**Chapter 11**

May 30 2017

**Chapter 11 Further Applications of Integration**
**11.1 The Center of Mass in a Discrete System**
- most of the ideas in this chapter are based on the simple physical concept of center of gravity
- this concept has implications for geometry
- we use it to arrive at a reasonable notion of the "center" of a general geometric figure
- a seesaw balances when w1d1 = w2d2
- this principle was discovered by Archimedes, and is known as the law of the lever
- E wk xk = 0
- we extend this by considering the x-axis as a weightless horizontal rod that pivots at point p
- by Archimedes' law
- the system of weights will exactly balance if
- E wk (xk - p) = 0
- whether this system is in equilibrium or not
- the sum measures the tendency of the system to turn in a clockwise direction about the pivot point p
- this sum is called the moment of the system about p
- the system is in equilibrium if this moment is zero
- it is easy to find a point p = x at which the system will balance
- with the property that the moment of the system about x is zero
- x = E wk xk / E wk
- this balancing point x is called the center of gravity of the given system of weights
- we now recall that the weight of a body at the surface of the earth is simply the force exerted on the body by the gravitational attraction of the earth
- F = mg, so wk = mk g, the g's cancel out
- x = E mk xk / E mk
- with the influence of gravity removed from the discussion in this way
- and the weights wk replaced by the masses mk
- it is customary to call the point x the center of mass of the system
- it is easy to extend these ideas to a two-dimensional system of masses mk located at points (xk, yk) in a horizontal xy-plane
- we define the moment of this system about the y-axis by
- My = E mk xk
- which is the sum of each of the masses multiplied by its signed distance from the y-axis
- if we think of the xy-plane as a weightless horizontal tray
- then in physical language the condition My = 0 means that this tray
- with the given distribution of masses
- will balance if it rests along a knife-edge along the y-axis
- similarly
- the moment of the system about the x-axis is
- Mx = E mk yk
- if we denote the total mass of all the particles in the system by m
- so m = E mk
- then the center of mass of the system is defined to be the point (x, y)
- where x = E mk xk / E mk = My / m
- and y = Mx / m
- the center of mass of our system can be interpreted in two ways
- first, if the equations are written in the form
- mx = My and my = Mx
- then we see that (x, y) is the point at which the entire mass m of the system can be concentrated without changing the moment about either axis
- the second interpretation depends on writing in the form
- E mk (xk - x) = 0 and E mk (yk - y) = 0
- so the tray will balance if it rests along any line through (x, y)
- it will also balance if supported by a sharp nail precisely at the point
**11.2 Centroids**
- the ideas discussed above apply to discrete systems of particles located at a finite number of points in a plane
- we now consider how integration can be used to generalize these ideas to a continuous distribution of mass throughout a region R in the xy-plane
- we shall think of R as a thin sheet of homogeneous material
- say a uniform metal plate
- whose density d (mass per unit area) is constant
- to define the moment of this plate about the y-axis
- we consider a thin vertical strip of height f(x) and width dx
- whose position in the region is specified by the variable x
- the area of this strip is f(x) dx
- its mass is d f(x) dx
- since all of its mass is essentially at the same distance x from the y-axis
- its moment about this axis is x d f(x) dx
- the total moment of hte plate about the y-axis is therefore obtained by allowing the strip to sweep across the region
- and by integrating, or adding together
- all these small contributions to the moment as x increases from a to b
- My = Sab x d f(x) dx
- we can approach this using the Leibnizian approach to integration as described in chapter 7
- similarly
- the moment of the plate about the x-axis is obtained by considering a thin horizontal strip of length g(y) and width dy
- Mx = Scd y d g(y) dy
- the total mass of the plate can be expressed in two ways
- m = Sab d f(x) dx = Scd d g(y) dy
- the center of mass (x, y) of the plate is
- x = My / m, y = Mx / m
- these formulas have the following remarkable feature
- since the density d is assumed to be constant
- it can be factored out and cancelled
- so x = Sab x f(x) dx / Sab f(x) dx
- the denominator is clearly the total area of the region
- the numerator is the moment of this area about the axis
- the center of mass is therefore determined solely by the geometric configuration of R
- it does not depend on the density of any mass this region may contain
- for this region the point is called the centroid of the region
- meaning a point like a center
- it will be convenient for our work if we simply to
- dA = f(x) dx and dA = g(y) dy
- so x = S x dA / S dA, y = S y dA / S dA
- in general, it appears that the centroid of a region must lie on a line of symmetry of the region
- if such a line exists
- further, if a region has two distinct lines of symmetry
- then the centroid must lie on both lines
- and is the point of intersection of these lines
- so in every case where a geometric figure has a center in the usual sense of the word
- this center is the centroid
- however, centroids are easily calculated for many regions that are not ordinarily considered to have centers at all
- so the centroid of a region is a far-reaching generalization of the concept of the center of a geometric figure
- we have discussed centroids of plane regions
- we can just as easily speak of the centroid of an arc in the xy-plane
- or of a region in three-dimensional space
**11.3 The Theorems of Pappus**
- two beautiful geometric theorems connecting centroids with solids and surfaces of revolution were discovered
- in the fourth century A.D. by Pappus of Alexandria
- the last of the great Greek mathematicians
- consider a plane region that lies completely on one side of a line in its plane
- if this region is revolved about the line as an axis
- the volume of the solid generated in this way
- equals the product of the area of the region and the distance travelled around the axis by its centroid
- consider an arc of a plane curve that lies completely on one side of a line in its plane
- if this arc is revolved about the line as an axis
- then the area of the surface generated in this way
- equals the product of the length of the arc and the distance travelled by its centroid
- apart from their aesthetic appeal
- the theorems of Pappus are useful in two ways
- when centroids are known from symmetry considerations
- we can use these theorems to find volumes and areas
- and also
- when the volumes and areas are known
- we can often use these theorems in reverse
- to determine the locations of centroids
**11.4 Moment of Inertia**
- consider a rigid body rotating about a fixed axis
- in order to study motions of this kind
- it is necessary to introduce the concept of the moment of inertia of the body about the axis
- consider a particle of mass m rotating around a fixed axis in a circle of radius r
- if its angular position is given by the angle angle
- then w = d angle / dt and alpha = dw / dt are its angular velocity and angular acceleration
- they are related to the corresponding linear quantities s, v, and a
- by s = r angle, v = r w, a = r alpha
- the twisting effect of the tangential force F is measured by its torque
- T = Fr
- which is the product of the force and the distance from its line of action to the axis
- in the same way the force produces linear acceleration in accordance with F = ma
- torque produces angular acceleration in accordance with T = I alpha
- where the constant of proportionality I is called the moment of inertia
- it can be thought of as a measure of the capacity of the system to resist angular acceleration
- in this sense it is the rotational analog of mass
- these remarks describe the conceptual role of the moment of inertia
- to discover what its definition must be in order to fit it for this role
- we transform T = I alpha by
- Fr = I a/r, mar = I a/r, so I = mr2
- in this section we are mainly concerned with learning how to use integration to calculate the moment of inertia about a given axis
- of a uniform thin sheet of material of constant density d (mass per unit area)
- our method is to imagine the plate divided into a large number of small pieces in such a way that each piece can be treated as a particle to which
- I = mr2 can be applied
- we then find the total moment of inertia by integrating
- or adding together
- the individual moments of inertia of all these pieces
- in addition to its importance in connection with the physics of rotating bodies
- the moment of inertia also has significant applications in structural engineering
- where it is found that the stiffness of a beam is proportional to the moment of inertia of a cross section of the beam
- about a horizontal axis through its centroid
- this fact is exploited in the design of the familiar steel girders called I-beams
- where flanges at the top and bottom of the beam
- as in the letter I
- increase the moment of inertia and hence the stiffness of the beam