Principles of Physics a Calculus Based Text

Raymond A. Serway, John W. Jewett, Jr.

Chapter 10

Rotational Motion - all with Video Answers

Educators

+ 1 more educators

Chapter Questions

Problem 1

During a certain time interval, the angular position of a swinging door is described by $\theta=5.00+10.0 t+2.00 t^{2}$ where $\theta$ is in radians and $t$ is in seconds. Determine the angular position, angular speed, and angular acceleration of the door $(\mathrm{a})$ at $t=0$ and $(\mathrm{b})$ at $t=3.00 \mathrm{s}$

Samantha Kamal

Samantha Kamal

Numerade Educator

Problem 2

A bar on a hinge starts from rest and rotates with an angular acceleration $\alpha=(10+6 t),$ where $\alpha$ is in $\operatorname{rad} / \mathrm{s}^{2}$ and $t$ is in seconds. Determine the angle in radians through which the bar turns in the first 4.00 s.

Ythan Reyes

Numerade Educator

Problem 3

A potter's wheel moves uniformly from rest to an angular speed of 1.00 rev $/ \mathrm{s}$ in 30.0 s. (a) Find its average angular acceleration in radians per second per second. (b) Would doubling the angular acceleration during the given period have doubled the final angular speed?

Samantha Kamal

Numerade Educator

Problem 4

A dentist's drill starts from rest. After 3.20 s of constant angular acceleration, it turns at a rate of $2.51 \times 10^{4} \mathrm{rev} / \mathrm{min}$
(a) Find the drill's angular acceleration. (b) Determine the angle (in radians) through which the drill rotates during this period.

Salamat Ali

Numerade Educator

Problem 5

The tub of a washer goes into its spin cycle, starting from rest and gaining angular speed steadily for $8.00 \mathrm{s}$, at which time it is turning at 5.00 rev/s. At this point, the person doing the laundry opens the lid, and a safety switch turns off the washer. The tub smoothly slows to rest in 12.0 s. Through how many revolutions does the tub turn while it is in motion?

Rehmat Kazmi

Numerade Educator

Problem 6

Why is the following situation impossible? Starting from rest, a disk rotates around a fixed axis through an angle of 50.0 rad in a time interval of 10.0 s. The angular acceleration of the disk is constant during the entire motion, and its final angular speed is $8.00 \mathrm{rad} / \mathrm{s}$

Rehmat Kazmi

Numerade Educator

Problem 7

An electric motor rotating a workshop grinding wheel at $1.00 \times 10^{2}$ rev $/$ min is switched off. Assume the wheel has a constant negative angular acceleration of magnitude $2.00 \mathrm{rad} / \mathrm{s}^{2} .$ (a) How long does it take the grinding wheel to stop? (b) Through how many radians has the wheel turned during the time interval found in part (a)?

Samantha Kamal

Numerade Educator

Problem 8

A centrifuge in a medical laboratory rotates at an angular speed of 3600 rev/min. When switched off, it rotates through 50.0 revolutions before coming to rest. Find the constant angular acceleration of the centrifuge.

Rehmat Kazmi

Numerade Educator

Problem 9

A Rotating wheel requires 3.00 s to rotate through 37.0 revolutions. Its angular speed at the end of the 3.00 -s interval is $98.0 \mathrm{rad} / \mathrm{s} .$ What is the constant angular acceleration of the wheel?

Rehmat Kazmi

Numerade Educator

Problem 10

A wheel $2.00 \mathrm{m}$ in diameter lies in a vertical plane and rotates about its central axis with a constant angular acceleration of $4.00 \mathrm{rad} / \mathrm{s}^{2} .$ The wheel starts at rest at $t=0,$ and the radius vector of a certain point $P$ on the rim makes an angle of $57.3^{\circ}$ with the horizontal at this time. At $t=2.00 \mathrm{s}$ find (a) the angular speed of the wheel and, for point $P$
(b) the tangential speed, (c) the total acceleration, and
(d) the angular position.

Mukesh Devi

Numerade Educator

Problem 11

A disk $8.00 \mathrm{cm}$ in radius rotates at a constant rate of 1200 rev/min about its central axis. Determine (a) its angular speed in radians per second, (b) the tangential speed at
a point $3.00 \mathrm{cm}$ from its center, (c) the radial acceleration of a point on the rim, and (d) the total distance a point on the rim moves in 2.00 s.

Rehmat Kazmi

Numerade Educator

Problem 12

Make an order-of-magnitude estimate of the number of revolutions through which a typical automobile tire turns in one year. State the quantities you measure or estimate and their values.

Rehmat Kazmi

Numerade Educator

Problem 13

A car traveling on a flat (unbanked), circular track accelerates uniformly from rest with a tangential acceleration of $1.70 \mathrm{m} / \mathrm{s}^{2} .$ The car makes it one-quarter of the way around the circle before it skids off the track. From these data, determine the coefficient of static friction between the car and the track.

Rehmat Kazmi

Numerade Educator

Problem 14

A car traveling on a flat (unbanked), circular track accelerates uniformly from rest with a tangential acceleration of
a. The car makes it one-quarter of the way around the circle before it skids off the track. From these data, determine the coefficient of static friction between the car and the track.

Rehmat Kazmi

Numerade Educator

Problem 15

A digital audio compact disc carries data, each bit of which occupies $0.6 \mu \mathrm{m}$ along a continuous spiral track from the inner circumference of the disc to the outside edge. A CD player turns the disc to carry the track counterclockwise above a lens at a constant speed of $1.30 \mathrm{m} / \mathrm{s}$. Find the required angular speed (a) at the beginning of the recording, where the spiral has a radius of $2.30 \mathrm{cm},$ and (b) at the end of the recording, where the spiral has a radius of $5.80 \mathrm{cm}$
(c) A full-length recording lasts for 74 min 33 s. Find the average angular acceleration of the disc. (d) Assuming that the acceleration is constant, find the total angular displacement of the disc as it plays. (e) Find the total length of the track.

Nicholas Mogoi

Numerade Educator

Problem 16

Figure $P 10.16$ shows the drive train of a bicycle that has wheels $67.3 \mathrm{cm}$ in diameter and pedal cranks $17.5 \mathrm{cm}$ long. The cyclist pedals at a steady cadence of 76.0 rev/min. The chain engages with a front sprocket $15.2 \mathrm{cm}$ in diameter and a rear sprocket $7.00 \mathrm{cm}$ in diameter. Calculate (a) the speed of a link of the chain relative to the bicycle frame, (b) the angular speed of the bicycle wheels, and (c) the speed of the bicycle relative to the road. (d) What pieces of data, if any, are not necessary for the calculations?

Rehmat Kazmi

Numerade Educator

Problem 17

Big Ben, the Parliament tower clock in London, has an hour hand $2.70 \mathrm{m}$ long with a mass of $60.0 \mathrm{kg}$ and a minute hand $4.50 \mathrm{m}$ long with a mass of $100 \mathrm{kg} \text { (Fig. } \mathrm{P} 10.17)$ Calculate the total rotational kinetic energy of the two hands about the axis of rotation.
(You may model the hands as long, thin rods rotated about one end. Assume the hour and minute hands are rotating at a constant rate of one revolution per 12 hours and 60 minutes, respectively.)

Rehmat Kazmi

Numerade Educator

Problem 18

Rigid rods of negligible mass lying along the $y$ axis connect three particles (Fig. P10.18). The system rotates about the $x$ axis with an angular speed of $2.00 \mathrm{rad} / \mathrm{s} .$ Find (a) the moment of inertia about the $x$ axis, $(\mathrm{b})$ the total rotational kinetic energy evaluated from $\frac{1}{2} I \omega^{2},(\mathrm{c})$ the tangential speed of each particle, and (d) the total kinetic energy evaluated from $\sum \frac{1}{2} m_{i} v_{i}^{2} .$ (e) Compare the answers for kinetic energy in parts (b) and (d).

Rehmat Kazmi

Numerade Educator

Problem 19

A war-wolf, or trebuchet, is a device used during the Middle Ages to throw rocks at castles and now sometimes used to fling large vegetables and pianos as a sport. A simple trebuchet is shown in Figure $\mathrm{P} 10.19 .$ Model it as a stiff rod of negligible mass, $3.00 \mathrm{m}$ long, joining particles of mass $m_{1}=0.120 \mathrm{kg}$ and $m_{2}=60.0 \mathrm{kg}$ at its ends. It can turn on a frictionless, horizontal axle perpendicular to the rod and $14.0 \mathrm{cm}$ from the large-mass particle. The operator releases the trebuchet from rest in a horizontal orientation. (a) Find the maximum speed that the small-mass object attains. (b) While the small-mass object is gaining speed, does it move with constant acceleration? (c) Does it move with constant tangential acceleration?
(d) Does the trebuchet move with constant angular acceleration? (e) Does it have constant momentum? (f) Does the trebuchet-Earth system have constant mechanical energy?

Rehmat Kazmi

Numerade Educator

Problem 20

As a gasoline engine operates, a flywheel turning with the crankshaft stores energy after each fuel explosion, providing the energy required to compress the next charge of fuel and air. For the engine of a certain lawn tractor, suppose a flywheel must be no more than $18.0 \mathrm{cm}$ in diameter. Its thickness, measured along its axis of rotation, must be no larger than $8.00 \mathrm{cm} .$ The flywheel must release energy $60.0 \mathrm{J}$ when its angular speed drops from 800 rev/min to 600 rev/min. Design a sturdy steel (density $7.85 \times 10^{3}$ $\mathrm{kg} / \mathrm{m}^{3}$ ) flywheel to meet these requirements with the smallest mass you can reasonably attain. Specify the shape and mass of the flywheel.

Rehmat Kazmi

Numerade Educator

Problem 21

Consider the system shown in Figure $\mathrm{P} 10.21$ with $m_{1}=20.0 \mathrm{kg}$ $m_{2}=12.5 \mathrm{kg}, R=0.200 \mathrm{m},$ and the mass of the pulley $M=5.00 \mathrm{kg} .$ Object $m_{2}$ is resting on the floor, and object $m_{1}$ is 4.00 $\mathrm{m}$ above the floor when it is released from rest. The pulley axis is frictionless. The cord is light, does not stretch, and does not slip on the pulley. (a) Calculate the time interval required for $m_{1}$ to hit the floor. (b) How would your answer change if the pulley were massless?

Rehmat Kazmi

Numerade Educator

Problem 22

The fishing pole in Figure $P 10.22$ makes an angle of $20.0^{\circ}$ with the horizontal. What is the torque exerted by the fish about an axis perpendicular to the page and passing through the angler's hand if the fish pulls on the fishing line with a force $\overrightarrow{\mathbf{F}}=100 \mathrm{N}$ at an angle $37.0^{\circ}$ below the horizontal? The force is applied at a point $2.00 \mathrm{m}$ from the angler's hands.

Rehmat Kazmi

Numerade Educator

Problem 23

Find the net torque on the wheel in Figure $P 10.23$ about the axle through $O$, taking $a=10.0 \mathrm{cm}$ and $b=25.0 \mathrm{cm}$

Rehmat Kazmi

Numerade Educator

Problem 24

Two vectors are given by $\overrightarrow{\mathbf{A}}=-3 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}-4 \hat{\mathbf{k}} \quad$ and $\overrightarrow{\mathbf{B}}=6 \hat{\mathbf{i}}-10 \hat{\mathbf{j}}+9 \hat{\mathbf{k}} .$ Evaluate the following quantities
$(\mathrm{a}) \cos ^{-1}[\overrightarrow{\mathbf{A}} \cdot \overrightarrow{\mathbf{B}} / A B]$ and $(\mathrm{b}) \sin ^{-1}[|\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}}| / A B] .$ (c) Which
give $(\mathrm{s})$ the angle between the vectors?

Katie Mcalpine

Numerade Educator

Problem 25

Given $\overrightarrow{\mathbf{M}}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{N}}=4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-2 \hat{\mathbf{k}},$ calculate the vector product $\overrightarrow{\mathbf{M}} \times \overrightarrow{\mathbf{N}}$

Katie Mcalpine

Numerade Educator

Problem 26

Use the definition of the vector product and the definitions of the unit vectors $\hat{\mathbf{i}}, \hat{\mathbf{j}},$ and $\hat{\mathbf{k}}$ to prove Equations 10.26 You may assume the $x$ axis points to the right, the $y$ axis up, and the $z$ axis horizontally toward you (not away from you). This choice is said to make the coordinate system a righthanded system.

Rehmat Kazmi

Numerade Educator

Problem 27

A force of $\overrightarrow{\mathbf{F}}=(2.00 \hat{\mathbf{i}}+3.00 \hat{\mathbf{j}}) \mathrm{N}$ is applied to an object that is pivoted about a fixed axle aligned along the $z$ coordinate axis. The force is applied at the point $\overrightarrow{\mathbf{r}}=(4.00 \hat{\mathbf{i}}+5.00 \hat{\mathbf{j}}) \mathrm{m}$ Find (a) the magnitude of the net torque about the $z$ axis and (b) the direction of the torque vector $\vec{\tau}$.

Rehmat Kazmi

Numerade Educator

Problem 28

A uniform beam resting on two pivots has a length $L=$ $6.00 \mathrm{m}$ and mass $M=90.0$ kg. The pivot under the left end exerts a normal force $n_{1}$ on the beam, and the second pivot located a distance $\ell=4.00 \mathrm{m}$ from the left end exerts a normal force $n_{2} .$ A woman of mass $m=55.0$ kg steps onto the left end of the beam and begins walking to the right as in Figure P10.28. The goal is to find the woman's position when the beam begins to tip. (a) What is the appropriate analysis model for the beam before it begins to tip? (b) Sketch a force diagram for the beam, labeling the gravitational and normal forces acting on the beam and placing the woman a distance $x$ to the right of the first pivot, which is the origin. (c) Where is the woman when the normal force $n_{1}$ is the greatest? (d) What is $n_{1}$ when the beam is about to tip? (e) Use Equation 10.27 to find the value of $n_{2}$ when the beam is about to tip. (f) Using the result of part (d) and Equation 10.28 , with torques computed around the second pivot, find the woman's position $x$ when the beam is about to tip. (g) Check the answer to part (e) by computing torques around the first pivot point.

Rehmat Kazmi

Numerade Educator

Problem 29

In exercise physiology studies, it is sometimes important to determine the location of a person's center of mass. This determination can be done with the arrangement shown in Figure $\mathrm{P} 10.29 .$ A light plank rests on two scales, which $\operatorname{read} F_{g 1}=380 \mathrm{N}$ and $F_{g 2}=$$320 \mathrm{N} .$ A distance of $1.65 \mathrm{m}$ separates the scales. How far from the woman's feet is her center of mass?

Rehmat Kazmi

Numerade Educator

Problem 30

Why is the following situation impossible? A uniform beam of mass $m_{b}=3.00 \mathrm{kg}$ and length $\ell=1.00 \mathrm{m}$ supports blocks with masses $m_{1}=5.00 \mathrm{kg}$ and $m_{2}=15.0 \mathrm{kg}$ at two positions as shown in Figure $\mathrm{P} 10.30$. The beam rests on two triangular blocks, with point $P$ a distance $d=0.300 \mathrm{m}$ to the right of the center of gravity of the beam. The position of the object of mass $m_{2}$ is adjusted along the length of the beam until the normal force on the beam at $O$ is zero.

Rehmat Kazmi

Numerade Educator

Problem 31

Figure $P 10.31$ shows a claw hammer being used to pull a nail out of a horizontal board. The mass of the hammer is 1.00 kg. A force of $150 \mathrm{N}$ is exerted horizontally as shown, and the nail does not yet move relative to the board. Find (a) the force exerted by the hammer claws on the nail and (b) the force exerted by the surface on the point of contact with the hammer head. Assume the force the hammer exerts on the nail is parallel to the nail.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 32

A uniform sign of weight $F_{g}$ and width $2 L$ hangs from a light, horizontal beam hinged at the wall and supported by a cable (Fig. $\mathrm{P} 10.32$ ). Determine (a) the tension in the cable and (b) the components of the reaction force exerted by the wall on the beam in terms of $F_{g}, d, L,$ and $\theta$

Rehmat Kazmi

Numerade Educator

Problem 33

A 15.0 -m uniform ladder weighing 500 N rests against a frictionless wall. The ladder makes a $60.0^{\circ}$ angle with the horizontal. (a) Find the horizontal and vertical forces the ground exerts on the base of the ladder when an 800 -N firefighter has climbed $4.00 \mathrm{m}$ along the ladder from the bottom. (b) If the ladder is just on the verge of slipping when the firefighter is $9.00 \mathrm{m}$ from the bottom, what is the coefficient of static friction between ladder and ground?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 34

A uniform ladder of length $L$ and mass $m_{1}$ rests against a frictionless wall. The ladder makes an angle $\theta$ with the horizontal. (a) Find the horizontal and vertical forces the ground exerts on the base of the ladder when a firefighter of mass $m_{2}$ has climbed a distance $x$ along the ladder from the bottom. (b) If the ladder is just on the verge of slipping when the firefighter is a distance $d$ along the ladder from the bottom, what is the coefficient of static friction between ladder and ground?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 35

The arm in Figure $\mathrm{P} 10.35$ weighs $41.5 \mathrm{N}$. The gravitational force on the arm acts through point $A$. Determine the magnitudes of the tension force $\mathbf{F}_{t}$ in the deltoid muscle and the force $\overrightarrow{\mathbf{F}}_{s}$ exerted by the shoulder on the humerus (upper-arm bone) to hold the arm in the position shown.

Rehmat Kazmi

Numerade Educator

Problem 36

A crane of mass $m_{1}=3000$ kg supports a load of mass $m_{2}=10000 \mathrm{kg}$ as shown in Figure $P 10.36 .$ The crane is pivoted with a frictionless pin at $A$ and rests against a smooth support at $B .$ Find the reaction forces at (a) point $A$ and $(\mathrm{b})$ point $B$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 37

An electric motor turns a flywheel through a drive belt that joins a pulley on the motor and a pulley that is rigidly attached to the flywheel as shown in Figure P10.37. The flywheel is a solid disk with a mass of $80.0 \mathrm{kg}$ and a radius $R=$ $0.625 \mathrm{m} .$ It turns on a frictionless axle. Its pulley has much smaller mass and a radius of $r=0.230 \mathrm{m}$. The tension $T_{u}$ in the upper (taut) segment of the belt is $135 \mathrm{N}$, and the flywheel has a clockwise angular acceleration of $1.67 \mathrm{rad} / \mathrm{s}^{2}$ Find the tension in the lower (slack) segment of the belt.

Rehmat Kazmi

Numerade Educator

Problem 38

This problem describes one experimental method for determining the moment of inertia of an irregularly shaped object such as the payload for a satellite. Figure P10.38 shows a counterweight of mass $m$ suspended by a cord wound around a spool of radius $r,$ forming part of a turntable supporting the object. The turntable can rotate without friction. When the counterweight is released from rest, it descends through a distance $h,$ acquiring a speed $v .$ Show that the moment of inertia $I$ of the rotating apparatus (including the turntable) is $m r^{2}\left(2 g h / v^{2}-1\right)$

Rehmat Kazmi

Numerade Educator

Problem 39

The combination of an applied force and a friction force produces a constant total torque of $36.0 \mathrm{N} \cdot \mathrm{m}$ on a wheel rotating about a fixed axis. The applied force acts for 6.00 s. During this time, the angular speed of the wheel increases from 0 to 10.0 rad/s. The applied force is then removed, and the wheel comes to rest in 60.0 s. Find
(a) the moment of inertia of the wheel, (b) the magnitude of the torque due to friction, and (c) the total number of revolutions of the wheel during the entire interval of 66.0 s.

Rehmat Kazmi

Numerade Educator

Problem 40

In Figure $P 10.40,$ the hanging object has a mass of $m_{1}=0.420 \mathrm{kg} ;$ the sliding block has a mass of $m_{2}=0.850 \mathrm{kg} ;$ and the pulley is a hollow cylinder with a mass of $M=0.350 \mathrm{kg}$ an inner radius of $R_{1}=$ $0.0200 \mathrm{m},$ and an outer radius of $R_{2}=0.0300 \mathrm{m}$ Assume the mass of the spokes is negligible. The coefficient of kinetic friction between the block and the horizontal surface is $\mu_{k}=0.250 .$ The pulley turns without friction on its axle. The light cord does not stretch and does not slip on the pulley. The block has a velocity of $v_{i}=0.820 \mathrm{m} / \mathrm{s}$ toward the pulley when it passes a reference point on the table. (a) Use energy methods to predict its speed after it has moved to a second point, $0.700 \mathrm{m}$ away. (b) Find the angular speed of the pulley at the same moment.

Rehmat Kazmi

Numerade Educator

Problem 41

A potter's wheel-a thick stone disk of radius $0.500 \mathrm{m}$ and mass $100 \mathrm{kg}-$ is freely rotating at $50.0 \mathrm{rev} / \mathrm{min} .$ The potter can stop the wheel in 6.00 s by pressing a wet rag against the rim and exerting a radially inward force of $70.0 \mathrm{N}$ Find the effective coefficient of kinetic friction between wheel and rag.

Rehmat Kazmi

Numerade Educator

Problem 42

A model airplane with mass $0.750 \mathrm{kg}$ is tethered to the ground by a wire so that it flies in a horizontal circle 30.0 $\mathrm{m}$ in radius. The airplane engine provides a net thrust of $0.800 \mathrm{N}$ perpendicular to the tethering wire. (a) Find the torque the net thrust produces about the center of the circle.
(b) Find the angular acceleration of the airplane. (c) Find the translational acceleration of the airplane tangent to its flight path.

Rehmat Kazmi

Numerade Educator

Problem 43

As shown in Figure $\mathrm{P} 10.43$ (page 348 ), two blocks are connected by a string of negligible mass passing over a pulley of radius $r=0.250 \mathrm{m}$ and moment of inertia $I$. The block on the frictionless incline is moving with a constant acceleration of magnitude $a=2.00 \mathrm{m} / \mathrm{s}^{2} .$ From this information, we wish to find the moment of inertia of the pulley. (a) What analysis model is appropriate for the blocks? (b) What analysis model is appropriate for the pulley? (c) From the analysis model in part (a), find the tension
$T_{1} \cdot(\mathrm{d})$ Similarly, find the tension $T_{2} .$ (e) From the analysis model in part (b), find a symbolic expression for the moment of inertia of the pulley in terms of the tensions $T_{1}$ and $T_{2},$ the pulley radius $r,$ and the acceleration $a .$ (f) Find the numerical value of the moment of inertia of the pulley.

Rehmat Kazmi

Numerade Educator

Problem 44

Consider two objects with $m_{1}>m_{2}$ connected by a light string that passes over a pulley having a moment of inertia of $I$ about its axis of rotation as shown in Figure $\mathrm{P} 10.44 .$ The string does not slip on the pulley or stretch. The pulley turns without friction. The two objects are released from rest separated by a vertical distance 2 $h$. (a) Use the principle of conservation of energy to find the translational speeds of the objects as they pass each other. (b) Find the angular speed of the pulley at this time.

Rehmat Kazmi

Numerade Educator

Problem 45

An object with a mass of $m=5.10 \mathrm{kg}$ is attached to the free end of a light string wrapped around a reel of radius $R=0.250 \mathrm{m}$ and mass $M=3.00 \mathrm{kg} .$ The reel is a solid disk, free to rotate in a vertical plane about the horizontal axis passing through its center as shown in Figure $\mathrm{P} 10.45 .$ The suspended object is released from rest $6.00 \mathrm{m}$ above the floor. Determine (a) the tension in the string, (b) the acceleration of the object, and (c) the speed with which the object hits the floor. (d) Verify your answer to part (c) by using the isolated system (energy) model.

Rehmat Kazmi

Numerade Educator

Problem 46

A playground merry-go-round of radius $R=2.00 \mathrm{m}$ has a moment of inertia $I=250 \mathrm{kg} \cdot \mathrm{m}^{2}$ and is rotating at 10.0 rev/min about a frictionless, vertical axle. Facing the axle, a 25.0 -kg child hops onto the merry-go-round and manages to sit down on the edge. What is the new angular speed of the merry-go-round?

Rehmat Kazmi

Numerade Educator

Problem 47

The position vector of a particle of mass $2.00 \mathrm{kg}$ as a function of time is given by $\overrightarrow{\mathbf{r}}=(6.00 \hat{\mathbf{i}}+5.00 t \hat{\mathbf{j}}),$ where $\overrightarrow{\mathbf{r}}$ is in meters and $t$ is in seconds. Determine the angular momentum of the particle about the origin as a function of time.

Katie Mcalpine

Numerade Educator

Problem 48

Heading straight toward the summit of Pike's Peak, an airplane of mass 12000 kg flies over the plains of Kansas at nearly constant altitude $4.30 \mathrm{km}$ with constant velocity $175 \mathrm{m} / \mathrm{s}$ west. (a) What is the airplane's vector angular momentum relative to a wheat farmer on the ground directly below the airplane? (b) Does this value change as the airplane continues its motion along a straight line? (c) What If $?$ What is its angular momentum relative to the summit of Pike's Peak?

Rehmat Kazmi

Numerade Educator

Problem 49

Big Ben (Fig. $\mathrm{P} 10.17$ ), the Parliament tower clock in London, has hour and minute hands with lengths of $2.70 \mathrm{m}$ and $4.50 \mathrm{m}$ and masses of $60.0 \mathrm{kg}$ and $100 \mathrm{kg}$, respectively. Calculate the total angular momentum of these hands about the center point. (You may model the hands as long, thin rods rotating about one end. Assume the hour and minute hands are rotating at a constant rate of one revolution per
12 hours and 60 minutes, respectively.)

Rehmat Kazmi

Numerade Educator

Problem 50

A disk with moment of inertia $I_{1}$ rotates about a frictionless, vertical axle with angular speed $\omega_{i} .$ A second disk, this one having moment of inertia $I_{2}$ and initially not rotating, drops $I_{1}$ onto the first disk (Fig. P10.50). Because of friction between the surfaces, the two eventually reach the same angular speed $\omega_{f} \cdot(\mathrm{a})$ Calculate $\omega_{f} \cdot(\mathrm{b})$ Calculate the ratio of the final to the initial rotational energy.

Rehmat Kazmi

Numerade Educator

Problem 51

A particle of mass $0.400 \mathrm{kg}$ is attached to the $100-\mathrm{cm}$ mark of a meterstick of mass 0.100 kg. The meterstick rotates on the surface of a frictionless, horizontal table with an angular speed of 4.00 rad/s. Calculate the angular momentum of the system when the stick is pivoted about an axis (a) perpendicular to the table through the $50.0-\mathrm{cm}$ mark and (b) perpendicular to the table through the 0 -cm mark.

Rehmat Kazmi

Numerade Educator

Problem 52

A space station is constructed in the shape of a hollow ring of mass $5.00 \times$ $10^{4}$ kg. Members of the crew walk on a deck formed by the inner surface of the outer cylindrical wall of the ring, with radius $r=100 \mathrm{m} .$ At rest when constructed, the ring is set rotating about its axisso that the people inside experience an effective free-fall acceleration equal to $g$. (See Fig. P10.52.) The rotation is achieved by firing two small rockets attached tangentially to opposite points on the rim of the ring. (a) What angular momentum does the space station acquire? (b) For what time interval must the rockets be fired if each exerts a thrust of $125 \mathrm{N}$ ?

Rehmat Kazmi

Numerade Educator

Problem 53

A puck of mass $m_{1}=80.0 \mathrm{g}$ and radius $r_{1}=4.00 \mathrm{cm}$ glides across an air table at a speed $v=1.50 \mathrm{m} / \mathrm{s}$ as shown in Figure P10.53a. It makes a glancing collision with a second puck of radius $r_{2}=6.00 \mathrm{cm}$ and mass $m_{2}=120 \mathrm{g}$ (initially at rest $)$ such that their rims just touch. Because their rims are coated with instant-acting glue, the pucks stick together and rotate after the collision (Fig. P10.53b). (a) What is the angular momentum of the system relative to the center of mass? (b) What is the angular speed about the center of mass?

Rehmat Kazmi

Numerade Educator

Problem 54

Why is the following situation impossible? A space station shaped like a giant wheel has a radius of $r=100 \mathrm{m}$ and a moment of inertia of $5.00 \times 10^{8} \mathrm{kg} \cdot \mathrm{m}^{2} .$ A crew of 150 people of average mass $65.0 \mathrm{kg}$ is living on the rim, and the station's rotation causes the crew to experience an apparent free-fall acceleration of $g$ (Fig. $P 10.52$ ). A research technician is assigned to perform an experiment in which a ball is dropped at the rim of the station every 15 minutes and the time interval for the ball to drop a given distance is measured as a test to make sure the apparent value of $g$ is correctly maintained. One evening, 100 average people move to the center of the station for a union meeting. The research technician, who has already been performing his experiment for an hour before the meeting, is disappointed that he cannot attend the meeting, and his mood sours even further by his boring experiment in which every time interval for the dropped ball is identical for the entire evening.

Rehmat Kazmi

Numerade Educator

Problem 55

The puck in Figure 10.25 has a mass of 0.120 kg. The distance of the puck from the center of rotation is originally $40.0 \mathrm{cm},$ and the puck is sliding with a speed of $80.0 \mathrm{cm} / \mathrm{s}$ The string is pulled downward $15.0 \mathrm{cm}$ through the hole in the frictionless table. Determine the work done on the puck. ( Suggestion: Consider the change of kinetic energy.)

Rehmat Kazmi

Numerade Educator

Problem 56

A student sits on a freely rotating stool holding two dumbbells, each of mass 3.00 kg (Fig. P10.56). When his arms are extended horizontally (Fig. $P 10.56 a),$ the dumbbells are $1.00 \mathrm{m}$ from the axis of rotation and the
student rotates with an angular speed of $0.750 \mathrm{rad} / \mathrm{s}$ The moment of inertia of the student plus stool is $3.00 \mathrm{kg} \cdot \mathrm{m}^{2}$ and is assumed to be constant. The student pulls the dumbbells inward horizontally to a position $0.300 \mathrm{m}$ from the rotation axis (Fig. P10.56b). (a) Find the new angular speed of the student.
(b) Find the kinetic energy of the rotating system before and after he pulls the dumbbells inward.

Rehmat Kazmi

Numerade Educator

Problem 57

A 60.0 -kg woman stands at the western rim of a hor- izontal turntable having a moment of inertia of $500 \mathrm{kg} \cdot \mathrm{m}^{2}$ and a radius of $2.00 \mathrm{m}$. The turntable is initially at rest and is free to rotate about a frictionless, vertical axle through its center. The woman then starts walking around the rim clockwise (as viewed from above the system) at a constant speed of $1.50 \mathrm{m} / \mathrm{s}$ relative to the Earth. Consider the womanturntable system as motion begins. (a) Is the mechanical energy of the system constant? (b) Is the momentum of the system constant? (c) Is the angular momentum of the system constant? (d) In what direction and with what angular speed does the turntable rotate? (e) How much chemical energy does the woman's body convert into mechanical energy of the woman-turntable system as the woman sets herself and the turntable into motion?

Dominador Tan

Numerade Educator

Problem 58

The angular momentum vector of a precessing gyroscope sweeps out a cone as shown in Figure $P 10.58 .$ The angular speed of the tip of the angular momentum vector, called its precessional frequency, is given by $\omega_{p}=\tau / L$ where $\tau$ is the magnitude of the torque on the gyroscope and $L$ is the magnitude of its angular momentum. In the motion called precession of the equinoxes, the Earth's axis of rotation precesses about the perpendicular to its orbital plane with a period of $2.58 \times 10^{4}$ yr. Model the Earth as a uniform sphere and calculate the torque on the Earth that is causing this precession.

Rehmat Kazmi

Numerade Educator

Problem 59

A cylinder of mass 10.0 kg rolls without slipping on a horizontal surface. At a certain instant, its center of mass has a speed of $10.0 \mathrm{m} / \mathrm{s}$. Determine (a) the translational kinetic energy of its center of mass, (b) the rotational kinetic energy about its center of mass, and (c) its total energy.

Dr. Valerie Bennett

Dr. Valerie Bennett

Numerade Educator

Problem 60

A uniform solid disk and a uniform hoop are placed side by side at the top of an incline of height $h$. (a) If they are released from rest and roll without slipping, which object reaches the bottom first? (b) Verify your answer by calculating their speeds when they reach the bottom in terms of $h$.

Rehmat Kazmi

Numerade Educator

Problem 61

A metal can containing condensed mushroom soup has mass $215 \mathrm{g}$, height $10.8 \mathrm{cm},$ and diameter $6.38 \mathrm{cm}$. It is placed at rest on its side at the top of a 3.00 -m-long incline that is at $25.0^{\circ}$ to the horizontal and is then released to roll straight down. It reaches the bottom of the incline after 1.50 s.
(a) Assuming mechanical energy conservation, calculate the moment of inertia of the can. (b) Which pieces of data, if any, are unnecessary for calculating the solution? (c) Why can't the moment of inertia be calculated from $I=\frac{1}{2} m r^{2}$ for the cylindrical can?

Rehmat Kazmi

Numerade Educator

Problem 62

A tennis ball is a hollow sphere with a thin wall. It is set rolling without slipping at $4.03 \mathrm{m} / \mathrm{s}$ on a horizontal section of a track as shown in Figure $\mathrm{P} 10.62$ It rolls around the inside of a vertical circular loop of radius $r=45.0 \mathrm{cm} .$ As the ball nears the bottom of the loop, the shape of the track deviates from a perfect circle so that the ball leaves the track at a point $h=20.0 \mathrm{cm}$ below the horizontal section. (a) Find the ball's speed at the top of the loop. (b) Demonstrate that the ball will not fall from the track at the top of the loop. (c) Find the ball's speed as it leaves the track at the bottom. What If? (d) Suppose that static friction between ball and track were negligible so that the ball slid instead of rolling. Would its speed then be higher, lower, or the same at the top of the loop? (e) Explain your answer to part (d).

Rehmat Kazmi

Numerade Educator

Problem 63

A spacecraft is in empty space. It carries on board a gyroscope with a moment of inertia of $I_{g}=20.0 \mathrm{kg} \cdot \mathrm{m}^{2}$ about the axis of the gyroscope. The moment of inertia of the spacecraft around the same axis is $I_{s}=5.00 \times 10^{5} \mathrm{kg} \cdot \mathrm{m}^{2} .$ Neither the spacecraft nor the gyroscope is originally rotating. The gyroscope can be powered up in a negligible period of time to an angular speed of 100 rad/s. If the orientation of the spacecraft is to be changed by $30.0^{\circ},$ for what time interval should the gyroscope be operated?

Katie Mcalpine

Numerade Educator

Problem 64

A mixing beater consists of three thin rods, each $10.0 \mathrm{cm}$ long. The rods diverge from a central hub, separated from each other by $120^{\circ},$ and all turn in the same plane. A ball is attached to the end of each rod. Each ball has crosssectional area $4.00 \mathrm{cm}^{2}$ and is so shaped that it has a drag coefficient of $0.600 .$ Calculate the power input required to spin the beater at 1000 rev $/ \min$ (a) in air and (b) in water.

Rehmat Kazmi

Numerade Educator

Problem 65

A long, uniform rod of length $L$ and mass $M$ is pivoted about a frictionless, horizontal pin through one end. The rod is released from rest in a vertical position as shown in Figure $P 10.65 .$ At the instant the rod is horizon-
tal, find (a) its angular speed, (b) the magnitude of its angular acceleration, (c) the $x$ and $y$ components of the acceleration of its center of mass, and (d) the components of the reaction force at the pivot.

Rehmat Kazmi

Numerade Educator

Problem 66

The hour hand and the minute hand of Big Ben, the Parliament tower clock in London, are $2.70 \mathrm{m}$ and $4.50 \mathrm{m}$ long and have masses of $60.0 \mathrm{kg}$ and $100 \mathrm{kg},$ respectively (see Fig. $\mathrm{P} 10.17$ ). (a) Determine the total torque due to the weight of these hands about the axis of rotation when the time reads (i) $3: 00,$ (ii) $5: 15,$ (iii) $6: 00,$ (iv) $8: 20,$ and
(v) $9: 45 .$ (You may model the hands as long, thin, uniform rods.) (b) Determine all times when the total torque about the axis of rotation is zero. Determine the times to the nearest second, solving a transcendental equation numerically.

Rehmat Kazmi

Numerade Educator

Problem 67

Two astronauts (Fig. P10.67), each having a mass of $75.0 \mathrm{kg},$ are connected by a $10.0-\mathrm{m}$ rope of negligible mass. They are isolated in space, orbiting their center of mass at speeds of $5.00 \mathrm{m} / \mathrm{s}$. Treating the astronauts as particles, calculate (a) the magnitude of the angular momentum of the two-astronaut system and (b) the rotational energy of the system. By pulling on the rope, one astronaut shortens the distance between them to $5.00 \mathrm{m}$
(c) What is the new angular momentum of the system?
(d) What are the astronauts' new speeds? (e) What is the new rotational energy of the system? (f) How much chemical potential energy in the body of the astronaut was converted to mechanical energy in the system when he shortened the rope?

Rehmat Kazmi

Numerade Educator

Problem 68

Two astronauts (Fig. $P 10.67$ ), each having a mass $M$ are connected by a rope of length $d$ having negligible mass. They are isolated in space, orbiting their center of mass at speeds $v$. Treating the astronauts as particles, calculate (a) the magnitude of the angular momentum of the two-astronaut system and (b) the rotational energy of the system. By pulling on the rope, one of the astronauts shortens the distance between them to $d / 2 .$ (c) What is the new angular momentum of the system? (d) What are the astronauts' new speeds? (e) What is the new rotational energy of the system? (f) How much chemical potential energy in the body of the astronaut was converted to mechanical energy in the system when he shortened the rope?

Rehmat Kazmi

Numerade Educator

Problem 69

When a person stands on tiptoe on one foot (a strenuous position $),$ the position of the foot is as shown in Figure $\mathrm{P} 10.69$ a. The total gravitational force $\overrightarrow{\mathbf{F}}_{g}$ on the body is supported by the normal force $\overrightarrow{\mathbf{n}}$ exerted by the floor on the toes of one foot. A mechanical model of the situation is shown in Figure $\mathrm{P} 10.69 \mathrm{b}$, where $\overrightarrow{\mathrm{T}}$ is the force exerted on the foot by the Achilles tendon and $\overrightarrow{\mathbf{R}}$ is the force exerted on the foot by the tibia. Find the values of $T, R,$ and $\theta$ when $F_{g}=700 \mathrm{N}$.

Sheh Lit Chang

University of Washington

Problem 70

A uniform, hollow, cylindrical spool has inside radius $R / 2,$ outside radius $R,$ and mass $M$ (Fig. $P 10.70$ ). It is mounted so that it rotates on a fixed, horizontal axle. A counterweight of mass $m$ is connected to the end of a string wound around the spool. The counterweight falls from rest at $t=0$ to a position $y$ at time $t$. Show that the torque due to the friction forces between spool and axle is
$$\tau_{f}=R\left[m\left(g-\frac{2 y}{t^{2}}\right)-M \frac{5 y}{4 t^{2}}\right]$$

Rehmat Kazmi

Numerade Educator

Problem 71

The reel shown in Figure $P 10.71$ has radius $R$ and moment of inertia $I$ One end of the block of mass $m$ is connected to a spring of force constant $k$ and the other end is fastened to a cord wrapped around the reel. The reel
axle and the incline are frictionless. The reel is wound counterclockwise so that the spring stretches a distance $d$ from its unstretched position and the reel is then released from rest. Find the angular speed of the reel when the spring is again unstretched.

Rehmat Kazmi

Numerade Educator

Problem 72

A block of mass $m_{1}=2.00 \mathrm{kg}$ and a block of mass $m_{2}=6.00 \mathrm{kg}$ are connected by a massless string over a pulley in the shape of a solid disk having radius $R=0.250 \mathrm{m}$ and mass $M=10.0 \mathrm{kg} .$ The fixed, wedge-shaped ramp makes an angle of $\theta=30.0^{\circ}$ as shown in Figure P10.72. The coefficient of kinetic friction is 0.360 for both blocks. (a) Draw force diagrams of both blocks and of the pulley. Determine
(b) the acceleration of the two blocks and (c) the tensions in the string on both sides of the pulley.

Rehmat Kazmi

Numerade Educator

Problem 73

A stepladder of negligible weight is constructed as shown in Figure $\mathrm{P} 10.73,$ with $A C=B C=\ell=4.00 \mathrm{m} .$ A painter of mass $m=70.0 \mathrm{kg}$ stands on the ladder $d=3.00 \mathrm{m}$ from the bottom. Assuming the floor is frictionless, find (a) the tension in the horizontal bar $D E$ connecting the two halves of the ladder,
(b) the normal forces at $A$ and $B$ and (c) the components of the reaction force at the single hinge $C$ that the left half of the ladder exerts on the right half. Suggestion: Treat the ladder as a single object, but also treat each half of the ladder separately.

Rehmat Kazmi

Numerade Educator

Problem 74

A stepladder of negligible weight is constructed as shown in Figure $P 10.73,$ with $A C=B C=\ell$. A painter of mass $m$ stands on the ladder a distance $d$ from the bottom. Assuming the floor is frictionless, find (a) the tension in the horizontal bar $D E$ connecting the two halves of the ladder, (b) the normal forces at $A$ and $B$, and (c) the components of the reaction force at the single hinge $C$ that the left half of the ladder exerts on the right half. Suggestion: Treat the ladder as a single object, but also treat each half of the ladder separately.

Rehmat Kazmi

Numerade Educator

Problem 75

A wad of sticky clay with mass $m$ and velocity $\overrightarrow{\mathbf{v}}_{i}$ is fired at a solid cylinder of mass $M$ and radius $R$ (Fig. P10.75). The cylinder is initially at rest and is mounted on a fixed horizontal axle that runs through its center of mass. The line of motion of the projectile is perpendicular to the axle and at a distance $d<R$ from the center.
(a) Find the angular speed of the system just after the clay strikes and sticks to the surface of the cylinder. (b) Is the mechanical energy of the clay-cylinder system constant in this process? Explain your answer. (c) Is the momentum of the clay-cylinder system constant in this process? Explain your answer.

Rehmat Kazmi

Numerade Educator

Problem 76

A common demonstration, illustrated in Figure $\mathrm{P} 10.76$ consists of a ball resting at one end of a uniform board of length $\ell$ that is hinged at the other end and elevated at an angle $\theta$. A light cup is attached to the board at $r_{c}$ so that it will catch the ball when the support stick is removed suddenly. (a) Show that the ball will lag behind the falling board when $\theta$ is less than $35.3^{\circ} .$ (b) Assuming the board is $1.00 \mathrm{m}$ long and is supported at this limiting angle, show that the cup must be $18.4 \mathrm{cm}$ from the moving end.

Rehmat Kazmi

Numerade Educator

Problem 77

The large quadriceps muscle in the upper leg terminates at its lower end in a tendon attached to the upper end of the tibia (Fig. P10.77a). The forces on the lower leg when the leg is extended are modeled as in Figure $\mathrm{P} 10.77 \mathrm{b}$, where $\mathbf{T}$ is the force in the tendon, $\overrightarrow{\mathbf{F}}_{g \operatorname{leg}}$ is the gravitational force acting on the lower leg, and $\overrightarrow{\mathbf{F}}_{g \text { foot }}$ is the gravitational force acting on the foot. Find $T$ when the tendon is at an angle of $\phi=$ $25.0^{\circ}$ with the tibia, assuming $F_{g, \operatorname{leg}}=30.0 \mathrm{N}, F_{g \text { foot }}=12.5 \mathrm{N}$ and the leg is extended at an angle $\theta=40.0^{\circ}$ with respect to the vertical. Also assume the center of gravity of the tibia is at its geometric center and the tendon attaches to the lower leg at a position one fifth of the way down the leg.

Rehmat Kazmi

Numerade Educator

Problem 78

A string is wound around a uniform disk of radius $R$ and mass $M$
The disk is released from rest with the string vertical and its top end tied to a fixed bar (Fig. $\mathrm{P} 10.78$ ). Show that (a) the tension in the string is one third of the weight of the disk, (b) the magnitude of the acceleration of the center of mass is $2 g / 3,$ and $(c)$ the speed of the center of mass is $(4 g h / 3)^{1 / 2}$ after the disk has descended through distance $h$. (d) Verify your answer to part (c) using the energy approach.

Rehmat Kazmi

Numerade Educator

Problem 79

Assume a person bends forward to lift a load "with his back" as shown in Figure $\mathrm{Pl} 0.79$ a. The spine pivots mainly at the fifth lumbar vertebra, with the principal supporting force provided by the erector spinalis muscle in the back. To see the magnitude of the forces involved, consider the model shown in Figure $\mathrm{P} 10.79 \mathrm{b}$ for a person bending forward to lift a 200 -N object. The spine and upper body are represented as
a uniform horizontal rod of weight $350 \mathrm{N}$, pivoted at the base of the spine. The erector spinalis muscle, attached at a point two-thirds of the way up the spine, maintains the position of the back. The angle between the spine and this muscle is $\theta=$ $12.0^{\circ} .$ Find (a) the tension $T$ in the back muscle and (b) the compressional force in the spine. (c) Is this method a good way to lift a load? Explain your answer, using the results of parts (a) and (b). (d) Can you suggest a better method to lift a load?

Brandy Heflin

Numerade Educator

Problem 80

Why is the following situation impossible? A worker in a factory pulls a cabinet across the floor using a rope as shown in Figure $\mathrm{P} 10.80$ a. The rope makes an angle $\theta=37.0^{\circ}$ with the floor and is tied $h_{1}=10.0 \mathrm{cm}$ from the bottom of the cabinet. The uniform rectangular cabinet has height $\ell=100 \mathrm{cm}$ and width $w=60.0 \mathrm{cm},$ and it weighs $400 \mathrm{N}$. The cabinet slides with constant speed when a force $F=300 \mathrm{N}$ is applied through the rope. The worker tires of walking backward. He fastens the rope to a point on the cabinet $h_{2}=65.0 \mathrm{cm}$ off the floor and lays the rope over his shoulder so that he can walk forward and pull as shown in Figure P10.80b. In this way, the rope again makes an angle of $\theta=37.0^{\circ}$ with the horizontal and again has a tension of 300 N. Using this technique, the worker is able to slide the cabinet over a long distance on the floor without tiring.

Rehmat Kazmi

Numerade Educator

Problem 81

A projectile of mass $m$ moves to the right with a speed $v_{i}$ (Fig. $P 10.81 \mathrm{a}$ ). The projectile strikes and sticks to the end of a stationary rod of mass $M$ length $d,$ pivoted about a frictionless axle perpendicular to the page through $O$ (Fig. $P 10.81 b) .$ We wish to find the fractional change of kinetic energy in the system due to the collision. (a) What is the appropriate analysis model to describe the projectile and the rod? (b) What is the angular momentum of the system before the collision about an axis through $O$ ?
(c) What is the moment of inertia of the system about an axis through $O$ after the projectile sticks to the rod? (d) If the angular speed of the system after the collision is $\omega$, what is the angular momentum of the system after the collision?
(e) Find the angular speed $\omega$ after the collision in terms of the given quantities. (f) What is the kinetic energy of the system before the collision? (g) What is the kinetic energy of the system after the collision? (h) Determine the fractional change of kinetic energy due to the collision.

Rehmat Kazmi

Numerade Educator

Problem 82

Figure $\mathrm{P} 10.82$ shows a vertical force applied tangentially to a uniform cylinder of weight $F_{E}$. The coefficient of static friction between the cylinder and all surfaces is $0.500 .$ The force $\overrightarrow{\mathbf{P}}$ is increased in magnitude until the cylinder begins to rotate. In terms of $F_{g}$, find the maximum force magnitude $P$ that can be applied without causing the cylinder to rotate. Suggestion: Show that both friction forces will be at their maximum values when the cylinder is on the verge of slipping.

Rehmat Kazmi

Numerade Educator

Problem 83

A solid sphere of mass $m$ and radius $r$ rolls without slipping along the track shown in Figure $\mathrm{P} 10.83$ It starts from rest with the lowest point of the sphere at height $h$ above the bottom of the loop of radius $R,$ much larger than $r$
(a) What is the minimum value of $h$ (in terms of $R$ ) such that the sphere completes the loop? (b) What are the force components on the sphere at the point $P$ if $h=3 R ?$

Rehmat Kazmi

Numerade Educator

Problem 84

A skateboarder with his board can be modeled as a particle of mass $76.0 \mathrm{kg}$, located at his center of mass, 0.500 $\mathrm{m}$ above the ground. As shown in Figure $\mathrm{P} 10.84$, the skateboarder starts from rest in a crouching position at one lip of a half-pipe (point ?). The half-pipe forms one half of a cylinder of radius $6.80 \mathrm{m}$ with its axis horizontal. On his descent, the skateboarder moves without friction and maintains his crouch so that his center of mass moves through one-quarter of a circle. (a) Find his speed at the bottom of the half-pipe (point ?). (b) Find his angular momentum lifting his center of gravity to $0.950 \mathrm{m}$ above the concrete (point ?). Explain why his angular momentum is constant in this maneuver, whereas the kinetic energy of his body is not constant. (d) Find his speed immediately after he stands up. (e) How much chemical energy in the skateboarder's legs was converted into mechanical energy in the skateboarder-Earth system when he stood up?

Rehmat Kazmi

Numerade Educator

Problem 85

When a gymnast performing on the rings executes the iron cross, he maintains the position at rest shown in Figure $\mathrm{P} 10.85$ a. In this maneuver, the gymnast's feet (not shown are off the floor. The primary muscles involved in supporting this position are the latissimus dorsi ("lats") and the pectoralis major ("pecs"). One of the rings exerts an upward force $\overrightarrow{\mathbf{F}}_{h}$ on a hand as shown in Figure $\mathrm{P} 10.85 \mathrm{b}$. The force $\overrightarrow{\mathbf{F}}_{\mathrm{s}}$ is exerted by the shoulder joint on the arm. The latissimus dorsi and pectoralis major muscles exert a total force $\overline{\mathbf{F}}_{m}$ on the arm. (a) Using the information in the figure, find the magnitude of the force $\overline{\mathbf{F}}_{m^{*}}$ (b) Suppose an athlete in training cannot perform the iron cross but can hold a position similar to the figure in which the arms make a $45^{\circ}$ angle with the horizontal rather than being horizontal. Why is this position easier for the athlete?

Rehmat Kazmi

Numerade Educator