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