Principal Axes and Moments in Rotational Symmetry

The result (10.7), that E mar = 0, can be paraphrased to say that the position vector of the CM relative to the CM is zero, and, in this form, is nearly obvious. Nevertheless, to be sure you understand the result, prove it by solving (10.4) for ea and substituting into the sum concerned.

To illustrate the result (10.18), that the total KE of a body is just the rotational KE relative to any point that is instantaneously at rest, do the following: Write down the KE of a uniform wheel (mass M) rolling with speed v along a flat road, as the sum of the energies of the CM motion and the rotation about the CM. Now write it as the energy of the rotation about the instantaneous point of contact with the road and show that you get the same answer. (The energy of rotation is 410)2. The moment of inertia of a uniform wheel about its center is I = z MR2. That about a point on the rim is /' = z MR2.)

Five equal point masses are placed at the five corners of a square pyramid whose square base is centered on the origin in the xy plane, with side L, and whose apex is on the z axis at a height H above the origin. Find the CM of the five-mass system.

The calculation of centers of mass or moments of inertia usually involves doing an integral, most often a volume integral, and such integrals are often best done in spherical polar coordinates (defined back in Figure 4.16). Prove that fdV f(r) = f r2dr f sin 0 d0 f d0 f (r, 0, 0). [Think about the small volume dV enclosed between r and r dr, 9 and 0 + d0, and 0 and 0 d0.] If the volume integral on the left runs over all space, what are the limits of the three integrals on the right?

A uniform solid hemisphere of radius R has its flat base in the xy plane, with its center at the origin. Use the result of 10.4 to find the center of mass. [Comment: This and the next two problems are intended to reactivate your skills at finding centers of mass by integration. In all cases, you will need to use the integral form of the definition (10.1) of the CM. If the mass is distributed through a volume (as here), the integral will be a volume integral with .]

(a) Find the CM of a uniform hemispherical shell of inner and outer radii a and b and mass M positioned as in Problem 10.5. [See the comment to Problem 10.5 and use the result of Problem 10.4.] (b) What becomes of your answer when a = 0? (c) What if b a?

A "rounded cone" is made by cutting out of a uniform sphere of radius R the volume with 0 < 00, where 8 is the usual angle measured from the polar axis and 00 is a constant between 0 and m. (a) Describe this cone and use the result of Problem 10.4 to find its volume. (b) Find its CM and comment on your results for the cases that 0o= 7 and 00 -- 0.

A uniform thin wire lies along the y axis between y = +L/2. It is now bent toward the left into an arc of a circle with radius R, leaving the midpoint at the origin and tangent to the y axis. Find the CM. [See the comment to Problem 10.5. In this case the integral is a one-dimensional integral.] Comment on your answer for the cases that R oo and that 27R = L.

The moment of inertia of a continuous mass distribution with density p is obtained by converting . the sum of (10.25) into the volume integral f p2 dm f p20 dv (Note the two forms of the Greek "rho": p = distance from z axis, p = mass density.) Find the moment of inertia of a uniform circular cylinder of radius R and mass M for rotation about its axis. Explain why the products of inertia are zero.

(a) A thin uniform rod of mass M and length L lies on the x axis with one end at the origin. Find its moment of inertia for rotation about the z axis. [Here the sum of (10.25) must be replaced by an integral of the form f x2,u dx where p, is the linear mass density, mass/length.] (b) What if the rod's center is at the origin?

(a) Use the result of Problem 10.4 to find the moment of inertia of a uniform solid sphere (mass M, radius R) for rotation about a diameter. (b) Do the same for a uniform hollow sphere whose inner and outer radii are a and b. [One slick way to do this is to think of the hollow sphere as a solid sphere of radius b from which you have removed a sphere of the same density but radius a.]

A triangular prism (like a box of Toblerone) of mass M, whose two ends are equilateral triangles parallel to the xy plane with side 2a, is centered on the origin with its axis along the z axis. Find its moment of inertia for rotation about the z axis. Without doing any integrals write down and explain its two products of inertia for rotation about the z axis.

A thin rod (of width zero, but not necessarily uniform) is pivoted freely at one end about the horizontal z axis, being free to swing in the xy plane (x horizontal, y vertically down). Its mass is m, its CM is a distance a from the pivot, and its moment of inertia (about the z axis) is I. (a) Write down the equation of motion Lz = rz and, assuming the motion is confined to small angles (measured from the downward vertical), find the period of this compound pendulum. ("Compound pendulum" is traditionally used to mean any pendulum whose mass is distributed as contrasted with a "simple pendulum," whose mass is concentrated at a single point on a massless arm.) (b) What is the length of the "equivalent" simple pendulum, that is, the simple pendulum with the same period?

A stationary space station can be approximated as a hollow spherical shell of mass 6 tonnes (6000 kg) and inner and outer radii of 5 m and 6 m. To change its orientation, a uniform flywheel (radius 10 cm, mass 10 kg) at the center is spun up quickly from rest to 1000 rpm. (a) How long will it take the station to rotate by 10 degrees? (b) What energy was needed for the whole operation? [To find the necessary moment of inertia, you could do Problem 10.11.]

(a) Write down the integral (as in Problem 10.9) for the moment of inertia of a uniform cube of side a and mass M, rotating about an edge, and show that it is equal to 3 Mae. (b) If I balance the cube on an edge in unstable equilibrium on a rough table, it will eventually topple and rotate until it hits the table. By considering the energy of the cube, find its angular velocity just before it hits the table. (Assume the edge does not slide on the table.)

Find the moment of inertia for a uniform cube of mass M and edge a as in Problem 10.15 and then do the following: The cube is sliding with velocity v across a flat horizontal frictionless table when it hits a straight very low step perpendicular to v, and the leading lower edge comes abruptly to rest. (a) By considering what quantities are conserved before, during, and after the brief collision, find the cube's angular velocity just after the collision. (b) Find the minimum speed v for which the cube rolls over after hitting the step.

Write down the integral for the moment of inertia of a uniform ellipsoid with surface (x /a )2 + (Y1b)2 (z/c)2 = 1 for rotation about the z axis. One simple way to do the integral is to make a change of variables to = xla,17 = y/b, and = z/c. Each of the two resulting integrals can be related to the corresponding integrals for a sphere (as in Problem 10.11). Do this. Check your answer for the case a = b =c.

Consider the rod of Problem 10.13. The rod is struck sharply with a horizontal force F which delivers an impulse F At = a distance b below the pivot. (a) Find the rod's angular momentum, and hence momentum, just after the impulse. (b) Find the impulse 17 delivered to the pivot. (c) For what value of b (call it bo) is ri = 0? (The distance bo defines the so-called "sweet spot." If the rod were a tennis racket and the pivot your hand, then if the ball hits the sweet spot, your hand would experience no impulse.)

Verify that the components of the vector r x (co x r) are given correctly by Equation (10.35). Do this both by working with components and by using the so-called BAC CAB rule, that is A x (B x C) = B(A C) C(A B).

Show that the inertia tensor is additive, in this sense: Suppose a body A is made up of two parts B and C. (For instance, a hammer is made up of a wooden handle wedged into a metal head.) Then IA = IB + I. Similarly, if A can be thought of as the result of removing C from B (as a hollow spherical shell is the result of removing a small sphere from inside a larger sphere), then IA = IB lc

The definition of the inertia tensor in Equations (10.37) and (10.38) has the rather ugly feature that the diagonal and off-diagonal elements are defined by completely different equations. Show that the two definitions can be combined into the single equation (which is slightly less messy in integral form) Iij = f Q(r26ii rirj)dV where 6ii is the Kronecker delta symbol I 1 i = j 6ii 10 i j.

A rigid body comprises 8 equal masses m at the corners of a cube of side a, held together by massless struts. (a) Use the definitions (10.37) and (10.38) to find the moment of inertia tensor I for rotation about a corner 0 of the cube. (Use axes along the three edges through 0.) (b) Find the inertia tensor of the same body but for rotation about the center of the cube. (Again use axes parallel to the edges.) Explain why in this case certain elements of I could be expected to be zero.

Consider a rigid plane body or "lamina," such as a flat piece of sheet metal, rotating about a point 0 in the body. If we choose axes so that the lamina lies in the xy plane, which elements of the inertia tensor I are automatically zero? Prove that 4, = Ixx + iyy

a) If I' denotes the moment of inertia tensor of a rigid body (mass M) about its CM, and I the corresponding tensor about a point P displaced from the CM by A = prove that /xx = /x7 mo2^2) and / = yz yz (10.117) and so forth. (These results, which generalize the parallel-axis theorem that you probably learned in introductory physics, mean that once you know the inertia tensor for rotation about the CM, calculating it for any other origin is trivially easy.) (b) Confirm that the results of Example 10.2 (page 381) fulfill the identities (10.117) [so that the calculations of part (a) of the example were actually unnecessary].

(a) Find all nine elements of the moment of inertia tensor with respect to the CM of a uniform cuboid (a rectangular brick shape) whose sides are 2a, 2b, and 2c in the x, y, and z directions and whose mass is M. Explain clearly why you could write down the off-diagonal elements without doing any integration. (b) Combine the results of part (a) and Problem 10.24 to find the moment of inertia tensor of the same cuboid with respect to the corner A at (a, b, c). (c) What is the angular momentum about A if the cuboid is spinning with angular velocity w around the edge through A and parallel to the x axis?

a) Prove that in cylindrical polar coordinates a volume integral takes the form f dV Pr), f p dp f c14) f dz f(p,O,z). (b) Show that the moment of inertia of a uniform solid cone pivoted at its tip and rotating about its axis is given by the integral (10.58), explaining clearly the limits of integration. Show that the integral evaluates to MR2. (c) Prove also that 4, = M(R2 4h2) as in Equation (10.61).

Find the inertia tensor for a uniform, thin hollow cone, such as an ice-cream cone, of mass M, height h, and base radius R, spinning about its pointed end.

Find the moment of inertia tensor I for the triangular prism of Problem 10.12 with height h. (If you did Problem 10.12, you've already done about half the work.) Your result should show that I has the form we've found for an axisymmetric body. This suggests what is true, that three-fold symmetry about an axis (symmetry under rotations of 120 degrees) is enough to ensure this form.

Prove that if the axes Ox, Oy, and Oz are principal axes of a certain rigid body, then the inertia tensor (with respect to these axes) is diagonal with the principal moments down the diagonal as in (10.66).

Consider a lamina, such as a flat piece of sheet metal, rotating about a point 0 in the body. Prove that the axis through 0 and perpendicular to the plane is a principal axis. [Hint: See Problem 10.23.]

Consider an arbitrary rigid body with an axis of rotational symmetry, which we'll call i. (a) Prove that the axis of symmetry is a principal axis. (b) Prove that any two directions x and Sr perpendicular to i and each other are also principal axes. (c) Prove that the principal moments corresponding to these two axes are equal: Al = X2.

Show that the principal moments of any rigid body satisfy X3 < + X2. [Hint: Look at the integrals that define these moments.] In particular, if X2, then X3 < 2.11. (b) For what shape of body is X3 = X1 + X2?

Here is a good exercise in vector identities and matrices, leading to some important general results: (a) For a rigid body made up of particles of mass ma, spinning about an axis through the origin with angular velocity co, prove that its total kinetic energy can be written as T = - ma[(wra)2 (co ra)2]

The inertia tensor I for a solid cube is given by (10.72). Verify that det(\(\mathbf{I}-\lambda \mathbf{1}\)) is as given in (10.73).

A rigid body consists of three masses fastened as follows: m at (a, 0, 0), 2m at (0, a, a), and 3m at (0, a, —a). (a) Find the inertia tensor I. (b) Find the principal moments and a set of orthogonal principal axes.

A rigid body consists of three masses fastened as follows: m at (a, 0, 0), 2m at (0, a, a), and 3m at (0, a, a). (a) Find the inertia tensor I. (b) Find the principal moments and a set of orthogonal principal axes.

A thin, flat, uniform metal triangle lies in the xy plane with its corners at (1, 0, 0), (0, 1, 0), and the origin. Its surface density (mass/area) is = 24. (Distances and masses are measured in unspecified units, and the number 24 was chosen to make the answer come out nicely.) (a) Find the triangle's inertia tensor I. (b) What are its principal moments and the corresponding axes?

Suppose that you have found three independent principal axes (directions e l , e2 , e3) and corresponding principal moments A 1 , A2, A3 of a rigid body whose moment of inertia tensor I (not diagonal) you had calculated. (You may assume, what is actually fairly easy to prove, that all of the quantities concerned are real.) (a) Prove that if A iAithen it is automatically the case that e i• e j= 0. (It may help to introduce a notation that distinguishes between vectors and matrices. For example, you could use an underline to indicate a matrix, so that a is the 3 x 1 matrix that represents the vector a, and the vector scalar product a • b is the same as the matrix product a b or b a. Then consider the number ii te j, which can be evaluated in two ways using the fact that both e, and e jare eigenvectors of I.) (b) Use the result of part (a) to show that if the three principal moments are all different, then the directions of three principal axes are uniquely determined. (c) Prove that if two of the principal moments are equal, A l= A2 say, then any direction in the plane of e land e2is also a principal axis with the same principal moment. In other words, when A l= A2 the corresponding principal axes are not uniquely determined. (d) Prove that if all three principal moments are equal, then any axis is a principal axis with the same principal moment.

Consider a top consisting of a uniform cone spinning freely about its tip at 1800 rpm. If its height is 10 cm and its base radius 2.5 cm, at what angular velocity will it precess?

Consider a top consisting of a uniform cone spinning freely about its tip at 1800 rpm. If its height is 10 cm and its base radius 2.5 cm, at what angular velocity will it precess?

Consider a lamina rotating freely (no torques) about a point 0 of the lamina. Use Euler's equations to show that the component of co in the plane of the lamina has constant magnitude. [Hint: Use the results of Problems 10.23 and 10.30. According to Problem 10.30, if you choose the direction e3 normal to the plane of the lamina, e3 points along a principal axis. Then what you have to prove is that the time derivative of col co22 is zero.]

I take a book that is 30 cm x 20 cm x 3 cm and is held shut by a rubber band, and I throw it into the air spinning about an axis that is close to the book's shortest symmetry axis at 180 rpm. What is the angular frequency of the small oscillations of its axis of rotation? What if I spin it about an axis close to the longest symmetry axis?

I throw a thin uniform circular disc (think of a frisbee) into the air so that it spins with angular velocity co about an axis which makes an angle a with the axis of the disc. (a) Show that the magnitude of co is constant. [Look at Equation (10.94).] (b) Show that as seen by me, the disc's axis precesses around the fixed direction of the angular momentum with angular velocity Qs = co-I4 3 sine a. (The results of Problems 10.23 and 10.46 will be useful.)

An axially symmetric space station (principal axis e3, and Xi = X2) is floating in free space. It has rockets mounted symmetrically on either side that are firing and exert a constant torque r about the symmetry axis. Solve Euler's equations exactly for co (relative to the body axis) and describe the motion. At t = 0 take co = (cow, 0, 130)-

Because of the earth's equatorial bulge, its moment about the polar axis is slightly greater that the other two moments, X3 = 1.00327A1 (but Xi = X2). (a) Show that the free precession described in Section 10.8 should have period 305 days. (As described in the text, the period of this "Chandler wobble" is actually more like 400 days.) (b) The angle between the polar axis and co is about 0.2 arc seconds. Use Equation (10.118) from Problem 10.46 to show that as seen from the space frame the period of this wobble should be about a day.

We saw in Section 10.8 that in the free precession of an axially symmetric body the three vectors e3 (the body axis), co, and L lie in a plane. As seen in the body frame, e3 is fixed, and co and L precess around e3 with angular velocity S2b = (03(X1 X3)/X1 As seen in the space frame L is fixed and co and e3 precess around L with angular velocity In In this problem you will find three equivalent expressions for Qs. (a) Argue that Sts = Sib + co. [Remember that relative angular velocities add like vectors.] (b) Bearing in mind that Stb is parallel to e3 prove that Qs = co sin a/ sin 0 where a is the angle between e3 and co and 6 is that between e3 and L (see Figure 10.9). (c) Thence prove that 2 + (x 2 ___ x2) sine s sin a L 3 1 3 / = = = sin 0 .11 X (10.118)

Imagine that this world is perfectly rigid, uniform, and spherical and is spinning about its usual axis at its usual rate. A huge mountain of mass 10-8 earth masses is now added at colatitude 60, causing the earth to begin the free precession described in Section 10.8. How long will it take the North Pole (defined as the northern end of the diameter along co) to move 100 miles from its current position? [Take the earth's radius to be 4000 miles.]

quation (10.97) gives the angular velocity of a body in terms of an unholy mixture of unit vectors. (a) Find co in terms of X, ST', and i. (b) Do the same in terms of el, e2, and e3.

Starting from Equation (10.100) for L, verify that L , is correctly given by Equations (10.102) and (10.103).

Equation (10.105) gives the kinetic energy in terms of Euler angles for a body with X1 = X2 Find the corresponding expression for a body whose three principal moments are all different.

Verify that the energy of a symmetric top can be written as E 162 + U eff (0) , where the effective potential energy is as given in (10.114).

Consider the rapid steady precession of a symmetric top predicted in connection with (10.112). (a) Show that in this motion the angular momentum L must be very close to the verti-cal. [Hint: Use (10.100) to write down the horizontal component Lhor of L. Show that if is given by the right side of (10.112), Lhor is exactly zero.] (b) Use this result to show that the rate of precession Q given in (10.112) agrees with the free precession rate Qs found in (10.06).

In the discussion of steady precession of a top in Section 10.10, the rates Q at which steady precession can occur were determined by the quadratic equation (10.110). In particular, we examined this equation for the case that w3 is very large. In this case you can write the equation as a02 bQ c = 0 where b is very large. (a) Verify that when b is very large, the two solutions of this equation are approximately c/b (which is small) and b/a (which is large). What precisely does the condition "b is very large" entail? (b) Verify that these give the two solutions claimed in (10.111) and (10.112).

Computer] The nutation of a top is controlled by the effective potential energy (10.114). Make a plot of Ueff (0) as follows: (a) First, since the second term of Ueff (9) is a constant, you can ignore it. Next, by choice of your units, you can take MgR = 1 = A1. The remaining parameters L, and L3 are genuinely independent parameters. To be definite set Lz = 10 and L3 = 8 and plot Ueff (0) as a function of O. (b) Explain clearly how you could use your graph to determine the angle 0, at which the top could precess steadily with 0 = constant. Find 0,, to three significant figures. (c) Find the rate of this steady precession, Q = cb, as given by (10.115). Compare with the approximate value of Q given by (10.112).

The analysis of the free precession of a symmtric body in Section 10.8 was based on Euler's equations. Obtain the same results using Euler's angles as follows: Since L is constant you may as well choose the space axis i so that L = Li (a) Use Equation (10.98) for z to write L in terms of the unit vectors e'1, e'2, and e3. (b) By comparing this expression with (10.100), obtain three equations for 0, 4, and . (c) Hence show that 0 and 4 are constant, and that the rate of precession of the body axis about the space axis i is Qs = L 0.1 as in (10.96). (d) Using (10.99) show that the angle between (I) and e3 is constant and that the three vectors L, co, and e3 are always coplanar.

An important special case of the motion of a symmetric top occurs when it spins about a vertical axis. Analyze this motion as follows: (a) By inspecting the effective PE (10.114), show that if at any time 0 = 0, then L3 and L, must be equal. (b) Set Lz = L3 = A30)3 and then make a Taylor expansion of (Jeff (0) about 0 = 0 to terms of order 02. (c) Show that if co3 > corm = 21/Mg RX i/A32, then the position 0 = 0 is stable, but if w3 < comir, it is unstable. (In practice, friction slows the top's spinning. Thus with w3 sufficiently fast, the vertical top is stable, but as it slows down the top will eventually lurch away from the vertical when co3 reaches comity)

(a) Find the Lagrangian for a symmetric top whose tip is free to slide on a frictionless, horizontal table. For generalized coordinates use the Euler angles (0,0,0 plus X and Y, where (X, Y, Z) is the CM position relative to a fixed point on the table. (Note that the vertical position Z is not an independent coordinate, since Z = R cos 0.) (b) Show that the CM motion of (X, Y) separates completely from the rotational motion. (c) Consider the two possible rates of steady precession (10.111) and (10.112) (for given 0 and w3). How do these differ in the present case from their corresponding values when the tip is held at a fixed pivot?