Classical Mechanics 0th Edition solutions

Classical Mechanics 0
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.

Classical Mechanics 0
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 ro

Classical Mechanics 0
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.

Classical Mechanics 0
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 sm

Classical Mechanics 0
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

Classical Mechanics 0
(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?

Classical Mechanics 0
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 yo

Classical Mechanics 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.] Comm

Classical Mechanics 0
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 cylinde

Classical Mechanics 0
(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'

Classical Mechanics 0
(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 spher

Classical Mechanics 0
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 explai

Classical Mechanics 0
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 d

Classical Mechanics 0
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 th

Classical Mechanics 0
(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 h

Classical Mechanics 0
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

Classical Mechanics 0
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

Classical Mechanics 0
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

Classical Mechanics 0
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).

Classical Mechanics 0
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 she

Classical Mechanics 0
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

Classical Mechanics 0
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

Classical Mechanics 0
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

Classical Mechanics 0
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 probabl

Classical Mechanics 0
(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 inte

Classical Mechanics 0
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 integrat

Classical Mechanics 0
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.

Classical Mechanics 0
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 abou

Classical Mechanics 0
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).

Classical Mechanics 0
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.]

Classical Mechanics 0
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 t

Classical Mechanics 0
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?

Classical Mechanics 0
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)

Classical Mechanics 0
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).

Classical Mechanics 0
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.

Classical Mechanics 0
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.

Classical Mechanics 0
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 trian

Classical Mechanics 0
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 quantiti

Classical Mechanics 0
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?

Classical Mechanics 0
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?

Classical Mechanics 0
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

Classical Mechanics 0
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 a

Classical Mechanics 0
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 prec

Classical Mechanics 0
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

Classical Mechanics 0
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 wobbl

Classical Mechanics 0
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

Classical Mechanics 0
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

Classical Mechanics 0
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.

Classical Mechanics 0
Starting from Equation (10.100) for L, verify that L , is correctly given by Equations (10.102) and (10.103).

Classical Mechanics 0
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.

Classical Mechanics 0
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).

Classical Mechanics 0
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.

Classical Mechanics 0
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 w

Classical Mechanics 0
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

Classical Mechanics 0
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 vec

Classical Mechanics 0
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

Classical Mechanics 0
(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 ind