Vector Mechanics for Engineers: Statics and Dynamics

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

Chapter 17

Plane Motion of Rigid Bodies: Energy and Momentum Methods - all with Video Answers

Educators

Section 1

Energy Methods for a Rigid Body

Problem 1

A 200-kg flywheel is at rest when a constant 300 N?m couple is applied. After executing 560 revolutions, the flywheel reaches its rated speed of 2400 rpm. Knowing that the radius of gyration of the flywheel is 400 mm, determine the average magnitude of the couple due to kinetic friction in the bearing.

Nathan Silvano

Nathan Silvano

Numerade Educator

Problem 2

The rotor of an electric motor has an angular velocity of 3600 rpm when the load and power are cut off. The 110-lb rotor, which has a centroidal radius of gyration of 9 in., then coasts to rest. Knowing that the kinetic friction of the rotor produces a couple with a magnitude of 2.5 lb?ft, determine the number of revolutions that the rotor executes before coming to rest.

Nathan Silvano

Numerade Educator

Problem 3

Two uniform disks of the same material are attached to a shaft as shown. Disk $A$ has a weight of $10 \mathrm{lb}$ and a radius of $r=6 \mathrm{in.}$ Disk $B$ is twice as thick as disk $A$. Knowing that a couple $\mathrm{M}$ with a magnitude of $22 \mathrm{lb} \cdot \mathrm{ft}$ is applied to disk $A$ when the system is at rest, determine the radius $n r$ of disk $B$ if the angular velocity of the system is to be 480 rpm after five revolutions.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 4

Two disks of the same material are attached to a shaft as shown. Disk A has a radius r and a thickness b, while disk B has a radius nr and a thickness 2b. A couple M with a constant magnitude is applied when the system is at rest and is removed after the system has executed two revolutions. Determine the value of n that results in the largest final speed for a point on the rim of disk B.

Nathan Silvano

Numerade Educator

Problem 5

The flywheel of a small punch rotates at 300 rpm. It is known that 1800 ft?lb of work must be done each time a hole is punched. It is desired that the speed of the flywheel after one punching be not less than 90 percent of the original speed of 300 rpm. (a) Determine the required moment of inertia of the flywheel. (b) If a constant 25-lb?ft couple is applied to the shaft of the flywheel, determine the number of revolutions that must occur between each punching, knowing that the initial velocity is to be 300 rpm at the start of each punching.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 6

The flywheel of a punching machine has a mass of 300 kg and a radius of gyration of 600 mm. Each punching operation requires 2500 J of work. (a) Knowing that the speed of the flywheel is 300 rpm just before a punching, determine the speed immediately after the punching. (b) If a constant 25-N?m couple is applied to the shaft of the flywheel, determine the number of revolutions executed before the speed is again 300 rpm.

Nathan Silvano

Numerade Educator

Problem 7

Disk $A,$ of weight 10 lb and radius $r=6 \mathrm{in.}$, is at rest when it is placed in contact with belt $B C,$ which moves to the right with a constant speed $v=40 \mathrm{ft} / \mathrm{s}$. Knowing that $\mu_{k}=0.20$ between the disk and the belt, determine the number of revolutions executed by the disk before it attains a constant angular velocity.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 8

The uniform $4-\mathrm{kg}$ cylinder $A$ with a radius of $r=150 \mathrm{mm}$ has an angular velocity of $\omega_{0}=50 \mathrm{rad} / \mathrm{s}$ when it is brought into contact with an identical cylinder $B$ that is at rest. The coefficient of kinetic friction at the contact point $D$ is $\mu_{k}$. After a period of slipping, the cylinders attain constant angular velocities of equal magnitude and opposite direction at the same time. Knowing that cylinder $A$ executes three revolutions before it attains a constant angular velocity and cylinder $B$ executes one revolution before it attains a constant angular velocity, determine (a) the final angular velocity of each cylinder, $(b)$ the coefficient of kinetic friction $\mu_{k} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 9

The 10 -in-radius brake drum is attached to a larger flywheel which is not shown. The total mass moment of inertia of the flywheel and drum is 16 lb. fit. $^{2}$ and the coefficient of kinetic friction between the drum and the brake shoe is $0.40 .$ Knowing that the initial angular velocity is 240 rpm clockwise, determine the force that must be exerted by the hydraulic cylinder if the system is to stop in 75 revolutions.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 10

Solve Prob. 9, assuming that the initial angular velocity of the flywheel is 240 rpm counterclockwise.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 11

Each of the gears $A$ and $B$ has a mass of $2.4 \mathrm{kg}$ and a radius of gyration of $60 \mathrm{mm}$, while gear $C$ has a mass of $12 \mathrm{kg}$ and a radius of gyration of $150 \mathrm{mm}$. A couple $\mathrm{M}$ of constant magnitude $10 \mathrm{N} \cdot \mathrm{m}$ is applied to gear $C .$ Determine $(a)$ the number of revolutions of gear $C$ required for its angular velocity to increase from 100 to $450 \mathrm{rpm},$ (b) the corresponding tangential force acting on gear $A .$

Nathan Silvano

Numerade Educator

Problem 12

Solve Prob. 11, assuming that the 10-N?m couple is applied to gear B.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 13

The gear train shown consists of four gears of the same thickness and of the same material; two gears are of radius $r,$ and the other two are of radius $n r$. The system is at rest when the couple $\mathbf{M}_{0}$ is applied to shaft $C .$ Denoting by $I_{0}$ the moment of inertia of a gear of radius $r,$ determine the angular velocity of shaft $A$ if the couple $\mathbf{M}_{0}$ is applied for one revolution of shaft $C .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 14

The double pullley shown has a mass of $15 \mathrm{kg}$ and a centroidal radius of gyration of $160 \mathrm{mm}$. Cylinder $A$ and block $B$ are attached to cords that are wrapped on the pulleys as shown. The coefficient of kinetic friction between block $B$ and the surface is $0.2 .$ Knowing that the system is at rest in the position shown when a constant force $P^{g}=200 \mathrm{N}$ is applied to cylinder $A$, determine $(a)$ the velocity of cylinder $A$ as it strikes the ground, $(b)$ the total distance that block $B$ moves before coming to rest.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 15

Gear $A$ has a mass of $1 \mathrm{kg}$ and a radius of gyration of $30 \mathrm{mm}$; gear $B$ has a mass of $4 \mathrm{kg}$ and a radius of gyration of $75 \mathrm{mm}$; gear $C$ has $\mathrm{a}$ mass of $9 \mathrm{kg}$ and a radius of gration of $100 \mathrm{mm}$. The system is at rest when a couple $\mathrm{M}_{0}$ of constant magnitude $4 \mathrm{N} \cdot \mathrm{m}$ is applied to gear $C .$ Assuming that no slipping occurs between the gears, determine the number of revolutions required for disk $A$ to reach an angular velocity of $300 \mathrm{rpm} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 16

A slender rod of length $l$ and weight $W$ is pivoted at one end as shown. It is released from rest in a horizontal position and swings freely. ( $a$ ) Determine the angular velocity of the rod as it passes through a vertical position and determine the corresponding reaction at the pivot. (b) Solve part $a$ for $W=1.8$ Ib and $l=3$ ft.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 17

A slender rod of length $l$ is pivoted about a point $C$ located at a distance $b$ from its center $G .$ It is released from rest in a horizontal position and swings freely. Determine $(a)$ the distance $b$ for which the angular velocity of the rod as it passes through a vertical position is maximum, $(b)$ the corresponding values of its angular velocity and of the reaction at $C .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 18

A slender $9-$lb rod can rotate in a vertical plane about a pivot at $B .$ A spring of constant $k=30 \mathrm{lb} / \mathrm{ft}$ and of unstretched length 6 in. is attached to the rod as shown. Knowing that the rod is released from rest in the position shown, determine its angular velocity after it has rotated through $90^{\circ} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 19

An adapted golf device attaches to a wheelchair to help people with mobility impairments play putt-putt. The stationary frame OD is attached to the wheelchair, and a club holder OB is attached to the pin at O. Holder OB is 6 in. long and weighs 8 oz, and the distance between $O$ and $D$ is $x=1$ ft. The putter shaft has a length of $f$ $L=36$ in. and weighs 10 oz, while the putter head at $A$ weighs 12 oz. Knowing that the 1 -lb/in. spring between $D$ and $B$ is unstretched when $\theta=90^{\circ}$ and that the putter is released from rest at $\theta=0$, determine the putter head speed when it hits the golf ball.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 20

A 160-lb gymnast is executing a series of full-circle swings on the horizontal bar. In the position shown, he has a small and negligible clockwise angular velocity and will maintain his body straight and rigid as he swings downward. Assuming that during the swing the centroidal radius of gyration of his body is 1.5 ft, determine his angular velocity and the force exerted on his hands after he has rotated through $(a) 90^{\circ},(b) 180^{\circ}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 21

A collar with a mass of $1 \mathrm{kg}$ is rigidly attached at a distance $d=300 \mathrm{mm}$ from the end of a uniform slender rod $A B .$ The rod has a mass of $3 \mathrm{kg}$ and is of length $L=600 \mathrm{mm}$. Knowing that the rod is released from rest in the position shown, determine the angular velocity of the rod after it has rotated through $90^{\circ} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 22

A collar with a mass of $1 \mathrm{kg}$ is rigidly attached to a slender rod $A B$ of mass $3 \mathrm{kg}$ and length $L=600 \mathrm{mm}$. The rod is released from rest in the position shown. Determine the distance $d$ for which the angular velocity of the rod is maximum after it has rotated through $90^{\circ} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 23

Two identical slender rods $A B$ and $B C$ are welded together to form an L-shaped assembly. The assembly is pressed against a spring at $D$ and released from the position shown. Knowing that the maximum angle of rotation of the assembly in its subsequent motion is $90^{\circ}$ counterclockwise, determine the magnitude of the angular velocity of the assembly as it passes through the position where rod $A B$ forms an angle of $30^{\circ}$ with the horizontal.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 24

The 30 -kg turbine disk has a centroidal radius of gyration of $175 \mathrm{mm}$ and is rotating clockwise at a constant rate of $60 \mathrm{rpm}$ when a small blade of weight $0.5 \mathrm{N}$ at point $A$ becomes loose and is thrown off. Neglecting friction, determine the change in the angular velocity of the turbine disk after it has rotated through ( $a$ ) $90^{\circ},(b) 270^{\circ}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 25

A rope is wrapped around a cylinder of radius r and mass m as shown. Knowing that the cylinder is released from rest, determine the velocity of the center of the cylinder after it has moved down-ward a distance s.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 26

Solve Prob. 25, assuming that the cylinder is replaced by a thin-walled pipe of radius r and mass m.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 27

Greek engineers had the unenviable task of moving large columns from the quarries to the city. One engineer, Chersiphron, tried several different techniques to do this. One method was to cut pivot holes into the ends of the stone and then use oxen to pull the column. The 4-ft diameter column weighs 12,000 lbs, and the team of oxen generates a
constant pull force of 1500 lbs on the center of the cylinder G. Knowing that the column starts from rest and rolls without slipping, determine (a) the velocity of its center G after it has moved 5 ft, (b) the minimum static coefficient of friction that will keep it from slipping.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 28

A small sphere of mass $m$ and radius $r$ is released from rest at $A$ and rolls without sliding on the curved surface to point $B$ where it leaves the surface with a horizontal velocity. Knowing that $a=1.5 \mathrm{m}$ and $b=1.2 \mathrm{m}$, determine $(a)$ the speed of the sphere as it strikes the ground at $C,(b)$ the corresponding distance $c .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 29

The mass center $G$ of a $3-\mathrm{kg}$ wheel of radius $R=180 \mathrm{mm}$ is located at a distance $r=60 \mathrm{mm}$ from its geometric center $C .$ The centroidal radius of gration of the wheel is $\bar{k}=90 \mathrm{mm}$. As the wheel rolls without sliding, its angular velocity is observed to vary. Knowing that $\omega=8 \mathrm{rad} / \mathrm{s}$ in the position shown, determine $(a)$ the angular velocity of the wheel when the mass center $G$ is directly above the geometric center $C,(b)$ the reaction at the horizontal surface at the same instant.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 30

A half-cylinder with mass $m$ and radius $r$ is released from rest in the position shown. Knowing that the half-cylinder rolls without sliding, determine ( $a$ ) its angular velocity after it has rolled through $90^{\circ},$ $(b)$ the reaction at the horizontal surface at the same instant. $[\text { Hint: Note that } G O=4 r / 3 \pi$ and that, by the parallel-axis theorem, $\left.\bar{I}=\frac{1}{2} m r^{2}-m(G O)^{2} .\right]$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 31

A sphere of mass m and radius r rolls without slipping inside a curved surface of radius R. Knowing that the sphere is released from rest in the position shown, derive an expression for (a) the linear velocity of the sphere as it passes through B, (b) the magnitude of the vertical reaction at that instant.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 32

Two uniform cylinders, each of weight $W=14$ lb and radius $r=5 \mathrm{in}$, are connected by a belt as shown. Knowing that at the instant shown the angular velocity of cylinder $B$ is 30 rad/s clockwise, determine (a) the distance through which cylinder $A$ will rise before the angular velocity of cylinder $B$ is reduced to 5 rad/s, (b) the tension in the portion of belt connecting the two cylinders.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 33

Two uniform cylinders, each of weight $W=14$ lb and radius $r=5$ in. are connected by a belt as shown. If the system is released from rest, determine (a) the velocity of the center of cylinder $A$ after it has moved through $3 \mathrm{ft},(b)$ the tension in the portion of belt connecting the two cylinders.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 34

A bar of mass $m=5 \mathrm{kg}$ is held as shown between four disks each of mass $m^{\prime}=2 \mathrm{kg}$ and radius $r=75 \mathrm{mm} .$ Knowing that the forces exerted on the disks are sufficient to prevent slipping and that the bar is released from rest, for each of the cases shown, determine the velocity of the bar after it has moved through the distance $h$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 35

The 1.5-kg uniform slender bar AB is connected to the 3-kg gear B that meshes with the stationary outer gear C. The centroidal radius of gyration of gear B is 30 mm. Knowing that the system is released from rest in the position shown, determine (a) the angular velocity of the bar as it passes through the vertical position, (b) the corresponding angular velocity of gear B.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 36

The motion of the uniform rod $A B$ is guided by small wheels of negligible mass that roll on the surface shown. If the rod is released from rest when $\theta=0,$ determine the velocities of $A$ and $B$ when $\theta=30^{\circ} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 37

A $5-m-$ -long ladder has a mass of $15 \mathrm{kg}$ and is placed against a house at an angle $\theta=20^{\circ} .$ Knowing that the ladder is released from rest, determine the angular velocity of the ladder and the velocity of end $A$ when $\theta=45^{\circ}$ Assume the ladder can slide freely on the horizontal ground and on the vertical wall.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 38

A long ladder of length $l,$ mass $m,$ and centroidal mass moment of inertia $\bar{I}$ is placed against a house at an angle $\theta=\theta_{0} .$ Knowing that the ladder is released from rest, determine the angular velocity of the ladder when $\theta=\theta_{2}$. Assume the ladder can slide freely on the horizontal ground and on the vertical wall.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 39

The ends of a $9-$ -lb rod $A B$ are constrained to move along slots cut in a vertical plate as shown. A spring of constant $k=3$ lb/in. is attached to end $A$ in such a way that its tension is zero when $\theta=0$. If the rod is released from rest when $\theta=50^{\circ}$, determine the angular velocity of the rod and the velocity of end $B$ when $\theta=0$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 40

The mechanism shown is one of two identical mechanisms attached to the two sides of a 200-lb uniform rectangular door. Edge ABC of the door is guided by wheels of negligible mass that roll in horizontal and vertical tracks. A spring with a constant of k 5 40 lb/ft is attached to wheel B. Knowing that the door is released from rest in the position $\theta=30^{\circ}$ with the spring unstretched, determine the velocity of wheel $A$ just as the door reaches the vertical position.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 41

The mechanism shown is one of two identical mechanisms attached to the two sides of a $200-$ buniform rectangular door. Edge $A B C$ of the door is guided by wheels of negligible mass that roll in horizontal and vertical tracks. A spring with a constant $k$ is attached to wheel $B$ in such a way that its tension is zero when $\theta=30^{\circ},$ Knowing that $B$ the door is released from rest in the posity of 0.6 and reaches the vertical position with an angular velocity of 0.6 rad/s, determine the spring constant $k .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 42

Each of the two rods shown is of length $L=1 \mathrm{m}$ and has a mass of $5 \mathrm{kg}$. Point $D$ is connected to a spring of constant $k=20 \mathrm{N} / \mathrm{m}$ and is constrained to move along a vertical slot. Knowing that the system is released from rest when rod $B D$ is horizontal and the spring connected to point $D$ is initially unstretched, determine the velocity of point $D$ when it is directly to the right of point $A$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 43

The 4-kg rod AB is attached to a collar of negligible mass at A and to a flywheel at B. The flywheel has a mass of 16 kg and a radius of gyration of 180 mm. Knowing that in the position shown the angular velocity of the flywheel is 60 rpm clockwise, determine the velocity of the flywheel when point B is directly below C.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 44

If in Prob. 43 the angular velocity of the flywheel is to be the same in the position shown and when point B is directly above C, determine the required value of its angular velocity in the position shown.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 45

The uniform rods $A B$ and $B C$ weigh $2.4 \mathrm{kg}$ and $4 \mathrm{kg}$, respectively, and the small wheel at $C$ is of negligible weight. If the wheel is moved slightly to the right and then released, determine the velocity of pin $B$ after rod $A B$ has rotated through $90^{\circ} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 46

The uniform rods $A B$ and $B C$ weigh 2.4 kg and 4 kg, respectively, and the small wheel at $C$ is of negligible weight. Knowing that in the position shown the velocity of wheel $C$ is $2 \mathrm{m} / \mathrm{s}$ to the right, determine the velocity of pin $B$ after rod $A B$ has rotated through $90^{\circ}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 47

The 80 -mm-radius gear shown has a mass of $5 \mathrm{kg}$ and a centroidal radius of gyration of $60 \mathrm{mm}$. The $4-\mathrm{kg}$ rod $A B$ is attached to the center of the gear and to a pin at $B$ that slides freely in a vertical slot. Knowing that the system is released from rest when $\theta=60^{\circ},$ determine the velocity of the center of the gear when $\theta=20^{\circ} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 48

Knowing that the maximum allowable couple that can be applied to a shaft is 15.5 kip?in., determine the maximum horsepower that can be transmitted by the shaft at (a) 180 rpm, (b) 480 rpm.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 49

Three shafts and four gears are used to form a gear train which will transmit 7.5 kW from the motor at A to a machine tool at F. (Bearings for the shafts are omitted from the sketch.) Knowing that the frequency of the motor is 30 Hz, determine the magnitude of the couple that is applied to shaft (a) AB, (b) CD, (c) EF.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 50

The shaft-disk-belt arrangement shown is used to transmit 2.4 kW from point A to point D. Knowing that the maximum allowable couples that can be applied to shafts AB and CD are 25 N?m and 80 N?m, respectively, determine the required minimum speed of shaft AB.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 51

The drive belt on a vintage sander transmits $1 / 2$ hp to a pulley that has a diameter of $d=4$ in. Knowing that the pulley rotates at 1450 rpm, determine the tension difference $T_{1}-T_{2}$ between the tight and slack sides of the belt.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator