Vector Mechanics for Engineers: Statics and Dynamics

Ferdinand P. Beer, E. Russell Johnston, Jr.,David F. Mazurek, Phillip J. Cornwell, Brian P. Self

Chapter 18

Kinetics of Rigid Bodies in Three Dimensions - all with Video Answers

Educators

Section 1

Energy and Momentum of a Rigid Body

Problem 1

A thin, homogeneous disk of mass $m$ and radius $r$ spins at the constant rate $\omega_{1}$ about an axle held by a fork-ended vertical rod that rotates at the constant rate $\omega_{2}$. Determine the angular momentum $\mathbf{H}_{G}$ of the disk about its mass center $G .$

Jonathan Ibarra

Jonathan Ibarra

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Problem 2

A thin rectangular plate of weight 15 lb rotates about its vertical
diagonal $A B$ with an angular velocity $\omega .$ Knowing that the $z$ axis is perpendicular to the plate and that $\omega$ is constant and equal to 5 rad/s, determine the angular momentum of the plate about its mass center $G$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 3

Two uniform rods $A B$ and $C E,$ each of weight 3 lb and length 2 ft, are welded to each other at their midpoints. Knowing that this assembly has an angular velocity of constant magnitude $\omega=12$ rad/s, determine the magnitude and direction of the angular momentum $\mathbf{H}_{D}$ of the assembly about $D .$

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 4

A homogeneous disk of weight $W=6$ lb rotates at the constant
rate $\omega_{1}=16$ rad/s with respect to arm $A B C,$ which is welded to a shaft $D C E$ rotating at the constant rate $\omega_{2}=8$ rad/s. Determine the angular momentum $\mathbf{H}_{A}$ of the disk about its center $A .$

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 5

A thin disk of mass $m=4$ kg rotates at the constant rate $\omega_{2}=15$ rad/s with respect to arm $A B C,$ which itself rotates at the constant rate $\omega_{1}=5$ rad/s about the $y$ axis. Determine the angular momentum of the disk about its center $C$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 6

A solid rectangular parallelepiped of mass $m$ has a square base of
side $a$ and a length $2 a$. Knowing that it rotates at the constant rate $\omega$ about its diagonal $A C^{\prime}$ and that its rotation is observed from $A$ as counterclockwise, determine (a) the magnitude of the angular momentum $\mathbf{H}_{G}$ of the parallelepiped about its mass center $G$, (b) the angle that $\mathbf{H}_{C}$ forms with the diagonal $A C^{\prime}$.

Jonathan Ibarra

Jonathan Ibarra

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Problem 7

Solve Prob. 18.6 , assuming that the solid rectangular parallelepiped
has been replaced by a hollow one consisting of six thin metal plates
welded together.

Jonathan Ibarra

Jonathan Ibarra

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Problem 8

A thin homogeneous disk with a mass $m$ and radius $r$ is mounted on the horizontal axle $A B$. The plane of the disk forms an angle of $\beta=20^{\circ}$ with the vertical. Knowing that the axle rotates with an angular velocity $\omega,$ determine the angle $\theta$ formed by the axle and the angular momentum of the disk about $G .$

Jonathan Ibarra

Jonathan Ibarra

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Problem 9

Determine the angular momentum $\mathbf{H}_{D}$ of the disk of Prob about point $D .$

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 10

Determine the angular momentum of the disk of Prob. 18.5 about
point $A .$

Jonathan Ibarra

Jonathan Ibarra

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Problem 11

Determine the angular momentum $\mathbf{H}_{o}$ of the disk of Sample Prob. 18.2 from the expressions obtained for its linear momentum $m \bar{v}$ and its angular momentum $\mathbf{H}_{G},$ using Eqs. (18.11). Verify that the result obtained is the same as that obtained by direct computation.

Jonathan Ibarra

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Problem 12

The $100-\mathrm{kg}$ projectile shown has a radius of gyration of $100 \mathrm{mm}$ about its axis of symmetry $G x$ and a radius of gyration of $250 \mathrm{mm}$ about the transverse axis $G y .$ Its angular velocity $\omega$ can be resolved into two components; one component, directed along $G x,$ measures the rate of spin of the projectile, while the other component, directed along $G D,$ measures its rate of precession. Knowing that $\theta=6^{\circ}$ and
that the angular momentum of the projectile about its mass center $G$ is $\mathbf{H}_{G}=\left(500 \mathrm{g} \cdot \mathrm{m}^{2} / \mathrm{s}\right) \mathrm{i}-\left(10 \mathrm{g} \cdot \mathrm{m}^{2} / \mathrm{s}\right) \mathrm{j},$ determine $(a)$ the rate of spin, $(b)$ the rate of precession.

Jonathan Ibarra

Jonathan Ibarra

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Problem 13

Determine the angular momentum $\mathrm{H}_{\mathrm{A}}$ of the projectile of Prob. 18.12 about the center $A$ of its base, knowing that its mass center $G$ has a velocity $\overline{\mathrm{v}}$ of $750 \mathrm{m} / \mathrm{s}$. Give your answer in terms of components respectively parallel to the $x$ and $y$ axes shown and to a third axis $z$ pointing toward you.

Jonathan Ibarra

Jonathan Ibarra

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Problem 14

(a) Show that the angular momentum $\mathbf{H}_{B}$ of a rigid body about point $B$ can be obtained by adding to the angular momentum $\mathbf{H}_{A}$ of that body about point $A$ the vector product of the vector $\mathbf{r}_{A / B}$ drawn from $B$ to $A$ and the linear momentum $m \overline{\mathbf{v}}$ of the body:
$$\mathbf{H}_{B}=\mathbf{H}_{A}+\mathbf{r}_{A / B} \times m \overline{\mathbf{v}}$$
(b) Further show that when a rigid body rotates about a fixed axis,
its angular momentum is the same about any two points $A$ and $B$ located on the fixed axis $\left(\mathrm{H}_{A}=\mathrm{H}_{B}\right)$ if, and only if, the mass center $G$ of the body is located on the fixed axis.

Jonathan Ibarra

Jonathan Ibarra

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Problem 15

Two L-shaped arms each have a mass of $5 \mathrm{kg}$ and are welded at the one-third points of the $600-\mathrm{mm}$ shaft $A B$ to form the assembly shown. Knowing that the assembly rotates at the constant rate of $360 \mathrm{rpm},$ determine $(a)$ the angular momentum $\mathrm{H}_{A}$ of the assembly about point $A,(b)$ the angle formed by $\mathrm{H}_{A}$ and $A B .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 16

For the assembly of Prob. 18.15 , determine $(a)$ the angular momentum $\mathbf{H}_{B}$ of the assembly about point $B,(b)$ the angle formed by $\mathbf{H}_{B}$ and $B A .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 17

A 10 -lb rod of uniform cross section is used to form the shaft shown. Knowing that the shaft rotates with a constant angular velocity $\omega$ of magnitude 12 rad/s, determine $(a)$ the angular momentum $\mathbf{H}_{G}$ of the shaft about its mass center $G,(b)$ the angle formed by $\mathbf{H}_{G}$ and
the axis $A B .$

Jonathan Ibarra

Jonathan Ibarra

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Problem 18

Determine the angular momentum of the shaft of Prob. 18.17 about
(a) point $A,(b)$ point $B$.

Jonathan Ibarra

Jonathan Ibarra

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Problem 19

Two triangular plates each have a mass of $8 \mathrm{kg}$ and are welded to a vertical shaft $A B .$ Knowing that the system rotates at the constant rate of $\omega=6 \mathrm{rad} / \mathrm{s}$, determine its angular momentum about $G .$

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 20

The assembly shown consists of two pieces of sheet aluminum with a
uniform thickness and total mass of $1.6 \mathrm{kg}$ welded to a light axle supported by bearings $A$ and $B$. Knowing that the assembly rotates with an angular velocity of constant magnitude $\omega=20 \mathrm{rad} / \mathrm{s}$, determine the angular momentum $\mathrm{H}_{G}$ of the assembly about point $G$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 21

One of the sculptures displayed on a university campus consists of a
hollow cube made of six aluminum sheets, each $1.5 \times 1.5 \mathrm{m}$, welded together and reinforced with internal braces of negligible weight. The cube is mounted on a fixed base at $A$ and can rotate freely about its vertical diagonal $A B .$ As she passes by this display on the way to a class in mechanics, an engineering student grabs corner $C$ of the cube and pushes it for $1.2 \mathrm{s}$ in a direction perpendicular to the plane $A B C$ with an average force of $50 \mathrm{N}$. Having observed that it takes $5 \mathrm{s}$ for the cube to complete one full revolution, she flips out her calculator and proceeds to determine the mass of the cube. What is the result of her calculation? (Hint: The perpendicular distance from the diagonal joining two vertices of a cube to any of its other six vertices can be obtained by multiplying
the side of the cube by $\sqrt{2 / 3}$.)

Jonathan Ibarra

Jonathan Ibarra

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Problem 22

If the aluminum cube of Prob. 18.21 were replaced by a cube of the
same size, made of six plywood sheets with mass $8 \mathrm{kg}$ each, how long would it take for that cube to complete one full revolution if the student pushed its corner $C$ in the same way that she pushed the corner of the aluminum cube?

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 23

A uniform rod of total mass $m$ is bent into the shape shown and is
suspended by a wire attached at $B$. The bent rod is hit at $D$ in a
direction perpendicular to the plane containing the rod (in the negative $z$ direction. Denoting the corresponding impulse by $\mathbf{F} \Delta t,$ determine ( $a$ ) the velocity of the mass center of the rod, $(b)$ the angular velocity of the rod.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 24

Solve Prob. 18.23 , assuming that the bent rod is hit at $C$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 25

Three slender rods, each of mass $m$ and length $2 a,$ are welded
together to form the assembly shown. The assembly is hit at $A$ in a
vertical downward direction. Denoting the corresponding impulse
by $\mathbf{F} \Delta t,$ determine immediately after the impact $(a)$ the velocity of the mass center $G,(b)$ the angular velocity of the rod.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 26

Solve Prob. $18.25,$ assuming that the assembly is hit at $B$ in the
negative $x$ direction.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 27

Two circular plates, each of mass $4 \mathrm{kg}$, are rigidly connected by a $\mathrm{rod} A B$ of negligible mass and are suspended from point $A$ as shown. Knowing that an impulse $\mathrm{F} \Delta t=-(2.4 \mathrm{N} \cdot \mathrm{s}) \mathrm{k}$ is applied at point $D,$ determine $(a)$ the velocity of the mass center $G$ of the assembly, $(b)$ the angular velocity of the assembly.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 28

Two circular plates, each of mass $4 \mathrm{kg}$, are rigidly connected by a $\mathrm{rod} A B$ of negligble mass and are suspended from point $A$ as shown. Knowing that an impulse $\mathrm{F} \Delta t=(2.4 \mathrm{N} \cdot \mathrm{s}) \mathrm{j}$ is applied at point $D,$ determine (a) the velocity of the mass center $G$ of the assembly, $(b)$ the angular velocity of the assembly.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 29

A circular plate of mass $m$ is falling with a velocity $\bar{v}_{0}$ and no angular velocity when its edge $C$ strikes an obstruction. A line passing the origin and parallel to the line $C G$ makes a $45^{\circ}$ angle with the $x$ -axis. Assuming the impact to be perfectly plastic $(e=0),$ determine the angular velocity of the plate immediately after the impact.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 30

For the plate of Prob. 18.29 , determine $(a)$ the velocity of its mass
center $G$ immediately after the impact, (b) the impulse exerted on
the plate by the obstruction during the impact.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 31

A square plate of side $a$ and mass $m$ supported by a ball-and-socket joint at $A$ is rotating about the $y$ axis with a constant angular velocity $\omega=\omega_{0} \mathrm{j}$ when an obstruction is suddenly introduced at $B$ in the $x y$ plane. Assuming the impact at $B$ to be perfectly plastic $(e=0),$ determine immediately after the impact ( a ) the angular velocity of the plate, (b) the velocity of its mass center $G .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 32

Determine the impulse exerted on the plate of Prob. 18.31 during the
impact by $(a)$ the obstruction at $B,(b)$ the support at $A .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 33

The coordinate axes shown represent the principal centroidal axes
of inertia of a 3000 -lb space probe whose radii of gyration are $k_{x}=1.375 \mathrm{ft}$, $k_{y}=1.425 \mathrm{ft}$, and $k_{z}=1.250 \mathrm{ft}$. The probe has no angular velocity when a $5-0 \mathrm{z}$ meteorite strikes one of its solar panels at point $A$ with a velocity $\mathrm{v}_{0}=(2400 \mathrm{ft} / \mathrm{s}) \mathrm{i}-(3000 \mathrm{ft} / \mathrm{s}) \mathrm{j}+$ $(3200 \mathrm{ft} / \mathrm{s}) \mathrm{k}$ relative to the probe. Knowing that the meteorite emerges on the other side of the panel with no change in the direction of its velocity, but with a speed reduced by 20 percent, determine the final angular velocity of the probe.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 34

The coordinate axes shown represent the principal centroidal axes
of inertia of a $3000-16$ space probe whooe radii of gyration are
$k_{x}=1.375 \mathrm{ft}, k_{y}=1.425 \mathrm{ft}$, and $k_{z}=1.250 \mathrm{ft}$. The probe has no angular velocity when a $5-0 \mathrm{z}$ meteorite strikes one of its solar panels at point $A$ and emerges on the other side of the panel with no change in the direction of its velocity, but with a speed reduced by 25 percent. Knowing that the final angular velocity of the probe
is $\omega=(0.05 \mathrm{rad} / \mathrm{s}) \mathrm{i}-(0.12 \mathrm{rad} / \mathrm{s}) \mathrm{j}+\omega_{\mathrm{e}} \mathrm{k}$ and that the $x$ -component of the resulting change in the velocity of the mass center of the probe is $-0.675 \mathrm{in} / \mathrm{s}$, determine $(a)$ the component $\omega_{z}$ of the final angular velocity of the probe, $(b)$ the relative velocity $\mathrm{v}_{0}$ with which the meteorite strikes the panel.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 35

A $1200-\mathrm{kg}$ satellite designed to study the sun has an angular velocity of $\omega_{0}=(0.050 \mathrm{rad} / \mathrm{s}) \mathrm{i}+(0.075 \mathrm{rad} / \mathrm{s}) \mathrm{k}$ when two small jets are activated at $A$ and $B$ in a direction parallel to the $y$ axis. Knowing that the coordinate axes are principal centroidal axes, that the radii of gyration of the satellite are $\vec{k}_{x}=1.120 \mathrm{m}, \vec{k}_{y}=1.200 \mathrm{m},$ and $\vec{k}_{z}=0.900 \mathrm{m},$ and that each jet produces a $50-\mathrm{N}$ thrust, determine (a) the required operating time of each jet if the angular velocity of the satellite is to be reduced to zero, (b) the resulting change in the velocity of the mass center $G .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 36

If jet $A$ in Prob. 18.35 is inoperative, determine $(a)$ the required
operating time of jet $B$ to reduce the $x$ -component of the angular
velocity of the satellite to zero, $(b)$ the resulting final angular velocity,
(c) the resulting change in the velocity of the mass center $G .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 37

Denoting, respectively, by $\omega, \mathbf{H}_{0},$ and $T$ the angular velocity, the angular momentum, and the kinetic energy of a rigid body with a a a a a a a fixed point $O,(a)$ prove that $\mathbf{H}_{o} \cdot \omega=2 T ;(b)$ show that the angle $\theta$
between $\omega$ and $\mathbf{H}_{O}$ will always be acute.

Jonathan Ibarra

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Problem 38

Show that the kinetic energy of a rigid body with a fixed point $O$ can be expressed as $T=\frac{1}{2} I_{O L} \omega^{2},$ where $\omega$ is the instantaneous angular velocity of the body and $I_{O L}$ is its moment of inertia about the line of action $O L$ of $\omega$. Derive this expression $(a)$ from Eqs. ( 9.46) (or Eq. B.19 in the Appendix and $(18.19),(b)$ by considering $T$ as the sum of the kinetic energies of particles $P_{i}$ describing circles of radius $\rho_{i}$ about line $O L$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 39

Determine the kinetic energy of the disk of Prob. 18.1

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 40

Determine the kinetic energy of the plate of Prob. $18.2 .$

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 41

Determine the kinetic energy of the assembly of Prob. 18.3

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 42

Determine the kinetic energy of the disk of Prob. 18.4

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 43

Determine the kinetic energy of the disk of Prob. 18.5

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 44

Determine the kinetic energy of the solid parallelepiped of Prob. $18.6 .$

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 45

Determine the kinetic energy of the hollow parallelepiped of Prob. $18.7 .$

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 46

Determine the kinetic energy of the disk of Prob. 18.8

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 47

Determine the kinetic energy of the assembly of Prob. 18.15

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 48

Determine the kinetic energy of the shaft of Prob. 18.17

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 49

Determine the kinetic energy of the assembly of Prob. 18.19

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Khoobchandra Agrawal

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Problem 50

Determine the kinetic energy imparted to the cube of Prob. 18.21

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 51

Determine the kinetic energy lost when edge $C$ of the plate of
Prob, $18,29$ hits the obstruction.

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 52

Determine the kinetic energy lost when the plate of Prob. 18.31 hits
the obstruction at $B .$

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 53

Determine the kinetic energy of the space probe of Prob. 18.33 in its
motion about its mass center after its collision with the meteorite.

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 54

Determine the kinetic energy of the space probe of Prob. 18.34 in its
motion about its mass center after its collision with the meteorite.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator