vessels known as lymphatic trunks . On the right side of the body, the right sides of the head, thorax, and right upper limb drain lymph fluid into the right subclavian vein via the right lymphatic duct (Figure \(\PageIndex{3}\)). On the left side of the body, the remaining portions of the body drain into the larger thoracic duct, which drains into the left subclavian vein. The thoracic duct itself begins just beneath the diaphragm in the cisterna chyli , a sac-like chamber that receives lymph from the lower abdomen, pelvis, and lower limbs by way of the left and right lumbar trunks and the intestinal trunk.
The overall drainage system of the body is asymmetrical (see Figure 21.4). The right lymphatic duct receives lymph from only the upper right side of the body. The lymph from the rest of the body enters the bloodstream through the thoracic duct via all the remaining lymphatic trunks. In general, lymphatic vessels of the subcutaneous tissues of the skin, that is, the superficial lymphatics, follow the same routes as veins, whereas the deep lymphatic vessels of the viscera generally follow the paths of arteries.
The Organization of Immune Function
The immune system is a collection of barriers, cells, and soluble proteins that interact and communicate with each other in extraordinarily complex ways. The modern model of immune function is organized into three phases based on the timing of their effects. The three temporal phases consist of the following:
- Barrier defenses such as the skin and mucous membranes, which act instantaneously to prevent pathogenic invasion into the body tissues
- The rapid but nonspecific innate immune response , which consists of a variety of specialized cells and soluble factors
- The slower but more specific and effective adaptive immune response , which involves many cell types and soluble factors, but is primarily controlled by white blood cells (leukocytes) known as lymphocytes , which help control immune responses
The cells of the blood, including all those involved in the immune response, arise in the bone marrow via various differentiation pathways from hematopoietic stem cells (Figure \(\PageIndex{4}\)). In contrast with embryonic stem cells, hematopoietic stem cells are present throughout adulthood and allow for the continuous differentiation of blood cells to replace those lost to age or function. These cells can be divided into three classes based on function:
- Phagocytic cells, which ingest pathogens to destroy them
- Lymphocytes, which specifically coordinate the activities of adaptive immunity
- Cells containing cytoplasmic granules, which help mediate immune responses against parasites and intracellular pathogens such as viruses
Lymphocytes: B Cells, T Cells, Plasma Cells, and Natural Killer Cells
As stated above, lymphocytes are the primary cells of adaptive immune responses (Table \(\PageIndex{1}\)). The two basic types of lymphocytes, B cells and T cells, are identical morphologically with a large central nucleus surrounded by a thin layer of cytoplasm. They are distinguished from each other by their surface protein markers as well as by the molecules they secrete. While B cells mature in red bone marrow and T cells mature in the thymus, they both initially develop from bone marrow. T cells migrate from bone marrow to the thymus gland where they further mature. B cells and T cells are found in many parts of the body, circulating in the bloodstream and lymph, and residing in secondary lymphoid organs, including the spleen and lymph nodes, which will be described later in this section. The human body contains approximately 10 12 lymphocytes.
B Cells
B cells are immune cells that function primarily by producing antibodies. An antibody is any of the group of proteins that binds specifically to pathogen-associated molecules known as antigens. An antigen is a chemical structure on the surface of a pathogen that binds to T or B lymphocyte antigen receptors. Once activated by binding to antigen, B cells differentiate into cells that secrete a soluble form of their surface antibodies. These activated B cells are known as plasma cells.
T Cells
The T cell , on the other hand, does not secrete antibody but performs a variety of functions in the adaptive immune response. Different T cell types have the ability to either secrete soluble factors that communicate with other cells of the adaptive immune response or destroy cells infected with intracellular pathogens. The roles of T and B lymphocytes in the adaptive immune response will be discussed further in this chapter.
Plasma Cells
Another type of lymphocyte of importance is the plasma cell. A plasma cell is a B cell that has differentiated in response to antigen binding, and has thereby gained the ability to secrete soluble antibodies. These cells differ in morphology from standard B and T cells in that they contain a large amount of cytoplasm packed with the protein-synthesizing machinery known as rough endoplasmic reticulum.
Natural Killer Cells
A fourth important lymphocyte is the natural killer cell, a participant in the innate immune response. A natural killer cell (NK) is a circulating blood cell that contains cytotoxic (cell-killing) granules in its extensive cytoplasm. It shares this mechanism with the cytotoxic T cells of the adaptive immune response. NK cells are among the body’s first lines of defense against viruses and certain types of cancer.
| Type of lymphocyte | Primary function |
|---|---|
| B lymphocyte | Generates diverse antibodies |
| T lymphocyte | Secretes chemical messengers |
| Plasma cell | Secretes antibodies |
| NK cell | Destroys virally infected cells |
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Primary Lymphoid Organs and Lymphocyte Development
Understanding the differentiation and development of B and T cells is critical to the understanding of the adaptive immune response. It is through this process that the body (ideally) learns to destroy only pathogens and leaves the body’s own cells relatively intact. The primary lymphoid organs are the bone marrow and thymus gland. The lymphoid organs are where lymphocytes mature, proliferate, and are selected, which enables them to attack pathogens without harming the cells of the body.
Bone Marrow
In the embryo, blood cells are made in the yolk sac. As development proceeds, this function is taken over by the spleen, lymph nodes, and liver. Later, the bone marrow takes over most hematopoietic functions, although the final stages of the differentiation of some cells may take place in other organs.  The B cell undergoes nearly all of its development in the red bone marrow, whereas the immature T cell, called a thymocyte , leaves the bone marrow and matures largely in the thymus gland.
Thymus
The thymus gland is a bilobed organ found in the space between the sternum and the aorta of the heart (Figure \(\PageIndex{6}\)). Connective tissue holds the lobes closely together but also separates them and forms a capsule.
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The connective tissue capsule further divides the thymus into lobules via extensions called trabeculae. The outer region of the organ is known as the cortex and contains large numbers of thymocytes with some epithelial cells, macrophages, and dendritic cells (two types of phagocytic cells that are derived from monocytes). The cortex is densely packed so it stains more intensely than the rest of the thymus (see Figure 21.7). The medulla, where thymocytes migrate before leaving the thymus, contains a less dense collection of thymocytes, epithelial cells, and dendritic cells.
Aging and the...
Immune System
By the year 2050, 25 percent of the population of the United States will be 60 years of age or older. The CDC estimates that 80 percent of those 60 years and older have one or more chronic disease associated with deficiencies of the immune systems. This loss of immune function with age is called immunosenescence. To treat this growing population, medical professionals must better understand the aging process. One major cause of age-related immune deficiencies is thymic involution, the shrinking of the thymus gland that begins at birth, at a rate of about three percent tissue loss per year, and continues until 35–45 years of age, when the rate declines to about one percent loss per year for the rest of one’s life. At that pace, the total loss of thymic epithelial tissue and thymocytes would occur at about 120 years of age. Thus, this age is a theoretical limit to a healthy human lifespan.
Thymic involution has been observed in all vertebrate species that have a thymus gland. Animal studies have shown that transplanted thymic grafts between inbred strains of mice involuted according to the age of the donor and not of the recipient, implying the process is genetically programmed. There is evidence that the thymic microenvironment, so vital to the development of naïve T cells, loses thymic epithelial cells according to the decreasing expression of the FOXN1 gene with age.
It is also known that thymic involution can be altered by hormone levels. Sex hormones such as estrogen and testosterone enhance involution, and the hormonal changes in pregnant people cause a temporary thymic involution that reverses itself, when the size of the thymus and its hormone levels return to normal, usually after lactation ceases. What does all this tell us? Can we reverse immunosenescence, or at least slow it down? The potential is there for using thymic transplants from younger donors to keep thymic output of naïve T cells high. Gene therapies that target gene expression are also seen as future possibilities. The more we learn through immunosenescence research, the more opportunities there will be to develop therapies, even though these therapies will likely take decades to develop. The ultimate goal is for everyone to live and be healthy longer, but there may be limits to immortality imposed by our genes and hormones.
Secondary Lymphoid Organs and their Roles in Active Immune Responses
Lymphocytes develop and mature in the primary lymphoid organs, but they mount immune responses from the secondary lymphoid organs . A naïve lymphocyte is one that has left the primary organ and entered a secondary lymphoid organ. Naïve lymphocytes are fully functional immunologically, but have yet to encounter an antigen to respond to. In addition to circulating in the blood and lymph, lymphocytes concentrate in secondary lymphoid organs, which include the lymph nodes, spleen, and lymphoid nodules. All of these tissues have many features in common, including the following:
- The presence of lymphoid follicles, the sites of the formation of lymphocytes, with specific B cell-rich and T cell-rich areas
- An internal structure of reticular fibers with associated fixed macrophages
- Germinal centers , which are the sites of rapidly dividing and differentiating B lymphocytes
- Specialized post-capillary vessels known as high endothelial venules ; the cells lining these venules are thicker and more columnar than normal endothelial cells, which allow cells from the blood to directly enter these tissues
Lymph Nodes
Lymph nodes function to remove debris and pathogens from the lymph, and are thus sometimes referred to as the “filters of the lymph” (Figure \(\PageIndex{7}\)). Any bacteria that infect the interstitial fluid are taken up by the lymphatic capillaries and transported to a regional lymph node. Dendritic cells and macrophages within this organ internalize and kill many of the pathogens that pass through, thereby removing them from the body. The lymph node is also the site of adaptive immune responses mediated by T cells, B cells, and accessory cells of the adaptive immune system. Like the thymus, the bean-shaped lymph nodes are surrounded by a tough capsule of connective tissue and are separated into compartments by trabeculae, the extensions of the capsule. In addition to the structure provided by the capsule and trabeculae, the structural support of the lymph node is provided by a series of reticular fibers laid down by fibroblasts.
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The major routes into the lymph node are via afferent lymphatic vessels (see Figure \(\PageIndex{7}\)). Cells and lymph fluid that leave the lymph node may do so by another set of vessels known as the efferent lymphatic vessels . Lymph enters the lymph node via the subcapsular sinus, which is occupied by dendritic cells, macrophages, and reticular fibers. Within the cortex of the lymph node are lymphoid follicles, which consist of germinal centers of rapidly dividing B cells surrounded by a layer of T cells and other accessory cells. As the lymph continues to flow through the node, it enters the medulla, which consists of medullary cords of B cells and plasma cells, and the medullary sinuses where the lymph collects before leaving the node via the efferent lymphatic vessels.
Spleen
In addition to the lymph nodes, the spleen is a major secondary lymphoid organ (Figure \(\PageIndex{8}\)). It is about 12 cm (5 in) long and is attached to the lateral border of the stomach via the gastrosplenic ligament. The spleen is a fragile organ without a strong capsule, and is dark red due to its extensive vascularization. The spleen is sometimes called the “filter of the blood” because of its extensive vascularization and the presence of macrophages and dendritic cells that remove microbes and other materials from the blood, including dying red blood cells. The spleen also functions as the location of immune responses to blood-borne pathogens.
The spleen is also divided by trabeculae of connective tissue, and within each splenic nodule is an area of red pulp, consisting of mostly red blood cells, and white pulp, which resembles the lymphoid follicles of the lymph nodes. Upon entering the spleen, the splenic artery splits into several arterioles (surrounded by white pulp) and eventually into sinusoids. Blood from the capillaries subsequently collects in the venous sinuses and leaves via the splenic vein. The red pulp consists of reticular fibers with fixed macrophages attached, free macrophages, and all of the other cells typical of the blood, including some lymphocytes. The white pulp surrounds a central arteriole and consists of germinal centers of dividing B cells surrounded by T cells and accessory cells, including macrophages and dendritic cells. Thus, the red pulp primarily functions as a filtration system of the blood, using cells of the relatively nonspecific immune response, and white pulp is where adaptive T and B cell responses are mounted.
Lymphoid Nodules
The other lymphoid tissues, the lymphoid nodules , have a simpler architecture than the spleen and lymph nodes in that they consist of a dense cluster of lymphocytes without a surrounding fibrous capsule. These nodules are located in the respiratory and digestive tracts, areas routinely exposed to environmental pathogens.
Tonsils are lymphoid nodules located along the inner surface of the pharynx and are important in developing immunity to oral pathogens (Figure \(\PageIndex{9}\)). The tonsil located at the back of the throat, the pharyngeal tonsil, is sometimes referred to as the adenoid when swollen. Such swelling is an indication of an active immune response to infection. Histologically, tonsils do not contain a complete capsule, and the epithelial layer invaginates deeply into the interior of the tonsil to form tonsillar crypts. These structures, which accumulate all sorts of materials taken into the body through eating and breathing, actually “encourage” pathogens to penetrate deep into the tonsillar tissues where they are acted upon by numerous lymphoid follicles and eliminated. This seems to be the major function of tonsils—to help children’s bodies recognize, destroy, and develop immunity to common environmental pathogens so that they will be protected in their later lives. Tonsils are often removed in those children who have recurring throat infections, especially those involving the palatine tonsils on either side of the throat, whose swelling may interfere with their breathing and/or swallowing.
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Mucosa-associated lymphoid tissue (MALT) consists of an aggregate of lymphoid follicles directly associated with the mucous membrane epithelia. MALT makes up dome-shaped structures found underlying the mucosa of the gastrointestinal tract, breast tissue, lungs, and eyes. Peyer’s patches, a type of MALT in the small intestine, are especially important for immune responses against ingested substances (Figure \(\PageIndex{10}\)). Peyer’s patches contain specialized endothelial cells called M (or microfold) cells that sample material from the intestinal lumen and transport it to nearby follicles so that adaptive immune responses to potential pathogens can be mounted. A similar process occurs involving MALT in the mucosa and submucosa of the appendix. A blockage of the lumen triggers these cells to elicit an inflammatory response that can lead to appendicitis.
Bronchus-associated lymphoid tissue (BALT) consists of lymphoid follicular structures with an overlying epithelial layer found along the bifurcations of the bronchi, and between bronchi and arteries. They also have the typically less-organized structure of other lymphoid nodules. These tissues, in addition to the tonsils, are effective against inhaled pathogens.


20.3: Linear Independence
Let \(S = \{v_1, v_2, \ldots, v_n\}\) be a set of vectors in a vector space \(V\text{.}\) If there exist scalars \(\alpha_1, \alpha_2 \ldots \alpha_n \in F\) such that not all of the \(\alpha_i\)'s are zero and
\[ \alpha_1 v_1 + \alpha_2 v_2 + \cdots + \alpha_n v_n = {\mathbf 0 }\text{,} \nonumber \]
then \(S\) is said to be linearly dependent . If the set \(S\) is not linearly dependent, then it is said to be linearly independent . More specifically, \(S\) is a linearly independent set if
\[ \alpha_1 v_1 + \alpha_2 v_2 + \cdots + \alpha_n v_n = {\mathbf 0 } \nonumber \]
implies that
\[ \alpha_1 = \alpha_2 = \cdots = \alpha_n = 0 \nonumber \]
for any set of scalars \(\{ \alpha_1, \alpha_2 \ldots \alpha_n \}\text{.}\)
Proposition \(20.9\)
Let \(\{ v_1, v_2, \ldots, v_n \}\) be a set of linearly independent vectors in a vector space. Suppose that
\[ v = \alpha_1 v_1 + \alpha_2 v_2 + \cdots + \alpha_n v_n = \beta_1 v_1 + \beta_2 v_2 + \cdots + \beta_n v_n\text{.} \nonumber \]
Then \(\alpha_1 = \beta_1, \alpha_2 = \beta_2, \ldots, \alpha_n = \beta_n\text{.}\)
- Proof
-
If
\[ v = \alpha_1 v_1 + \alpha_2 v_2 + \cdots + \alpha_n v_n = \beta_1 v_1 + \beta_2 v_2 + \cdots + \beta_n v_n\text{,} \nonumber \]
then
\[ (\alpha_1 - \beta_1) v_1 + (\alpha_2 - \beta_2) v_2 + \cdots + (\alpha_n - \beta_n) v_n = {\mathbf 0}\text{.} \nonumber \]
Since \(v_1, \ldots, v_n\) are linearly independent, \(\alpha_i - \beta_i = 0\) for \(i = 1, \ldots, n\text{.}\)
The definition of linear dependence makes more sense if we consider the following proposition.
Proposition \(20.10\)
A set \(\{ v_1, v_2, \dots, v_n \}\) of vectors in a vector space \(V\) is linearly dependent if and only if one of the \(v_i\) 's is a linear combination of the rest.
- Proof
-
Suppose that \(\{ v_1, v_2, \dots, v_n \}\) is a set of linearly dependent vectors. Then there exist scalars \(\alpha_1, \ldots, \alpha_n\) such that
\[ \alpha_1 v_1 + \alpha_2 v_2 + \cdots + \alpha_n v_n = {\mathbf 0 }\text{,} \nonumber \]
with at least one of the \(\alpha_i\)'s not equal to zero. Suppose that \(\alpha_k \neq 0\text{.}\) Then
\[ v_k = - \frac{\alpha_1}{\alpha_k} v_1 - \cdots - \frac{\alpha_{k - 1}}{\alpha_k} v_{k-1} - \frac{\alpha_{k + 1}}{\alpha_k} v_{k + 1} - \cdots - \frac{\alpha_n}{\alpha_k} v_n\text{.} \nonumber \]
Conversely, suppose that
\[ v_k = \beta_1 v_1 + \cdots + \beta_{k - 1} v_{k - 1} + \beta_{k + 1} v_{k + 1} + \cdots + \beta_n v_n\text{.} \nonumber \]
Then
\[ \beta_1 v_1 + \cdots + \beta_{k - 1} v_{k - 1} - v_k + \beta_{k + 1} v_{k + 1} + \cdots + \beta_n v_n = {\mathbf 0}\text{.} \nonumber \]
The following proposition is a consequence of the fact that any system of homogeneous linear equations with more unknowns than equations will have a nontrivial solution. We leave the details of the proof for the end-of-chapter exercises.
Proposition \(20.11\)
Suppose that a vector space \(V\) is spanned by \(n\) vectors. If \(m \gt n\text{,}\) then any set of \(m\) vectors in \(V\) must be linearly dependent
A set \(\{ e_1, e_2, \ldots, e_n \}\) of vectors in a vector space \(V\) is called a basis for \(V\) if \(\{ e_1, e_2, \ldots, e_n \}\) is a linearly independent set that spans \(V\text{.}\)
Example \(20.12\)
The vectors \(e_1 = (1, 0, 0)\text{,}\) \(e_2 = (0, 1, 0)\text{,}\) and \(e_3 =(0, 0, 1)\) form a basis for \({\mathbb R}^3\text{.}\) The set certainly spans \({\mathbb R}^3\text{,}\) since any arbitrary vector \((x_1, x_2, x_3)\) in \({\mathbb R}^3\) can be written as \(x_1 e_1 + x_2 e_2 + x_3 e_3\text{.}\)
Solution
Also, none of the vectors \(e_1, e_2, e_3\) can be written as a linear combination of the other two; hence, they are linearly independent. The vectors \(e_1, e_2, e_3\) are not the only basis of \({\mathbb R}^3\text{:}\) the set \(\{ (3, 2, 1), (3, 2, 0), (1, 1, 1) \}\) is also a basis for \({\mathbb R}^3\text{.}\)
Example \(20.13\)
Let \({\mathbb Q}( \sqrt{2}\, ) = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \}\text{.}\) The sets
Solution
\(\{1, \sqrt{2}\, \}\) and \(\{1 + \sqrt{2}, 1 - \sqrt{2}\, \}\) are both bases of \({\mathbb Q}( \sqrt{2}\, )\text{.}\)
From the last two examples it should be clear that a given vector space has several bases. In fact, there are an infinite number of bases for both of these examples. In general, there is no unique basis for a vector space. However, every basis of \({\mathbb R}^3\) consists of exactly three vectors, and every basis of \({\mathbb Q}(\sqrt{2}\, )\) consists of exactly two vectors. This is a consequence of the next proposition.
Proposition \(20.14\)
Let \(\{ e_1, e_2, \ldots, e_m \}\) and \(\{ f_1, f_2, \ldots, f_n \}\) be two bases for a vector space \(V\text{.}\) Then \(m = n\text{.}\)
- Proof
-
Since \(\{ e_1, e_2, \ldots, e_m \}\) is a basis, it is a linearly independent set. By Proposition 20.11, \(n \leq m\text{.}\) Similarly, \(\{ f_1, f_2, \ldots, f_n \}\) is a linearly independent set, and the last proposition implies that \(m \leq n\text{.}\) Consequently, \(m = n\text{.}\)
If \(\{ e_1, e_2, \ldots, e_n \}\) is a basis for a vector space \(V\text{,}\) then we say that the dimension of \(V\) is \(n\) and we write \(\dim V =n\text{.}\) We will leave the proof of the following theorem as an exercise.
Theorem \(20.15\)
Let \(V\) be a vector space of dimension \(n\text{.}\)
- If \(S = \{v_1, \ldots, v_n \}\) is a set of linearly independent vectors for \(V\text{,}\) then \(S\) is a basis for \(V\text{.}\)
- If \(S = \{v_1, \ldots, v_n \}\) spans \(V\text{,}\) then \(S\) is a basis for \(V\text{.}\)
-
If
\(S = \{v_1, \ldots, v_k \}\)
is a set of linearly independent vectors for
\(V\)
with
\(k \lt n\text{,}\)
then there exist vectors
\(v_{k + 1}, \ldots, v_n\)
such that
\[ \{v_1, \ldots, v_k, v_{k + 1}, \ldots, v_n \} \nonumber \]
is a basis for \(V\text{.}\)