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University Physics with Modern Physics In SI Units

Hugh D Young; Roger A Freedman

Chapter 10

Dynamics of Rotational Motion - all with Video Answers

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Chapter Questions

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Problem 1

Calculate the torque (magnitude and direction) about point $O$ due to the force $\vec{F}$ in each of the cases sketched in Fig. E10.1. In each case, both the force $\overrightarrow{\boldsymbol{F}}$ and the rod lie in the plane of the page, the rod has length $4.00 \mathrm{~m}$, and the force has magnitude $F=15.0 \mathrm{~N}$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
04:46

Problem 2

Calculate the net torque about point $O$ for the two forces applied as in Fig. E10.2. The rod and both forces are in the plane of the page.

Mohit Khurana
Mohit Khurana
Texas A&M University
03:20

Problem 3

A square metal plate $0.180 \mathrm{~m}$ on each side is pivoted about an axis through point $O$ at its center and perpendicular to the plate (Fig. E10.3). Calculate the net torque about this axis due to the three forces shown in the figure if the magnitudes of the forces are $F_{1}=24.0 \mathrm{~N}, \quad F_{2}=15.8 \mathrm{~N},$
and $F_{3}=15.5 \mathrm{~N}$. The plate and all forces are in the plane of the page.

Penny Riley
Penny Riley
Numerade Educator
04:26

Problem 4

Three forces are applied to a wheel of radius $0.350 \mathrm{~m}$, as shown in Fig. E10.4. One force is perpendicular to the rim, one is tangent to it, and the other one makes a $40.0^{\circ}$ angle with the radius. What is the net torque on the wheel due to these three forces for an axis perpendicular to the wheel and passing through its center?

Mohit Khurana
Mohit Khurana
Texas A&M University
04:29

Problem 5

One force acting on a machine part is $\overrightarrow{\boldsymbol{F}}=(-5.00 \mathrm{~N}) \hat{\imath}+$
$(4.00 \mathrm{~N}) \hat{\jmath}$. The vector from the origin to the point where the force is applied is $\overrightarrow{\boldsymbol{r}}=(-0.450 \mathrm{~m}) \hat{\imath}+(0.150 \mathrm{~m}) \hat{\jmath}$. (a) In a sketch, show $\overrightarrow{\boldsymbol{r}}, \overrightarrow{\boldsymbol{F}},$
and the origin. (b) Use the right-hand rule to determine the direction of the torque. (c) Calculate the vector torque for an axis at the origin produced by this force. Verify that the direction of the torque is the same as you obtained in part ( $(\mathrm{b})$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:08

Problem 6

A metal bar is in the $x y$ -plane with one end of the bar at the origin. A force $\overrightarrow{\boldsymbol{F}}=(7.71 \mathrm{~N}) \hat{\imath}+(-3.08 \mathrm{~N}) \hat{\jmath}$ is applied to the bar at
the point $x=2.47 \mathrm{~m}, y=3.19 \mathrm{~m}$. (a) In terms of unit vectors $\hat{\imath}$ and $\hat{\jmath}$, what is the position vector $\vec{r}$ for the point where the force is applied?
(b) What are the magnitude and direction of the torque with respect to the origin produced by $\overrightarrow{\boldsymbol{F}}$ ?

Vishal Gupta
Vishal Gupta
Numerade Educator
01:51

Problem 7

A machinist is using a wrench to loosen a nut. The wrench is $25.0 \mathrm{~cm}$ long, and he exerts a $17.0 \mathrm{~N}$ force at the end of the handle at $37^{\circ}$ with the handle (Fig. E10.7). (a) What torque does the machinist exert about the center of the nut? (b) What is the maximum torque he could exert with a force of this magnitude, and how should the force be oriented?

Andrew C
Andrew C
Numerade Educator
08:18

Problem 8

A uniform disk with mass $40.0 \mathrm{~kg}$ and radius $0.200 \mathrm{~m}$ is pivoted at its center about a horizontal, frictionless axle that is stationary. The disk is initially at rest, and then a constant force $F=30.0 \mathrm{~N}$ is applied tangent to the rim of the disk. (a) What is the magnitude $v$ of the tangential velocity of a point on the rim of the disk after the disk has turned through 0.200 revolution? (b) What is the magnitude $a$ of the resultant acceleration of a point on the rim of the disk after the disk has turned through 0.200 revolution?

Mohit Khurana
Mohit Khurana
Texas A&M University
01:44

Problem 9

The flywheel of an engine has moment of inertia $2.00 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about its rotation axis. What constant torque is required to bring it up to an angular speed of 450 rev $/$ min in $7.90 \mathrm{~s}$, starting from rest?

Zhaojie Xu
Zhaojie Xu
Numerade Educator
15:17

Problem 10

A cord is wrapped around the rim of a solid uniform wheel $0.290 \mathrm{~m}$ in radius and of mass $7.80 \mathrm{~kg}$. A steady horizontal pull of $32.0 \mathrm{~N}$ to the right is exerted on the cord, pulling it off tangentially from the wheel. The wheel is mounted on frictionless bearings on a horizontal axle through its center. (a) Compute the angular acceleration of the wheel and the acceleration of the part of the cord that has already been pulled off the wheel. (b) Find the magnitude and direction of the force that the axle exerts on the wheel. (c) Which of the answers in parts (a) and (b) would change if the pull were upward instead of horizontal?

Andrija Isakov
Andrija Isakov
Numerade Educator
04:50

Problem 11

A machine part has the shape of a solid uniform sphere of mass $225 \mathrm{~g}$ and diameter $3.00 \mathrm{~cm} .$ It is spinning about a frictionless axle through its center, but at one point on its equator it is scraping against metal, resulting in a friction force of $0.0200 \mathrm{~N}$ at that point.
(a) Find its angular acceleration.
(b) How long will it take to decrease its rotational speed by $22.5 \mathrm{rad} / \mathrm{s} ?$

Abhishek Jana
Abhishek Jana
Numerade Educator
07:14

Problem 12

A stone is suspended from the free end of a wire that is wrapped around the outer rim of a pulley, similar to what is shown in Fig. $10.10 .$ The pulley is a uniform disk with mass $10.8 \mathrm{~kg}$ and radius $45.0 \mathrm{~cm}$ and turns on frictionless bearings. You measure that the stone travels $12.4 \mathrm{~m}$ in the first $2.50 \mathrm{~s}$ starting from rest. Find
(a) the mass of the stone and (b) the tension in the wire.

Stephanie Larson
Stephanie Larson
Numerade Educator
05:51

Problem 13

A $2.00 \mathrm{~kg}$ textbook rests on a frictionless, horizontal surface. A cord attached to the book passes over a pulley whose diameter is $0.150 \mathrm{~m}$, to a hanging book with mass $3.00 \mathrm{~kg} .$ The system is released from rest, and the books are observed to move $1.20 \mathrm{~m}$ in $0.800 \mathrm{~s}$.
(a) What is the tension in each part of the cord? (b) What is the moment of inertia of the pulley about its rotation axis?

Zachary Brauchler
Zachary Brauchler
Numerade Educator
09:58

Problem 14

A $15.0 \mathrm{~kg}$ bucket of water is suspended by a very light rope wrapped around a solid uniform cylinder $0.300 \mathrm{~m}$ in diameter with mass $12.0 \mathrm{~kg}$. The cylinder pivots on a frictionless axle through its center. The bucket is released from rest at the top of a well and falls $10.0 \mathrm{~m}$ to the water. (a) What is the tension in the rope while the bucket is falling? (b) With what speed does the bucket strike the water? (c) What is the time of fall? (d) While the bucket is falling, what is the force exerted on the cylinder by the axle?

Mohit Khurana
Mohit Khurana
Texas A&M University
03:49

Problem 15

A wheel rotates without friction about a stationary horizontal axis at the center of the wheel. A constant tangential force equal to $74.0 \mathrm{~N}$ is applied to the rim of the wheel. The wheel has radius $0.130 \mathrm{~m}$. Starting from rest, the wheel has an angular speed of $14.5 \mathrm{rev} / \mathrm{s}$ after $3.78 \mathrm{~s}$. What is the moment of inertia of the wheel?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:15

Problem 16

A $12.0 \mathrm{~kg}$ box resting on a horizontal, frictionless surface is attached to a $5.00 \mathrm{~kg}$ weight by a thin, light wire that passes over a frictionless pulley (Fig. E10.16). The pulley has the shape of a uniform solid disk of mass $2.10 \mathrm{~kg}$ and diameter $0.560 \mathrm{~m}$. After the system is released, find (a) the tension in the wire on both sides of the pulley, (b) the acceleration of the box, and (c) the horizontal and vertical components of the force that the axle exerts on the pulley.

Penny Riley
Penny Riley
Numerade Educator
04:17

Problem 17

A solid cylinder with radius $0.140 \mathrm{~m}$ is mounted on a frictionless, stationary axle that lies along the cylinder axis. The cylinder is initially at rest. Then starting at $t=0$ a constant horizontal force of $3.00 \mathrm{~N}$ is applied tangential to the surface of the cylinder. You measure the angular displacement $\theta-\theta_{0}$ of the cylinder as a function of the time $t$ since the force was first applied. When you plot $\theta-\theta_{0}$ (in radians) as a function of $t^{2}$ (in $\mathrm{s}^{2}$ ), your data lie close to a straight line. If the slope of this line is $16.0 \mathrm{rad} / \mathrm{s}^{2},$ what is the moment of inertia of the cylinder for rotation about the axle?

Andrew C
Andrew C
Numerade Educator
04:00

Problem 18

Two spheres are rolling without slipping on a horizontal floor. They are made of different materials, but each has mass $5.00 \mathrm{~kg}$ and radius $0.120 \mathrm{~m}$. For each the translational speed of the center of mass is $4.00 \mathrm{~m} / \mathrm{s}$. Sphere $A$ is a uniform solid sphere and sphere $B$ is a thin-walled, hollow sphere. How much work, in joules, must be done on each sphere to bring it to rest? For which sphere is a greater magnitude of work required? Explain. (The spheres continue to roll without slipping as they slow down.)

Andrew C
Andrew C
Numerade Educator
13:40

Problem 19

A $2.20 \mathrm{~kg}$ hoop $1.20 \mathrm{~m}$ in diameter is rolling to the right without slipping on a horizontal floor at a steady $2.60 \mathrm{rad} / \mathrm{s}$. (a) How fast is its center moving? (b) What is the total kinetic energy of the hoop? (c) Find the velocity vector of each of the following points, as viewed by a person at rest on the ground: (i) the highest point on the hoop; (ii) the lowest point on the hoop; (iii) a point on the right side of the hoop, midway between the top and the bottom. (d) Find the velocity vector for each of the points in part (c), but this time as viewed by someone moving along with the same velocity as the hoop.

Abhishek Jana
Abhishek Jana
Numerade Educator
13:40

Problem 20

A $2.20 \mathrm{~kg}$ hoop $1.20 \mathrm{~m}$ in diameter is rolling to the right without slipping on a horizontal floor at a steady $2.60 \mathrm{rad} / \mathrm{s}$.
(a) How fast is its center moving? (b) What is the total kinetic energy of the hoop? (c) Find the velocity vector of each of the following points, as viewed by a person at rest on the ground:
(i) the highest point on the hoop;
(ii) the lowest point on the hoop; (iii) a point on the right side of the hoop, midway between the top and the bottom.
(d) Find the velocity vector for each of the points in part (c), but this time as viewed by someone moving along with the same velocity as the hoop.

Abhishek Jana
Abhishek Jana
Numerade Educator
04:14

Problem 21

What fraction of the total kinetic energy is rotational for the following objects rolling without slipping on a horizontal surface? (a) A uniform solid cylinder; (b) a uniform sphere; (c) a thin-walled, hollow sphere;
(d) a hollow cylinder with outer radius $R$ and inner radius $R / 2$.

Abhishek Jana
Abhishek Jana
Numerade Educator
05:45

Problem 22

A string is wrapped several times around the rim of a small hoop with radius $8.00 \mathrm{~cm}$ and mass $0.180 \mathrm{~kg}$. The free end of the string is held in place and the hoop is released from rest (Fig. E10.22). After the hoop has descended $75.0 \mathrm{~cm},$ calculate (a) the angular speed of the rotating hoop and (b) the speed of its center.

Mohit Khurana
Mohit Khurana
Texas A&M University
02:18

Problem 23

A solid ball is released from rest and slides down a hillside that slopes downward at $70.0^{\circ}$ from the horizontal. (a) What minimum value must the coefficient of static friction between the hill and ball surfaces have for no slipping to occur?
(b) Would the coefficient of friction calculated in part
(a) be sufficient to prevent a hollow ball (such as a football) from slipping? Justify your answer. (c) In part (a), why did we use the coefficient of static friction and not the coefficient of kinetic friction?

Penny Riley
Penny Riley
Numerade Educator
08:13

Problem 24

A hollow, spherical shell with mass $2.00 \mathrm{~kg}$ rolls without slipping down a $38.0^{\circ}$ slope.
(a) Find the acceleration, the friction force, and the minimum coefficient of static friction needed to prevent slipping. (b) How would your answers to part (a) change if the mass were doubled to $4.00 \mathrm{~kg}$ ?

Andrew C
Andrew C
Numerade Educator
02:57

Problem 25

A $386 \mathrm{~N}$ wheel comes off a moving truck and rolls without slipping along a highway. At the bottom of a hill it is rotating at $27.1 \mathrm{rad} / \mathrm{s}$. The radius of the wheel is $0.632 \mathrm{~m}$, and its moment of inertia about its rotation axis is $0.800 \mathrm{MR}^{2}$. Friction does work on the wheel as it rolls up the hill to a stop, a height $h$ above the bottom of the hill; this work has absolute value $3506 \mathrm{~J}$. Calculate $h$.

Penny Riley
Penny Riley
Numerade Educator
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Problem 26

A uniform marble rolls down a symmetrical bowl, starting from rest at the top of the left side. The top of each side is a distance $h$ above the bottom of the bowl. The left half of the bowl is rough enough to cause the marble to roll without slipping, but the right half has no friction because it is coated with oil. (a) How far up the smooth side will the marble go, measured vertically from the bottom?
(b) How high would the marble go if both sides were as rough as the left side?
(c) How do you account for the fact that the marble goes higher with friction on the right side than without friction?

Mohit Khurana
Mohit Khurana
Texas A&M University
01:45

Problem 27

At a typical bowling alley the distance from the line where the ball is released (foul line) to the first pin is $18.29 \mathrm{~m}$. Estimate the time it takes the ball to reach the pins after you release it if it rolls without slipping and has a constant translational speed. Assume that the mass of the ball is $5.44 \mathrm{~kg}$ and its diameter $21.6 \mathrm{~cm}$. (a) Use your estimate to calculate the rotation rate of the ball, in rev/s. (b) What is its total kinetic energy in joules and what fraction of the total is its rotational kinetic energy? Ignore the finger holes and treat the bowling ball as a uniform sphere.

Penny Riley
Penny Riley
Numerade Educator
03:02

Problem 28

Two uniform solid balls are rolling without slipping at a constant speed. Ball 1 has twice the diameter, half the mass, and one-third the speed of ball 2 . The kinetic energy of ball 2 is $27.0 \mathrm{~J}$. What is the kinetic energy of ball 1 ?

Andrew C
Andrew C
Numerade Educator
02:17

Problem 29

A thin, light string is wrapped around the outer rim of a uniform hollow cylinder of mass $4.80 \mathrm{~kg}$ having inner and outer radii as shown in Fig. E10.29. The cylinder is then released from rest.
(a) How far must the cylinder fall before its center is moving at $6.44 \mathrm{~m} / \mathrm{s} ?(\mathrm{~b})$ If you just dropped this cylinder without any string, how fast would its center be moving when it had fallen the distance in part (a)? (c) Why do you get two different answers when the cylinder falls the same distance in both cases?

Penny Riley
Penny Riley
Numerade Educator
05:46

Problem 30

A Ball Rolling Uphill. A bowling ball rolls without slipping up a ramp that slopes upward at an angle $\beta$ to the horizontal (see Example 10.7 in Section 10.3 ). Treat the ball as a uniform solid sphere, ignoring the finger holes. (a) Draw the free-body diagram for the ball. Explain why the friction force must be directed uphill.
(b) What is the acceleration of the center of mass of the ball? (c) What minimum coefficient of static friction is needed to prevent slipping?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:02

Problem 31

A size -5 football of diameter $22.6 \mathrm{~cm}$ and mass $426 \mathrm{~g}$ rolls up a hill without slipping, reaching a maximum height of $5.00 \mathrm{~m}$ above the base of the hill. We can model this ball as a thin-walled hollow sphere. (a) At what rate was it rotating at the base of the hill? (b) How much rotational kinetic energy did it have then? Neglect rolling friction and assume the system's total mechanical energy is conserved.

Penny Riley
Penny Riley
Numerade Educator
03:04

Problem 32

An engine delivers $130 \mathrm{~kW}$ to an aircraft propeller at 2400 rev $/$ min. (a) How much torque does the aircraft engine provide?
(b) How much work does the engine do in one revolution of the propeller?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:28

Problem 33

A playground merry-go-round has radius $2.20 \mathrm{~m}$ and moment of inertia $2400 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about a vertical axle through its center, and it turns with negligible friction. (a) A child applies an $21.0 \mathrm{~N}$ force tangentially to the edge of the merry-go-round for $17.0 \mathrm{~s}$. If the merrygo-round is initially at rest, what is its angular speed after this $17.0 \mathrm{~s}$ interval? (b) How much work did the child do on the merry-go-round?
(c) What is the average power supplied by the child?

Penny Riley
Penny Riley
Numerade Educator
03:19

Problem 34

An electric motor consumes $10.8 \mathrm{~kJ}$ of electrical energy in $1.00 \mathrm{~min}$. If one-third of this energy goes into heat and other forms of internal energy of the motor, with the rest going to the motor output, how much torque will this engine develop if you run it at $2300 \mathrm{rpm} ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
10:23

Problem 35

A $2.00 \mathrm{~kg}$ grinding wheel is in the form of a solid cylinder of radius $0.100 \mathrm{~m}$. (a) What constant torque will bring it from rest to an angular speed of 1000 rev $/ \min$ in 3.0 s? (b) Through what angle has it turned during that time? (c) Use Eq. (10.21) to calculate the work done by the torque. (d) What is the grinding wheel's kinetic energy when it is rotating at 1000 rev/min? Compare your answer to the result in part (c).

Andrija Isakov
Andrija Isakov
Numerade Educator
02:48

Problem 36

A $2.00 \mathrm{~kg}$ grinding wheel is in the form of a solid cylinder of radius $0.100 \mathrm{~m}$. (a) What constant torque will bring it from rest to an angular speed of $1000 \mathrm{rev} / \mathrm{min}$ in $3.0 \mathrm{~s}$ ? (b) Through what angle has it turned during that time? (c) Use Eq. (10.21) to calculate the work done by the torque. (d) What is the grinding wheel's kinetic energy when it is rotating at 1000 rev/min? Compare your answer to the result in part (c).

Penny Riley
Penny Riley
Numerade Educator
01:38

Problem 37

A $4.00 \mathrm{~kg}$ rock has a horizontal velocity of magnitude $12.0 \mathrm{~m} / \mathrm{s}$ when it is at point $P$ in Fig. E10.37. (a) At this instant, what are the magnitude and direction of its angular momentum relative to point $O ?(b)$ If the only force acting on the rock is its weight, what is the rate of change (magnitude and direction) of its angular momentum at this instant?

Penny Riley
Penny Riley
Numerade Educator
03:06

Problem 38

A woman with mass $55 \mathrm{~kg}$ is standing on the rim of a large disk that is rotating at $0.47 \mathrm{rev} / \mathrm{s}$ about an axis through its center. The disk has mass $119 \mathrm{~kg}$ and radius $3.5 \mathrm{~m} .$ Calculate the magnitude of the total angular momentum of the woman-disk system. (Assume that you can treat the woman as a point.)

Vishal Gupta
Vishal Gupta
Numerade Educator
03:04

Problem 39

Find the magnitude of the angular momentum of the second hand on a clock about an axis through the center of the clock face. The clock hand has a length of $15.0 \mathrm{~cm}$ and a mass of $6.00 \mathrm{~g}$. Take the second hand to be a slender rod rotating with constant angular velocity about one end.

Abhishek Jana
Abhishek Jana
Numerade Educator
01:20

Problem 40

(a) Calculate the magnitude of the angular momentum of the earth in a circular orbit around the sun. Is it reasonable to model it as a particle? (b) Calculate the magnitude of the angular momentum of the earth due to its rotation around an axis through the north and south poles, modeling it as a uniform sphere. Consult Appendix $\mathrm{B}$ and the astronomical data at the back of the book.

Penny Riley
Penny Riley
Numerade Educator
02:17

Problem 41

A hollow, thin-walled sphere of mass $12.0 \mathrm{~kg}$ and diameter $49.0 \mathrm{~cm}$ is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by $\theta(t)=A t^{2}+B t^{4},$ where $A$ has numerical value 1.10 and $B$ has numerical value $1.60 .$ (a) What are the units of the constants $A$ and $B ?$ (b) At the time $4.00 \mathrm{~s}$, find (i) the angular momentum of the sphere and (ii) the net torque on the sphere.

Penny Riley
Penny Riley
Numerade Educator
03:13

Problem 42

A small block on a frictionless, horizontal surface has a mass of $2.40 \times 10^{-2} \mathrm{~kg} .$ It is attached to a massless cord passing through a hole in the surface (Fig. E10.42). The block is originally revolving at a distance of $0.300 \mathrm{~m}$ from the hole with an angular speed of $1.95 \mathrm{rad} / \mathrm{s}$. The cord is then pulled from below, shortening the radius of the circle in which the block revolves to $0.150 \mathrm{~m}$. Model the block as a particle.
(a) Is the angular momentum of the block conserved? Why or why not?
(b) What is the new angular speed?
(c) Find the change in kinetic energy of the block. (d) How much work was done in pulling the cord?

Penny Riley
Penny Riley
Numerade Educator
01:17

Problem 43

Under some circumstances, a star can collapse into an extremely dense object made mostly of neutrons and called a neutron star. The density of a neutron star is roughly $10^{14}$ times as great as that of ordinary solid matter. Suppose we represent the star as a uniform, solid, rigid sphere, both before and after the collapse. The star's initial radius was $9.0 \times 10^{5} \mathrm{~km}$ (comparable to our sun); its final radius is $16 \mathrm{~km}$. If the original star rotated once in 32 days, find the angular speed of the neutron star.

Penny Riley
Penny Riley
Numerade Educator
02:29

Problem 44

A diver comes off a board with arms straight up and legs straight down, giving her a moment of inertia about her rotation axis of $18 \mathrm{~kg} \cdot \mathrm{m}^{2}$. She then tucks into a small ball, decreasing this moment of inertia to $3.6 \mathrm{~kg} \cdot \mathrm{m}^{2}$. While tucked, she makes two complete revolutions in $1.0 \mathrm{~s}$. If she hadn't tucked at all, how many revolutions would she have made in the $1.5 \mathrm{~s}$ from board to water?

Averell Hause
Averell Hause
Carnegie Mellon University
01:56

Problem 45

The Spinning Figure Skater. The outstretched hands and arms of a figure skater preparing for a spin can be considered a slender rod pivoting about an axis through its center (Fig. E10.45). When the skater's hands and arms are brought in and wrapped around his body toexecute the spin, the hands and arms can be considered a thinwalled, hollow cylinder. His hands and arms have a combined mass of $7.0 \mathrm{~kg}$. When outstretched, they span $1.7 \mathrm{~m} ;$ when wrapped, they form a cylinder of radius $24 \mathrm{~cm}$. The moment of inertia about the rotation axis of the remainder of his body is constant and equal to $0.40 \mathrm{~kg} \cdot \mathrm{m}^{2}$. If his original angular speed is $0.30 \mathrm{rev} / \mathrm{s}$, what is his final angular speed?

Penny Riley
Penny Riley
Numerade Educator
01:42

Problem 46

A solid wood door $1.00 \mathrm{~m}$ wide and $2.00 \mathrm{~m}$ high is hinged along one side and has a total mass of $42.0 \mathrm{~kg}$. Initially open and at rest, the door is struck at its center by a handful of sticky mud with mass $0.500 \mathrm{~kg},$ traveling perpendicular to the door at $13.0 \mathrm{~m} / \mathrm{s}$ just before impact. Find the final angular speed of the door. Does the mud make a significant contribution to the moment of inertia?

Penny Riley
Penny Riley
Numerade Educator
02:20

Problem 47

A large wooden turntable in the shape of a flat uniform disk has a radius of $2.00 \mathrm{~m}$ and a total mass of $140 \mathrm{~kg}$. The turntable is initially rotating at $4.00 \mathrm{rad} / \mathrm{s}$ about a vertical axis through its center. Suddenly, a $80.0 \mathrm{~kg}$ parachutist makes a soft landing on the turntable at a point near the outer edge. (a) Find the angular speed of the turntable after the parachutist lands. (Assume that you can treat the parachutist as a particle.) (b) Compute the kinetic energy of the system before and after the parachutist lands. Why are these kinetic energies not equal?

Penny Riley
Penny Riley
Numerade Educator
03:21

Problem 48

A steroid Collision! Suppose that an asteroid traveling straight toward the center of the earth were to collide with our planet at the equator and bury itself just below the surface. What would have to be the mass of this asteroid, in terms of the earth's mass $M,$ for the day to become $22.0 \%$ longer than it presently is as a result of the collision? Assume that the asteroid is very small compared to the earth and that the earth is uniform throughout.

Penny Riley
Penny Riley
Numerade Educator
04:54

Problem 49

A small $10.0 \mathrm{~g}$ bug stands at one end of a thin uniform bar that is initially at rest on a smooth horizontal table. The other end of the bar pivots about a nail driven into the table and can rotate freely, without friction. The bar has mass $50.0 \mathrm{~g}$ and is $100 \mathrm{~cm}$ in length. The bug jumps off in the horizontal direction, perpendicular to the bar, with a speed of $20.0 \mathrm{~cm} / \mathrm{s}$ relative to the table. (a) What is the angular speed of the bar just after the frisky insect leaps? (b) What is the total kinetic energy of the system just after the bug leaps?
(c) Where does this energy come from?

Abhishek Jana
Abhishek Jana
Numerade Educator
04:54

Problem 50

A small $10.0 \mathrm{~g}$ bug stands at one end of a thin uniform bar that is initially at rest on a smooth horizontal table. The other end of the bar pivots about a nail driven into the table and can rotate freely, without friction. The bar has mass $50.0 \mathrm{~g}$ and is $100 \mathrm{~cm}$ in length. The bug jumps off in the horizontal direction, perpendicular to the bar, with a speed of $20.0 \mathrm{~cm} / \mathrm{s}$ relative to the table. (a) What is the angular speed of the bar just after the frisky insect leaps? (b) What is the total kinetic energy of the system just after the bug leaps?
(c) Where does this energy come from?

Abhishek Jana
Abhishek Jana
Numerade Educator
03:28

Problem 51

You live on a planet far from ours. Based on extensive communication with a physicist on earth, you have determined that all laws of physics on your planet are the same as ours and you have adopted the same units of seconds and meters as on earth. But you suspect that the value of $g,$ the acceleration of an object in free fall near the surface of your planet, is different from what it is on earth. To test this, you take a solid uniform cylinder and let it roll down an incline. The vertical height $h$ of the top of the incline above the lower end of the incline can be varied. You measure the speed $v_{\mathrm{cm}}$ of the center of mass of the cylinder when it reaches the bottom for various values of $h$. You plot $v_{\mathrm{cm}}^{2}$ (in $\mathrm{m}^{2} / \mathrm{s}^{2}$ ) versus $h$ (in $\mathrm{m}$ ) and find that your data lie close to a straight line with a slope of $6.42 \mathrm{~m} / \mathrm{s}^{2}$. What is the value of $g$ on your planet?

Andrew C
Andrew C
Numerade Educator
01:48

Problem 52

A uniform, $4.0 \mathrm{~kg},$ square, solid wooden gate $1.5 \mathrm{~m}$ on each side hangs vertically from a frictionless pivot at the center of its upper edge. A $1.1 \mathrm{~kg}$ raven flying horizontally at $4.5 \mathrm{~m} / \mathrm{s}$ flies into this door at its center and bounces back at $2.0 \mathrm{~m} / \mathrm{s}$ in the opposite direction. (a) What is the angular speed of the gate just after it is struck by the unfortunate raven? (b) During the collision, why is the angular momentum conserved but not the linear momentum?

Penny Riley
Penny Riley
Numerade Educator
04:12

Problem 53

A teenager is standing at the rim of a large horizontal uniform wooden disk that can rotate freely about a vertical axis at its center. The mass of the disk (in $\mathrm{kg}$ ) is $M$ and its radius (in $\mathrm{m}$ ) is $R$. The mass of the teenager (in $\mathrm{kg}$ ) is $m .$ The disk and teenager are initially at rest. The teenager then throws a large rock that has a mass (in kg) of $m_{\text {rock. }}$ As it leaves the thrower's hands, the rock is traveling horizontally with speed $v$ (in $\mathrm{m} / \mathrm{s}$ ) relative to the earth in a direction tangent to the rim of the disk. The teenager remains at rest relative to the disk and so rotates with it after throwing the rock. In terms of $M, R, m, m_{\text {rock }}$ and $v,$ what is the angular speed of the disk? Treat the teenager as a point mass.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:43

Problem 54

A uniform solid disk made of wood is horizontal and rotates freely about a vertical axle at its center. The disk has radius $0.600 \mathrm{~m}$ and mass $1.60 \mathrm{~kg}$ and is initially at rest. A bullet with mass $0.0200 \mathrm{~kg}$ is fired horizontally at the disk, strikes the rim of the disk at a point perpendicular to the radius of the disk, and becomes embedded in its rim, a distance of $0.600 \mathrm{~m}$ from the axle. After being struck by the bullet, the disk rotates at $4.00 \mathrm{rad} / \mathrm{s}$. What is the horizontal velocity of the bullet just before it strikes the disk?

Andrew C
Andrew C
Numerade Educator
02:17

Problem 55

The Hubble Space Telescope is stabilized to within an angle of about 2 -millionths of a degree by means of a series of gyroscopes that spin at $19,200 \mathrm{rpm}$. Although the structure of these gyroscopes is actually quite complex, we can model each of the gyroscopes as a thin-walled cylinder of mass $2.0 \mathrm{~kg}$ and diameter $5.0 \mathrm{~cm}$, spinning about its central axis. How large a torque would it take to cause these gyroscopes to precess through an angle of $1.20 \times 10^{-6}$ degree during a 5.50 hour exposure of a galaxy?

Penny Riley
Penny Riley
Numerade Educator
01:24

Problem 56

A Gyroscope on the Moon. A certain gyroscope precesses at a rate of $0.70 \mathrm{rad} / \mathrm{s}$ when used on earth. If it were taken to a lunar base, where the acceleration due to gravity is $0.165 g,$ what would be its precession rate?

Zhaojie Xu
Zhaojie Xu
Numerade Educator
07:05

Problem 57

You are riding your bicycle on a city street, and you are staying a constant distance behind a car that is traveling at the speed limit of $45 \mathrm{~km} / \mathrm{h}$. Estimate the diameters of the bicycle wheels and sprockets and use these estimated quantities to calculate the number of revolutions per minute made by the large sprocket to which the pedals are attached. Do a Web search if you aren't familiar with the parts of a bicycle.

Andrija Isakov
Andrija Isakov
Numerade Educator
02:22

Problem 58

You are riding your bicycle on a city street, and you are staying a constant distance behind a car that is traveling at the speed limit of $45 \mathrm{~km} / \mathrm{h}$. Estimate the diameters of the bicycle wheels and sprockets and use these estimated quantities to calculate the number of revolutions per minute made by the large sprocket to which the pedals are attached. Do a Web search if you aren't familiar with the parts of a bicycle.

Penny Riley
Penny Riley
Numerade Educator
04:20

Problem 59

A grindstone in the shape of a solid disk with diameter $0.520 \mathrm{~m}$ and a mass of $50.0 \mathrm{~kg}$ is rotating at $810 \mathrm{rev} / \mathrm{min}$. You press an ax against the rim with a normal force of $190 \mathrm{~N}$ (Fig. $\mathrm{P} 10.58$ ), and the grindstone comes to rest in $7.10 \mathrm{~s}$. Find the coefficient of friction between the ax and the grindstone. You can ignore friction in the bearings.

Nishant Kumar
Nishant Kumar
Numerade Educator
06:11

Problem 60

A rests on a horizontal tabletop. A light horizontal rope is attached to it and passes over a pulley, and block $B$ is suspended from the free end of the rope. The light rope that connects the two blocks does not slip over the surface of the pulley (radius $0.080 \mathrm{~m}$ ) because the pulley rotates on a frictionless axle. The horizontal surface on which block $A$ (mass $2.50 \mathrm{~kg}$ ) moves is frictionless. The system is released from rest, and block $B$ (mass $6.00 \mathrm{~kg}$ ) moves downward $1.80 \mathrm{~m}$ in $2.00 \mathrm{~s}$. (a) What is the tension force that the rope exerts on block $B$ ?
(b) What is the tension force on block $A ?$ (c) What is the moment of inertia of the pulley for rotation about the axle on which it is mounted?

Andrew C
Andrew C
Numerade Educator
07:51

Problem 61

A thin, uniform, 3.80 $\mathrm{kg}$ bar, $80.0 \mathrm{~cm}$ long, has very small $2.50 \mathrm{~kg}$ balls glued on at either end (Fig. P10.61). It is supported horizontally by a thin, horizontal, frictionless axle passing through its center and perpendicular to the bar. Suddenly the right-hand ball becomes detached and falls off, but the other ball remains glued to the bar. (a) Find the angular acceleration of the bar just after the ball falls off. (b) Will the angular acceleration remain constant as the bar continues to swing? If not, will it increase or decrease?
(c) Find the angular velocity of the bar just as it swings through its vertical position.

Abhishek Jana
Abhishek Jana
Numerade Educator
01:24

Problem 62

Example 10.7 discusses a uniform solid sphere rolling without slipping down a ramp that is at an angle $\beta$ above the horizontal. Now consider the same sphere rolling without slipping up the ramp. (a) In terms of $g$ and $\beta,$ calculate the acceleration of the center of mass of the sphere. Is your result larger or smaller than the acceleration when the sphere rolls down the ramp, or is it the same?
(b) Calculate the friction force (in terms of $M, g,$ and $\beta$ ) for the sphere to roll without slipping as it moves up the incline. Is the result larger, smaller, or the same as the friction force required to prevent slipping as the sphere rolls down the incline?

Penny Riley
Penny Riley
Numerade Educator
01:38

Problem 63

The Atwood's Machine. Figure $\mathbf{P} 10.63$ illustrates an Atwood's machine. Find the lincar accelerations of blocks $A$ and $B,$ the angular acceleration of the wheel $C$, and the tension in each side of the cord if there is no slipping between the cord and the surface of the wheel. Let the masses of blocks $A$ and $B$ be $3.50 \mathrm{~kg}$ and $2.00 \mathrm{~kg},$ respectively, the moment of inertia of the wheel about its axis be $0.400 \mathrm{~kg} \cdot \mathrm{m}^{2},$ and the radius of the wheel be $0.100 \mathrm{~m}$.

Penny Riley
Penny Riley
Numerade Educator
03:08

Problem 64

The mechanism shown in Fig. $\mathbf{P} 10.64$ is used to raise a crate of supplies from a ship's hold. The crate has total mass $50 \mathrm{~kg} .$ A rope is wrapped around a wooden cylinder that turns on a metal axle. The cylinder has radius $0.25 \mathrm{~m}$ and moment of inertia $I=2.9 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about the axle. The
crate is suspended from the free end of the rope. One end of the axle pivots on frictionless bearings; a crank handle is attached to the other end. When the crank is turned, the end of the handle rotates about the axle in a vertical circle of radius $0.12 \mathrm{~m},$ the cylinder turns, and the crate is raised. What magnitude of the force $\vec{F}$ applied tangentially to the rotating crank is required to raise the crate with an acceleration of $1.40 \mathrm{~m} / \mathrm{s}^{2} ?$ (You can ignore the mass of the rope as well as the moments of inertia of the axle and the crank.)

Eric Mockensturm
Eric Mockensturm
Numerade Educator
11:55

Problem 65

A solid uniform sphere and a thin-walled, hollow sphere have the same mass $M$ and radius $R .$ If they roll without slipping up a ramp that is inclined at an angle $\beta$ above the horizontal and if both have the same $v_{\mathrm{cm}}$ before they start up the incline, calculate the maximum height above their starting point reached by each object. Which object reaches the greater height, or do both of them reach the same height?

Andrew C
Andrew C
Numerade Educator
01:45

Problem 66

A block with mass $m=5.00 \mathrm{~kg}$ slides down a surface inclined $36.9^{\circ}$ to the horizontal (Fig. P10.66). The coefficient of kinetic friction is 0.26 . A string attached to the block is wrapped around a flywheel on a fixed axis at $O$. The flywheel has mass $6.25 \mathrm{~kg}$ and moment of inertia $0.500 \mathrm{~kg} \cdot \mathrm{m}^{2}$ with respect to the axis of rotation. The string pulls without slipping at a perpendicular distance of $0.400 \mathrm{~m}$ from that axis. (a) What is the acceleration of the block down the plane?
(b) What is the tension in the string?

Dominador Tan
Dominador Tan
Numerade Educator
04:41

Problem 67

A wheel with radius $0.0600 \mathrm{~m}$ rotates about a horizontal frictionless axle at its center. The moment of inertia of the wheel about the axle is $2.50 \mathrm{~kg} \cdot \mathrm{m}^{2}$. The wheel is initially at rest. Then at $t=0 \mathrm{a}$ force $F=(5.00 \mathrm{~N} / \mathrm{s}) t$ is applied tangentially to the wheel and the wheel starts to rotate. What is the magnitude of the force at the instant when the wheel has turned through 8.00 revolutions?

Andrew C
Andrew C
Numerade Educator
04:09

Problem 68

A lawn roller in the form of a thin-walled, hollow cylinder with mass $M$ is pulled horizontally with a constant horizontal force $F$ applied by a handle attached to the axle. If it rolls without slipping, find the acceleration and the friction force.

Mohit Khurana
Mohit Khurana
Texas A&M University
01:57

Problem 69

Two weights are connected by a very light, flexible cord that passes over an $70.0 \mathrm{~N}$ frictionless pulley of radius $0.400 \mathrm{~m}$. The pulley is a solid uniform disk and is supported by a hook connected to the ceiling (Fig. $\mathbf{P} \mathbf{1 0 . 6 9}$ ). What force does the ceiling exert on the hook?

Penny Riley
Penny Riley
Numerade Educator
01:01

Problem 70

A large uniform horizontal turntable rotates freely about a vertical axle at its center. You measure the radius of the turntable to be $3.00 \mathrm{~m}$. To determine the moment of inertia $I$ of the turn table about the axle, you start the turntable rotating with angular speed $\omega,$ which you measure. You then drop a small object of mass $m$ onto the rim of the turntable. After the object has come to rest relative to the turntahle, you measure the angular speed $\omega_{\mathrm{f}}$ of the rotating turntahle. You plot the quantity $\left(\omega-\omega_{\mathrm{f}}\right) / \omega_{\mathrm{f}}$ (with both $\omega$ and $\omega_{\mathrm{f}}$ in $\mathrm{rad} / \mathrm{s}$ ) as a function of $m$ (in $\mathrm{kg}$ ). You find that your data lie close to a straight line that has slope $0.250 \mathrm{~kg}^{-1}$. What is the moment of inertia $I$ of the turntable?

Dominador Tan
Dominador Tan
Numerade Educator
08:42

Problem 71

The Yo-yo. A yo-yo is made from two uniform disks, each with mass $m$ and radius $R$, connected by a light axle of radius $b$. A light, thin string is wound several times around the axle and then held stationary while the yo-yo is released from rest, dropping as the string unwinds. Find the linear acceleration and angular acceleration of the yo-yo and the tension in the string.

Abhishek Jana
Abhishek Jana
Numerade Educator
01:36

Problem 72

A thin-walled, hollow spherical shell of mass $m$ and radius $r$ starts from rest and rolls without slipping down a track (Fig. P10.72). Points $A$ and $B$ are on a circular part of the track having radius $R$. The diameter of the shell is very small compared to $h_{0}$ and $R$, and the work done by rolling friction is negligible. (a) What is the minimum height $h_{0}$ for which this shell will make a complete loop-the-loop on the circular part of the track? (b) How hard does the track push on the shell at point $B,$ which is at the same level as the center of the circle? (c) Suppose that the track had no friction and the shell was released from the same height $h_{0}$ you found in part (a). Would it make a complete loop-theloop? How do you know? (d) In part (c), how hard does the track push on the shell at point $A,$ the top of the circle? How hard did it push on the shell in part (a)?

Dominador Tan
Dominador Tan
Numerade Educator
03:49

Problem 73

A basketball (which can be closely modeled as a hollow spherical shell) rolls down a mountainside into a valley and then up the opposite side, starting from rest at a height $H_{0}$ above the bottom. In Fig. $\mathbf{P} 10.73,$ the rough part of the terrain prevents slipping while the smooth part has no friction. Neglect rolling friction and assume the system's total mechanical energy is conserved. (a) How high, in terms of $H_{0}$, will the ball go up the other side? (b) Why doesn't the ball return to height $H_{0} ?$ Has it lost any of its original potential energy?

Andrew C
Andrew C
Numerade Educator
01:54

Problem 74

A solid uniform ball rolls without slipping up a hill (Fig. $\mathbf{P} \mathbf{1 0} . \mathbf{7} \mathbf{4}$ ). At the top of the hill, it is moving horizontally, and then it goes over the vertical cliff. Neglect rolling friction and assume the system's total mechanical energy is conserved. (a) How far from the foot of the cliff does the ball land, and how fast is it moving just before it lands? (b) Notice that when the balls lands, it has a greater translational speed than when it was at the bottom of the hill. Does this mean that the ball somehow gained energy? Explain!

Dominador Tan
Dominador Tan
Numerade Educator
01:33

Problem 75

A solid, uniform, spherical boulder starts from rest and rolls down a 50.0 -m-high hill, as shown in Fig. $\mathbf{P 1 0 . 7 5}$. The top half of the hill is rough enough to cause the boulder to roll without slipping, but the lower half is covered with ice and there is no friction. What is the translational speed of the boulder when it reaches the bottom of the hill? Neglect rolling friction and assume the system's total mechanical energy is conserved.

Dominador Tan
Dominador Tan
Numerade Educator
09:04

Problem 76

You are designing a system for moving aluminum cylinders from the ground to a loading dock. You use a sturdy wooden ramp that is $6.00 \mathrm{~m}$ long and inclined at $37.0^{\circ}$ above the horizontal. Each cylinder is fitted with a light, frictionless yoke through its center, and a light (but strong) rope is attached to the yoke. Each cylinder is uniform and has mass 460 $\mathrm{kg}$ and radius $0.300 \mathrm{~m}$. The cylinders are pulled up the ramp by applying a constant force $\overrightarrow{\boldsymbol{F}}$ to the free end of the rope. $\overrightarrow{\boldsymbol{F}}$ is parallel to the surface of the ramp and exerts no torque on the cylinder. The coefficient of static friction between the ramp surface and the cylinder is $0.120 .$ (a) What is the largest magnitude $\overrightarrow{\boldsymbol{F}}$ can have so that the cylinder still rolls without slipping as it moves up the ramp? (b) If the cylinder starts from rest at the bottom of the ramp and rolls without slipping as it moves up the ramp, what is the shortest time it can take the cylinder to reach the top of the ramp?

Mohit Khurana
Mohit Khurana
Texas A&M University
01:51

Problem 77

A $43.0-\mathrm{cm}$ -diameter wheel, consisting of a rim and six spokes, is constructed from a thin, rigid plastic material having a linear mass density of $25.0 \mathrm{~g} / \mathrm{cm}$. This wheel is released from rest at the top of a hill $51.0 \mathrm{~m}$ high.
(a) How fast is it rolling when it reaches the bottom of the hill? (b) How would your answer change if the linear mass density and the diameter of the wheel were each doubled?

Penny Riley
Penny Riley
Numerade Educator
02:11

Problem 78

A 43.0 -cm-diameter wheel, consisting of a rim and six spokes, is constructed from a thin, rigid plastic material having a linear mass density of $25.0 \mathrm{~g} / \mathrm{cm}$. This wheel is released from rest at the top of a hill $51.0 \mathrm{~m}$ high. (a) How fast is it rolling when it reaches the bottom of the hill? (b) How would your answer change if the linear mass density and the diameter of the wheel were each doubled?

Penny Riley
Penny Riley
Numerade Educator
04:04

Problem 79

A uniform solid cylinder with mass $M$ and radius $2 R$ rests on a horizontal tabletop. A string is attached by a yoke to a frictionless axle through the center of the cylinder so that the cylinder can rotate about the axle. The string runs over a disk-shaped pulley with mass $M$ and radius $R$ that is mounted on a frictionless axle through its center. A block of mass $M$ is suspended from the free end of the string (Fig. $\mathbf{P 1 0 . 7 9}$ ). The string doesn't slip over the pulley surface, and the cylinder rolls without slipping on the tabletop. Find the magnitude of the acceleration of the block after the system is released from rest.

Andrew C
Andrew C
Numerade Educator
06:48

Problem 80

A $5.00 \mathrm{~kg}$ ball is dropped from a height of $12.0 \mathrm{~m}$ above one end of a uniform bar that pivots at its center. The bar has mass 8.00 $\mathrm{kg}$ and is $4.00 \mathrm{~m}$ in length. At the other end of the bar sits another 5.00 $\mathrm{kg}$ ball, unattached to the bar. The dropped ball sticks to the bar after the collision. How high will the other ball go after the collision?

Abhishek Jana
Abhishek Jana
Numerade Educator
04:09

Problem 81

A uniform rod of length $L$ rests on a frictionless horizontal surface. The rod pivots about a fixed frictionless axis at one end. The rod is initially at rest. A bullet traveling parallel to the horizontal surface and perpendicular to the rod with speed $v$ strikes the rod at its center and becomes embedded in it. The mass of the bullet is one-fourth the
(b) What mass of the rod. (a) What is the final angular speed of the rod? is the ratio of the kinetic energy of the system after the collision to the kinetic energy of the bullet before the collision?

Abhishek Jana
Abhishek Jana
Numerade Educator
04:09

Problem 82

A uniform rod of length $L$ rests on a frictionless horizontal surface. The rod pivots about a fixed frictionless axis at one end. The rod is initially at rest. A bullet traveling parallel to the horizontal surface and perpendicular to the rod with speed $v$ strikes the rod at its center and becomes embedded in it. The mass of the bullet is one-fourth the mass of the rod. (a) What is the final angular speed of the rod? (b) What is the ratio of the kinetic energy of the system after the collision to the kinetic energy of the bullet before the collision?

Abhishek Jana
Abhishek Jana
Numerade Educator
04:46

Problem 83

In your job as a mechanical engineer you are designing a flywheel and clutch-plate system like the one in Example $10.11 .$ Disk $A$ is made of a lighter material than disk $B$, and the moment of inertia of disk $A$ about the shaft is one-third that of disk $B$. The moment of inertia of the shaft is negligible. With the clutch disconnected, $A$ is brought up to an angular speed $\omega_{0} ; B$ is initially at rest. The accelerating torque is then removed from $A,$ and $A$ is coupled to $B$. (Ignore bearing friction.) The design specifications allow for a maximum of $2400 \mathrm{~J}$ of thermal energy to be developed when the connection is made. What can be the maximum value of the original kinetic energy of disk $A$ so as not to exceed the maximum allowed value of the thermal energy?

Abhishek Jana
Abhishek Jana
Numerade Educator
05:27

Problem 84

A local ice hockey team has asked you to design an apparatus for measuring the speed of the hockey puck after a slap shot. Your design is a 2.00 -m-long, uniform rod pivoted about one end so that it is free to rotate horizontally on the ice without friction. The $0.800 \mathrm{~kg}$ rod has a light basket at the other end to catch the $0.163 \mathrm{~kg}$ puck. The puck slides across the ice with velocity $\overrightarrow{\boldsymbol{v}}$ (perpendicular to the rod), hits the basket, and is caught. After the collision, the rod rotates. If the rod makes one revolution every $0.736 \mathrm{~s}$ after the puck is caught, what was the puck's speed just before it hit the rod?

Mohit Khurana
Mohit Khurana
Texas A&M University
05:52

Problem 85

A $500.0 \mathrm{~g}$ bird is flying horizontally at $2.25 \mathrm{~m} / \mathrm{s}$, not paying much attention, when it suddenly flies into a stationary vertical bar, hitting it $25.0 \mathrm{~cm}$ below the top (Fig. P10.85). The bar is uniform, $0.750 \mathrm{~m}$ long, has a mass of $1.50 \mathrm{~kg}$, and is hinged at its base. The collision stuns the bird so that it just drops to the ground afterward (but soon recovers to fly happily away).What is the angular velocity of the bar (a) just after it is hit by the bird and (b) just as it reaches the ground?

Abhishek Jana
Abhishek Jana
Numerade Educator
04:26

Problem 86

A small block with mass $0.200 \mathrm{~kg}$ is attached to a string passing through a hole in a frictionless, horizontal surface(see Fig. E10.42). The block is originally revolving in a circle with a radius of $0.790 \mathrm{~m}$ about the hole with a tangential speed of $3.70 \mathrm{~m} / \mathrm{s}$. The string is then pulled slowly from below, shortening the radius of the circle in which the block revolves. The breaking strength of the string is $31.0 \mathrm{~N}$. What is the radius of the circle when the string breaks?

Vishal Gupta
Vishal Gupta
Numerade Educator
01:17

Problem 87

A $50 \mathrm{~kg}$ runner runs around the edge of a horizontal turntable mounted on a vertical, frictionless axis through its center. The runner's velocity relative to the earth has magnitude $3.0 \mathrm{~m} / \mathrm{s}$. The turntable is rotating in the opposite direction with an angular velocity of magnitude $0.22 \mathrm{rad} / \mathrm{s}$ relative to the earth. The radius of the turntable is $2.6 \mathrm{~m}$, and its moment of inertia about the axis of rotation is $84 \mathrm{~kg} \cdot \mathrm{m}^{2}$. Find the final angular velocity of the system if the runner comes to rest relative to the turntable. (You can model the runner as a particle.)

Penny Riley
Penny Riley
Numerade Educator
05:40

Problem 88

The 2017 Aston Martin DB11 can be equipped either with a $\mathrm{V} 12$ engine or with a V8 engine. The $\mathrm{V} 12$ engine is reported to produce a maximum power of $447 \mathrm{~kW}$ at $6500 \mathrm{rpm}$ and a maximum torque of $700 \mathrm{~N} \cdot \mathrm{m}$ from $1500 \mathrm{rpm}$. (a) Calculate the torque, at $6500 \mathrm{rpm}$. Is your answer smaller than the specified maximum value? (b) Calculate the power at $4500 \mathrm{rpm}$. Is your answer smaller than the specified maximum value? (c) The V8 engine is reported to produce $375 \mathrm{kw}$ at $6000 \mathrm{rpm}$. What is the torque at $6000 \mathrm{rpm} ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
14:47

Problem 89

You have one object of each of these shapes, all with mass $0.840 \mathrm{~kg}$ : a uniform solid cylinder, a thin-walled hollow cylinder, a uniform solid sphere, and a thin-walled hollow sphere. You release each object from rest at the same vertical height $h$ above the bottom of a long wooden ramp that is inclined at $35.0^{\circ}$ from the horizontal. Each object rolls without slipping down the ramp. You measure the time $t$ that it takes each one to reach the bottom of the ramp; Fig. $\mathrm{P} 10.89$ shows the results.
(a) From the bar graphs, identify objects $A$ through $D$ by shape. (b) Which of objects $A$ through $D$ has the greatest total kinetic energy at the bottom of the ramp, or do all have the same kinetic energy? (c) Which of objects $A$ through $D$ has the greatest rotational kinetic energy $\frac{1}{2} I \omega^{2}$ at the bottom of the ramp, or do all have the same rotational kinetic energy? (d) What minimum coefficient of static friction is required for all four objects to roll without slipping?

Abhishek Jana
Abhishek Jana
Numerade Educator
06:10

Problem 90

You are testing a small flywheel (radius $0.166 \mathrm{~m}$ ) that will be used to store a small amount of energy. The flywheel is pivoted with low-friction bearings about a horizontal shaft through the flywheel's center. A thin, light cord is wrapped multiple times around the rim of the flywheel. Your lab has a device that can apply a specified horizontal force $\overrightarrow{\boldsymbol{F}}$ to the free end of the cord. The device records both the magnitude of that force as a function of the horizontal distance the end of the cord has traveled and the time elapsed since the force was first applied. The flywheel is initially at rest. (a) You start with a test run to determine the flywheel's moment of inertia $I$. The magnitude $F$ of the force is a constant $25.0 \mathrm{~N},$ and the end of the rope moves $8.35 \mathrm{~m}$ in $2.00 \mathrm{~s}$. What is $I$ ? (b) In a second test, the flywheel again starts from rest but the free end of the rope travels $6.00 \mathrm{~m} ;$ Fig. $\mathrm{P} 10.90$ shows the force magnitude $F$ as a function of the distance $d$ that the end of the rope has moved. What is the kinetic energy of the flywheel when $d=6.00 \mathrm{~m} ?$ (c) What is the angular speed of the flywheel, in rev/min, when $d=6.00 \mathrm{~m}$ ?

Keshav Singh
Keshav Singh
Numerade Educator
05:42

Problem 91

A block with mass $m$ is revolving with linear speed $v_{1}$ in a circle of radius $r_{1}$ on a frictionless horizontal surface (see Fig. E10.42). The string is slowly pulled from below until the radius of the circle in which the block is revolving is reduced to $r_{2}$.
(a) Calculate the tension $T$ in the string as a function of $r$, the distance of the block from the hole. Your answer will be in terms of the initial velocity $v_{1}$ and the radius $r_{1}$. (b) Use $W=\int_{r_{1}}^{r_{2}} \overrightarrow{\boldsymbol{T}}(r) \cdot d \overrightarrow{\boldsymbol{r}}$ to calculate the work done by $\overrightarrow{\boldsymbol{T}}$ when $r$ changes from $r_{1}$ to $r_{2}$. (c) Compare the results of part (b) to the change in the kinetic energy of the block.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:11

Problem 92

When an object is rolling without slipping, the rolling friction force is much less than the friction force when the object is sliding; a coin will roll on its edge much farther than it will slide on its flat side (see Section 5.3 ). When an object is rolling without slipping on a horizontal surface, we can approximate the friction force to be zero, so that $a_{x}$ and $\alpha_{z}$ are approximately zero and $v_{x}$ and $\omega_{z}$ are approximately constant. Rolling without slipping means $v_{x}=r \omega_{z}$ and $a_{x}=r \alpha_{z}$. If an object is set in motion on a surface without these equalities, sliding (kinetic) friction will act on the object as it slips until rolling without slipping is established. A solid cylinder with mass $M$ and radius $R$, rotating with angular speed $\omega_{0}$ about an axis through its center, is set on a horizontal surface for which the kinetic friction coefficient is $\mu_{\mathrm{k}}$. (a) Draw a free-body diagram for the cylinder on the surface. Think carefully about the direction of the kinetic friction force on the cylinder. Calculate the accelerations $a_{x}$ of the center of mass and $\alpha_{z}$ of rotation about the center of mass. (b) The cylinder is initially slipping completely, so initially $\omega_{z}=\omega_{0}$ but $v_{x}=0 .$ Rolling without slipping sets in when $v_{x}=r \omega_{z} .$ Calculate the distance the cylinder rolls before slipping stops. (c) Calculate the work done by the friction force on the cylinder as it moves from where it was set down to where it begins to roll without slipping.

Dominador Tan
Dominador Tan
Numerade Educator
01:09

Problem 93

A demonstration gyroscope wheel is constructed by removing the tire from a bicycle wheel $0.640 \mathrm{~m}$ in diameter, wrapping lead wire around the rim, and taping it in place. The shaft projects $0.210 \mathrm{~m}$ at each side of the wheel, and a woman holds the ends of the shaft in her hands. The mass of the system is $8.50 \mathrm{~kg}$; its entire mass may be assumed to be located at its rim. The shaft is horizontal, and the wheel is spinning about the shaft at $5.10 \mathrm{rev} / \mathrm{s}$. Find the magnitude and direction of the force each hand exerts on the shaft (a) when the shaft is at rest; (b) when the shaft is rotating in a horizontal plane about its center at $5.20 \times 10^{-2} \mathrm{rev} / \mathrm{s} ;(\mathrm{c})$ when the shaft is rotating in a horizontal plane about its center at $0.315 \mathrm{rev} / \mathrm{s}$. (d) At what rate must the shaft rotate in order that it may be supported at one end only?

Dominador Tan
Dominador Tan
Numerade Educator
02:26

Problem 94

The moment of inertia of the empty turntable is $1.5 \mathrm{~kg} \cdot \mathrm{m}^{2}$ With a constant torque of $2.5 \mathrm{~N} \cdot \mathrm{m},$ the turntable-person system takes $3.0 \mathrm{~s}$ to spin from rest to an angular speed of $1.0 \mathrm{rad} / \mathrm{s}$. What is the person's moment of inertia about an axis through her center of mass? Ignore friction in the turntable axle.
(a) $2.5 \mathrm{~kg} \cdot \mathrm{m}^{2}$
(b) $6.0 \mathrm{~kg} \cdot \mathrm{m}^{2}$
(c) $7.5 \mathrm{~kg} \cdot \mathrm{m}^{2}$
(d) $9.0 \mathrm{~kg} \cdot \mathrm{m}^{2}$

Andrew C
Andrew C
Numerade Educator
01:00

Problem 95

While the turntable is being accelerated, the person suddenly extends her legs. What happens to the turntable? (a) It suddenly speeds up; (b) it rotates with constant speed;
(c) its acceleration decreases;
(d) it suddenly stops rotating.

Abhishek Jana
Abhishek Jana
Numerade Educator
02:18

Problem 96

A doubling of the torque produces a greater angular acceleration. Which of the following would do this, assuming that the tension in the rope doesn't change?
(a) Increasing the pulley diameter by a factor of $\sqrt{2} ;(b)$ increasing the pulley diameter by a factor of $2 ;(c)$ increasing the pulley diameter by a factor of $4 ;$ (d) decreasing the pulley diameter by a factor of $\sqrt{2}$

Mohit Khurana
Mohit Khurana
Texas A&M University
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Problem 97

If the body's center of mass were not placed on the rotational axis of the turntable, how would the person's measured moment of inertia compare to the moment of inertia for rotation about the center of mass? (a) The measured moment of inertia would be too large; (b) the measured moment of inertia would be too small;
(c) the two moments of inertia would be the same; (d) it depends on where the body's center of mass is placed relative to the center of the turntable.

fX
Fenglin Xi
Numerade Educator