Classical Dynamics of Particles and Systems

Jerry B. Marion

Chapter 13 Dynamics of Rigid Bodies - all with Video Answers

Educators

Chapter Questions

00:58

Problem 1

Calculate the moments of inertia $I_1, I_2$, and $I_3$ for a homogeneous sphere of radius $R$ and mass $M$.

Paul Gabriel

Numerade Educator

01:00

Calculate the moments of inertia $I_1, I_2$, and $I_3$ for a homogeneous cone of mass $M$ whose height is $h$ and whose base has a radius $R$. Choose the $x_3$-axis along the axis of symmetry of the cone. Choose the origin at the apex of the cone and calculate the elements of the inertia tensor. Then make a transformation such that the center of mass of the cone becomes the origin and find the principal moments of inertia.

Manik Pulyani

Numerade Educator

Problem 3

Calculate the moments of inertia $I_1, I_2$, and $I_3$, for a homogeneous ellipsoid of mass $M$, the lengths of the axes being $2 a>2 b>2 c$.

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01:36

Problem 4

Consider a thin rod of length $l$ and mass $m$ which is pivoted about one end. Calculate the moment of inertia. Find the point at which, if all of the mass were concentrated, the moment of inertia about the pivot axis would be the same as the real moment of inertia. The distance of this point from the pivot is called the radius of gyration.

Manish Jain

Numerade Educator

05:59

Problem 5

(a) Find the height at which a billiard ball should be struck so that it will roll with no initial slipping. (b) Calculate the optimum height of the rail of a billiard table. On what basis is the calculation predicated?

Averell Hause

Carnegie Mellon University

02:10

Problem 6

Given two spheres of the same diameter and same mass, but one of which is solid and the other is a hollow shell, describe in detail a nondestructive experiment to determine which is solid and which is hollow.

Samantha Baker

Numerade Educator

03:48

Problem 7

A homogeneous disk of radius $R$ and mass $M$ rolls without slipping on a horizontal surface and is attracted to a point which lies at a distance $d$ below the plane. If the force of attraction is proportional to the distance from the center of mass of the disk to the force center, find the frequency of small oscillations around the position of equilibrium.

Shoukat Ali

Other Schools

05:30

Problem 8

A door is constructed of a thin homogeneous slab of material ; it has a width of 1 meter. If the door is opened through $90^{\circ}$ it is found that upon release it closes itself in 2 seconds. Assume that the hinges are frictionless and show that the line of hinges must make an angle of approximately $3^{\circ}$ with the vertical.

Shoukat Ali

Other Schools

02:06

Problem 9

A homogencous slab of thickness $a$ is placed atop a fixed cylinder of radius $R$ whose axis is horizontal. Show that the condition for stable equilibrium of the slab under the assumption that there is no slipping is $R>a / 2$. What is the frequency of small oscillations? Sketch the potential energy $U$ as a function of the angular displacement $\theta$. Show that there is a minimum at $\theta=0$ for $R>a / 2$ but not for $R<a / 2$.

Carson Merrill

Numerade Educator

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Problem 10

A solid sphere of mass $M$ and radius $R$ rotates freely in space with an angular velocity $\omega$ about a fixed diameter. A particle of mass $m$, initially at one pole, moves with a constant velocity $v$ along a great circle of the sphere. Show that, when the particle has reached the other pole, the rotation of the sphere will have been retarded by an angle
$$
\alpha=\frac{\pi \omega R}{v}\left(1-\sqrt{\frac{2 M}{2 M+5 m}}\right)
$$

How many times must a man walk around the Earth in order to change the length of a day by 1 second?

Susan Hallstrom

Numerade Educator

01:55

Problem 11

A homogeneous cube, each edge of which has a length $l$, is initially in a position of unstable equilibrium with one edge in contact with a horizontal planc. The cube is then given a small displacement and allowed to fall. Show that the angular velocity of the cube when one face strikes the plane is given by
$$
\omega^2=A \frac{q}{l}(\sqrt{2}-1)
$$
where $A=3 / 2$ if the edge cannot slide on the plane and where $A=12 / 5$ if sliding can occur without friction.

Penny Riley

Numerade Educator

02:54

Problem 12

Show that none of the principal moments of inertia can exceed the sum of the other two.

Sarah Mccrumb

Numerade Educator

03:04

Problem 13

Show that the principal moments of inertia are all equal for any regular polyhedron whose center is at the origin of the coordinate system. Find the radius of the homogeneous sphere (of the same mass) that has moments of inertia equal to those of a regular tetrahedron.

Sarah Mccrumb

Numerade Educator

02:25

Problem 14

If a physical pendulum has the same period of oscillation when pivoted about either of two points of unequal distances from the center of mass, show that the length of the simple pendulum which has the same period is equal to the separation of the pivot points. Such a physical pendulum is called Kater's reversible pendulum and at one time provided the most accurate way (to about 1 part in $10^5$ ) in which measurements of the acceleration of gravity could be made.* Discuss the advantages of Kater's pendulum over a simple pendulum for such a purpose.

Surendra Kumar

Numerade Educator

Problem 15

Consider the following inertia tensor:
$$
\{\because\}=\left\{\begin{array}{ccc}
\frac{1}{2}(A+B) & \frac{1}{2}(A-B) & 0 \\
\frac{1}{2}(A-B) & \frac{1}{2}(A+B) & 0 \\
0 & 0 & C
\end{array}\right\}
$$

Perform a rotation of the coordinate system by an angle 0 about the $x_3$-axis. Evaluate the transformed tensor elements and show that the choice $\theta=\pi / 4$ renders the inertia tensor diagonal with elements $A, B$, and $C$.

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01:42

Problem 16

Consider a thin homogencous plate which lies in the $x_1-x_2$ plane. Show that the inertia tensor takes the form
$$
\{\}=\left\{\begin{array}{rrc}
A & -C & 0 \\
-C & B & 0 \\
0 & 0 & A+B
\end{array}\right\}
$$

Hunza Gilgit

Numerade Educator

Problem 17

If, in the previous problem, the coordinate axes are rotated through an angle $\theta$ about the $x_3$-axis, show that the new inertia tensor is
$$
\{1\}=\left\{\begin{array}{ccc}
A^{\prime} & -C^{\prime} & 0 \\
-C^{\prime} & B^{\prime} & 0 \\
0 & 0 & A^{\prime}+B^{\prime}
\end{array}\right\}
$$
where
$$
\begin{aligned}
& A^{\prime}=A \cos ^2 \theta-C \sin 2 \theta+B \sin ^2 \theta \\
& B^{\prime}=A \sin ^2 \theta+C \sin 2 \theta+B \cos ^2 \theta \\
& C^{\prime}=C \cos 2 \theta-\frac{1}{2}(B-A) \sin 2 \theta
\end{aligned}
$$
and hence, show that the $x_1$ - and $x_2$-axes become principal axes if the angle of rotation is
$$
0=\frac{1}{2} \tan ^{-1}\left(\frac{2 C}{B-A}\right)
$$

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01:09

Problem 18

Consider a plane homogeneous plate of density $\rho$ which is bounded by the logarithmic spiral $r=k e^{2 U}$ and the radii $\theta=0$ and $\theta=\pi$. Obtain the inertia tensor for the origin at $r=0$ if the plate lies in the $x_1-x_2$ plane. Perform a rotation of the coordinate axes to obtain the principal moments of inertia and use the results of the previous problem to show that they are
$$
I_1^{\prime}=\rho k^4 P(Q+R) ; \quad I_2^{\prime}=\rho k^4 P(Q-R) ; \quad I_3^{\prime}=I_1^{\prime}+I_2^{\prime}
$$
where
$$
P=\frac{e^{4 \pi x}-1}{16\left(1+4 \alpha^2\right)} ; \quad Q=\frac{1+4 \alpha^2}{2 \alpha} ; \quad R=\frac{1+2 \alpha^2}{\sqrt{1+4 \alpha^2}}
$$

Hast Aggarwal

Numerade Educator

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Problem 19

The proofs represented by (a) Eqs. (13.35)-(13.42), and (b) Eqs. (13.45) (13.51) were expressed entirely in the summation convention. Rewrite these proofs in matrix notation.

Victor Salazar

Numerade Educator

Problem 20

The trace of a tensor is defined as the sum of the diagonal elements:
$$
\operatorname{tr}\{1\} \equiv \sum_k I_{k k}
$$
Show, by performing a similarity transformation, that the trace is an invariant quantity; i.e., that
$$
\operatorname{tr}\{1\}=\operatorname{tr}\left\{I^{\prime}\right\}
$$
where $\{1\}$ is the tensor in one coordinate system and $\{1\}\}$ is the tensor in a coordinate system rotated with respect to the first system. Verify this result for the different forms of the inertia tensor for a cube that are given in several examples in the text.

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Problem 21

Show by the same method used for the previous problem, that the determinant of the elements of a tensor is an invariant quantity. Verify this result also for the case of the cube.

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21:19

Problem 22

Find the frequency of small oscillations for a thin homogeneous plate if the motion takes place in the plane of the plate and if the plate has the shape of an equilateral triangle and is suspended (a) from the midpoint of one side, and (b) from one apex.

Maria Gabriela Cota Moreira

Numerade Educator

01:37

Problem 23

Consider a thin disk composed of two homogeneous halves connected along a diameter of the disk. If one half has density $\rho$ and the other has density $2 \rho$, find the expression for the Lagrangian when the disk rolls without slipping along a horizontal surface.

Penny Riley

Numerade Educator

Problem 24

Obtain the components of the angular velocity vector $\omega$ [see Eq. (13.69)] directly from the transformation matrix $\lambda$ [Eq. (13.66)].

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Problem 25

Show from Fig. 13-8c that the components of $\omega$ along the fixed $\left(x_i^{\prime}\right)$ axes are
$$
\begin{aligned}
\omega_1^{\prime} & =\dot{\theta} \cos \varphi+\dot{\psi} \sin \theta \sin \varphi \\
\omega_2^{\prime} & =\dot{\theta} \sin \varphi-\dot{\psi} \sin \theta \cos \varphi \\
\omega_3^{\prime} & =\dot{\psi} \cos \theta+\dot{\varphi}
\end{aligned}
$$

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Problem 26

Show for the case of the force-free motion of a symmetrical top that $\mathbf{L}, \boldsymbol{\omega}$, and $\omega_3 \mathbf{e}_3$ are coplanar. (See Fig. 13-10.)

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10:07

Problem 27

Show that the situation depicted in Fig. 13-10 actually refers to the force-free motion of a symmetrical top which is a prolate spheroid, i.e., $I_3<I_{12}$, whereas for the case of an oblate spheroid, i.e., $I_{12}<I_3$, the $x_3$-axis would lie between $\mathbf{L}$ and $\omega$ (i.e., the body cone would revolve inside the fixed cone).

Sarah Mccrumb

Numerade Educator

Problem 28

Refer to the discussion of the symmetrical top in Section 13.10. Investigate the equation for the turning points of the nutational motion by setting $\hat{\theta}=0$ in Eq. (13.110). Show that the resulting equation is a cubic in $\cos \theta$ and has two real roots and one imaginary root for $\theta$.

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Problem 29

In vestigate the motion of the symmetrical top discussed in Section 13.10 for the case in which the axis of rotation is vertical (i.e., the $x_3^{\prime}$ - and $x_3$-axes coincide). Show that the motion is either stable or unstable depending upon whether the quantity $4 I_{12} \mathrm{Mgh} / I_3^2 \omega_3^2$ is greater than or less than unity. Sketch the effective potential $V(\theta)$ for the two cases and point out the features of these curves that determine whether or not the motion is stable. If the top is set to spinning in the stable configuration, what will be the effect as friction gradually reduces the value of $\omega_3$ ? (This is the case of the "sleeping top.")

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01:25

Problem 30

Consider a thin homogeneous plate with principal moments of inertia
$$
\begin{aligned}
I_1 & \text { along the principal axis } x_1 \\
I_2>I_1 & \text { along the principal axis } x_2 \\
I_3=I_1+I_2 & \text { along the principal axis } x_3
\end{aligned}
$$

Let the origins of the $x_i$ and $x_i^i$ systems coincide and be located at the center of mass, $O$, of the plate. At time $t=0$, the plate is set rotating in a force-free manner with an angular velocity $\Omega$ about an axis inclined at an angle $\alpha$ from the plane of the plate and perpendicular to the $x_2$-axis. Let $I_1 / I_2 \equiv \cos 2 \beta$, and show that at time $t$ the angular velocity about the $x_2$-axis is
$$
\omega_2(t)=\Omega \cos \alpha \tanh (\Omega t \sin \beta)
$$

Narayan Hari

Numerade Educator