Mechanics Berkeley Physics

Charles Kittel, Walter D. Knight, Malvin A. Ruderman, A. Carl Helmholz, Burton J. Moyer

Chapter 8

Elementary Dynamics of Rigid Bodies - all with Video Answers

Educators

Chapter Questions

Problem 1

Parallel axis theorem. Beginning with the fact that the moment of inertia of a thin disk about a diametral axis is $\frac{1}{4} m a^{2}$, employ the parallel axis theorem to prove that for a solid circular cylinder of mass $M$, radius $a$, and length $L$, the moment of inertia about a transverse axis through the center of mass is $M a^{2} / 4+M L^{2} / 12$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 2

Additivity of moments of inertia. Using the principle that moments of inertia are simply additive, calculate the moment of inertia about the central axis of the cylindrical object in Fig. $8.15$ if its mass is $M$, its radius $a$, the radius of each of the four cylindrical voids is $a / 3$, and the axis of each void is at distance $a / 2$ from the central axis.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 3

Moment of inertia of solid sphere. Show that the moment of inertia about a diameter of a solid sphere is $\frac{2}{5} M r^{2} .$ This can be simply done by considering the sphere to be a stack of circular disks of infinitesimal thickness fitting within a spherical bounding surface.

Baskar P

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Problem 4

Moments of inertia of triangle. Three equal mass points at the vertices of an equilateral triangle (see Fig. $8.16$ ) are joined by a rigid triangular sheet of negligible mass.
(a) Find the moment of inertia $I_{z}$ about the normal axis through the center $C$
(b) Evaluate $I_{y}$ for the $y$ axis as shown.
(c) By invoking the perpendicular axis theorem, evaluate $I_{x^{\prime}}$

Surendra Kumar

Numerade Educator

Problem 5

Square plate: equality of moments. Prove that the moment of inertia of a rigid square plate about a diagonal axis in its plane is the same as that about an axis in the plane through the center, parallel to edges of the square. (The perpendicular axis theorem, together with symmetry, allows you to prove this without any calculation.)

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Problem 6

Rolling rigid bodies. A solid cylinder, a thin-walled cylindrical shell, a solid sphere, and a thin-walled spherical shell are all rolled down an inclined plane sloped at angle $\theta .$ Each object has the same radius $R .$ Find the acceleration of each.

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Problem 7

Rolling of hollow sphere. A hollow sphere, with inside radius $R_{1}$ and outside radius $R_{2}$, rolls without slipping down an inclined plane at angle $\theta$ from the horizontal.
(a) Find its angular and linear accelerations.
(b) At its lower end the plane merges into a curved transition that finally becomes a horizontal plane. With what speed will the object be moving on the final horizontal plane if it started from rest on the inclined plane with its center at height $h$ above the final horizontal plane? (Use conservation of energy.)

Surendra Kumar

Numerade Educator

Problem 8

Frictional torque. A heavy flywheel in the form of a solid cylinder of radius $50 \mathrm{~cm}$, thickness $20 \mathrm{~cm}$, and mass $1200 \mathrm{~kg}$ rotates freely on bearings at an initial rate of $150 \mathrm{rps}$. It is to be brought to rest by a friction brake, in which a brake shoe is pressed against the periphery of the flywheel with a force equivalent to a $40-\mathrm{kg}$ weight. The coefficient of friction between the braking surfaces is $0.4$ and is assumed to be independent of relative surface speed.
(a) Through what angle will the flywheel turn in coming to rest if the brake is steadily applied?
(b) How long will it be in coming to rest?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 9

Compound pendulum: equivalent length. Prove that a uniform rod of length $L$, hanging as a compound pendulum from a pivot at one end, has the same frequency for small oscillations as a simple pendulum whose length is $2 L / 3$.

Surendra Kumar

Numerade Educator

Problem 10

Center of percussion. Consider a rigid rod of length $L$, suspended from one end by a pivot at $P .$ A force $F$ acting for a brief period (i.e., an impulsive force) is to be applied to set the rod into pendulum motion as shown in Fig. $8.17 .$ The support arrangement at $P$ is very fragile, and it is necessary to apply $F$ at such a distance $x$ that no reaction force occurs at $P .$ Find the value of $x$ to meet this requirement. This position is called the center of percussion for the point of suspension
P. (Hint: The effect of $F$ will be to accelerate the center of mass and to give angular acceleration about $P$ by its moment with respect to $P$. Compatibility of these accelerations, assuming no reaction force at $P$, will specify the value of $x$ in terms of $L .$

Surendra Kumar

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Problem 11

Unbalanced rigid body. A thin rim or hoop of mass $M$ and radius $R$ is mounted with massless spokes so as to rotate freely in the vertical plane about a horizontal axis through its center. A particle of mass $m$ is fastened to the rim, causing the system to hang at rest with $m$ at the bottom. Find the frequency of small oscillations. Also find the maximum angular velocity attained if the system is released from a stationary condition with $m$ at the top.

Surendra Kumar

Numerade Educator

Problem 12

Reversible pendulum. Prove that for a compound pendulum there are two support distances $l_{1}$ and $l_{2}$ from the center of mass that will produce the same frequency of small oscillations and that these distances are related by
$$
l_{1} l_{2}=\frac{I_{c}}{M}
$$
Furthermore show that if we have located such a pair of conjugate points and measured their common frequency $\omega$, we may obtain the value of $g$ from
$$
g=\omega^{2}\left(l_{1}+l_{2}\right)
$$
(This is a technique called the reversible pendulum method for measuring $g .$ The support points are on a straight line passing through the center of mass on either side of it; so $l_{1}+l_{2}$ is simply the distance between support points. The position of the center of mass thus is not required.)

Surendra Kumar

Numerade Educator

Problem 13

Rotating torque. A rectangular plate of mass $M$, with sides $a$ and $b$, is rotated with angular velocity $\omega$ about a fixed axis along a diagonal. Evaluate the rotating torque vector that the bearings must apply to the plate to hold it in this mode of rotation. Draw a good diagram showing the angular momentum vector. Express it as a vector.

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Problem 14

Lack of dynamical balance. $\Lambda$ uniform thin rod of mass $M$ and length $L$ is rotated about a transverse axis through its center. The axis is supposed to be perpendicular to the rod, but through an imperfection it deviates from this by a small angle $\delta .$ Find the rotating torque vector required if the rotation is with angular velocity $\omega$, Express the angular momentum vector and show it in a diagram.

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Problem 15

Gyroscope. A certain gyroscope consists of a solid cylinder with radius $a=4 \mathrm{~cm} .$ It is supported by a massless stem whose tip is pivoted freely at a point $5 \mathrm{~cm}$ from the center of mass of the cylinder. It is observed to be moving in steady precession at an angle of inclination from vertical, and the precession occurs at one complete circular excursion every $3 \mathrm{~s}$. Evaluate the angular velocity of spin of the gyroscope about its own axis.

Surendra Kumar

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Problem 16

Angular acceleration. A solid cylinder of mass $2.0 \mathrm{~kg}$ and radius $4.0 \mathrm{~cm}$ is constrained to rotate about its axis, which is horizontal. A string is wrapped around it and one end hanging freely has a mass of $150 \mathrm{~g}$ attached (see Fig. $8.7$ ). Find the linear acceleration of the mass, the angular acceleration of the cylinder, the tension in the string, and the vertical force keeping the cylinder up.

Surendra Kumar

Numerade Educator

Problem 17

Rotation of gyroscope. Figure $8.18$ represents a gyroscope wheel seen from one side, with its axle mounted in bearings $A$ and $B$. It is spinning with angular velocity as shown, the near side of the wheel moving downward. Upward support forces exist equally at $A$ and $B$.
(a) It is now desired to reorient the wheel to place $A$ directly over $B$, without moving the center of mass of the system. Describe the additional forces, besides support, to be applied at $A$ and $B$.
(b) If instead of placing $A$ over $B$ we had wished to bring $A$ out toward the viewer, and $B$ behind $A$, describe the forces we should apply at $A$ and $B$.

Surendra Kumar

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Problem 18

18. Torques about center of mass. A cylinder of mass $M_{1}$ radius $R_{1}$; with axis horizontal, is constrained to rotate abou its axis. A string wrapped around this cylinder is also wrappe around a cylinder of mass $M_{2}$, radius $R_{2}$, which is free unwind and fall with its axis horizontal as in Fig. 8.19. Fint in the approximation of vertical string the
(a) Acceleration of the center of mass of $M_{2}$.
(b) Angular acceleration of $M_{2}$.
(c) Angular acceleration of $M_{1}$.
(d) Tension in the string. If one takes moments about point $P$ of the figure, what is the "fictitious force" at the center of mass of the second cylinder?

Surendra Kumar

Numerade Educator

Problem 19

Minimum coefficient of friction. For a symmetric body to roll without slipping down an inclined plane, show that
$$
\mu \geq \frac{\tan \theta}{M R^{2} / I_{c}+1}
$$
where the symbols have the usual meanings.

Baskar P

Numerade Educator