A bicycle odometer (which measures distano4ns; +ef cqrchok-r.2-p1soce traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on asp1h2 erq-rnoo+4 os-fk c.;c bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular ve1dv5qoc + .,azthh00huigcd) locity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration ch5hhcdtd1 h ,cg0)vq.+a0zuoi ange?

Could a nonrigid body be described by a ma4yviq/nycw ;jv:9 tc: kn1(single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater toryerz;lj 6xk)*gy x,y(que than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficulc-69(v v)wf xlr xy,hlt to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer c-y wxl,v()vlf h6xr9this.

A 21-speed bicycle has seven sprockets at the rear whypc6 zy;sg 4l1eel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gears 41;cpy g6ylz is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have si)quoau- ffg 8zn) y);lender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of massgqinaof) z ;8fyu-u) ) is advantageous.

Why do tightrope wal*c+h k+; u,jrhbv6d jvkers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zed f hsny9 yee5n8xbpj;9r, t37ro? If the net torque on a system is zero, i9sb9rn xj8yynp, 7tedfh3 5 e;s the net force zero?

Two inclines have the same height but make diffe9)z.wu ai /rphrent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of9rzu.whp /ai) the ball at the bottom be greater? Explain.

Two solid spheres simulta 2af: .c0qqxyrwn1y; wq1iuk5neously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first5yn02qafuq.c: ;kw xyiq11wr ? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start f+dnr azu9q7of1wn-/) tqor xc4 8jg+prom rest at the top of an incline. Which reacunpx+81 wq-acgn+t4r fqro9z 7) do/ jhes the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum aifs2;qn5n r+vnd angular momentum are conserved. Yet most moving or rotating objects eventually slow down and sts5i2f;+nnrqv op. Explain.

If there were a great migration of peopq1w*29gazz 4lpb w 9 +pblfr6 23cjxvple toward the Earth’s equator, how would this affect the length v6b3pz q4+lj1pxwc2z*pawf 9 lrbg92of the day?

Can the diver of Fig. 8–29 do a somersault wipi ucair6tgu,pi8j+ tk4;8;cw3s3 huthout having any initial rotation when she leaves the bosi8uapuru ;;4p6 3c jiwc3+gt8t kh,iard?

The moment of inertia of a rotating solid disk about an axis thr n/ p1:d8u 7qqdrurk4aough its center of mass:/ukq8q nd1p7 4ru rda is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg m6j9 jumdv 317 0j*cqunf ys)ymass in each outstretched hand. If you suddenly drop the masses, willf0ydn6 1s mjcuj7uqjyvm *9 )3 your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and tyruva/ddr: ;: he other is solid. Describe an r :;/d rv:uyadexperiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In wh-8p j ms(m)ko uat q30x;ncu4nat direction is the linear velocity of a point on the top of the wheel? If the anguu t mn)op3 s0;mc-qu4(8jnaxklar acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edgexz1tu68ud :c y eb*zu* of a large freely rotating turntable. What happens if you walk towardt*x duuuc 8e 6z:1yb*z the center?

A shortstop may leap into the7 z.6u1 q6mvlc1t ilep air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look qultp lie.7vz c6q6m 11uickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss w+g dxm;md qd67bvcedg0r+ s03 hy a helicopter must have more than one rotor (or propeller). Discuss one or more ways the secondxdbqdsd6g3m0c 0r+ 7 + ;vegdm propeller can operate to keep the helicopter stable.

  Express the following anglesoc;wp5 2 *mvd5fcz-h n in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coiby ivi e50o s98e:/hsk ak(les+;4fpmncidence. Calculate, using the information insid+y(s4pvbf is;lk/h5e o0i8k: mea9ese the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diver 6 k*zzv5;sogjges at anjs6k *z5vzg; o angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500s 3gl;,ww/mrl rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the bladw/llwr;m sg3, es slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to uw/ nxk2 xo1g rsl*9 lgr*4qfdr1;s7xanother child. If the ball makes 15.0 revolutions, what is its diamelrw9of 17xx / kgqxlgnsr*rud1;s2 4 *ter?    m

  A bicycle with tires 68 cmsdy1q;wic/t6yy 0 z 11wz+ahj t7 gh)s in diameter travels 8.0 km. How many revolutions do the wheel7/s yjy1z1ay th1cdg0;6qwh tzsw ) +is make?    $rev$

  (a) A grinding wheel 0.35 m in di1:p kz+sgf:qunkz 1mu5 ;x yg5ameter rotates at 2500 rpm. Calculate its angular velocity m:gqzxzy kk5n5 : g+;11sfpuuin $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makesf 010ze.npbvad gwle3/b07d,:z cz n v one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a chi0c0 /7beng n zp .:3dzevvd,wafz1lb0ld seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit aroub2 oooat9254tqk7 nle)znzd6 ovx4a.nd the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sp5d 1muq+y2 ngveed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 n+8 uk ltzaw):kd-(ldcm from the axis of ro8-nl (dtazdwk k)+ul :tation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from; rj +-+hkcmekbuen1qhnw480 130 rpm to 280 rpm in 4.0 s. Determinjku8e+mrh 1 wbn-k qc4h+0e;ne
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiuafv, p. .r+rf,b6 grkos $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts abofb;z* g3cz1ov ard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.1z *vbfc3;goz5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Throug w )4q--c)(u y7y )aqof5wbemtufad9dv:bp:ch how many revolutions did it turn indw:e )5 oqudvybq)bc)7au-cm4 - fy: tw9(paf this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculd94m1k jsd, omate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying q, ud9(2i -bib cuia*ohighspeed jets in a whirling “human cent2bcbq9(* i,u -iudoa irifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter acceleem g20xj7 wb/nrates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have j7weng0 b2 x/mtraveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min )s cij.jt-fb;It turns 1500 revolutions )js;tf.cj- ib before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly fjf() hiabl/. n o0yi/from 95km/h to 45km/h The tires havelfoh.i)/ f/0(ji nyab a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the ca ssr5g.f6o/zomu q i-p92kpy/r reduces its speed uniformly from6m.p2 /uozqk5 s9i-pysrf /og 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weikjh xpselx,1 - fe.2g 3z.r3haght on each pedal when climbing a hill. The pedals rotate in -rxkz,.h 3.3g2e1phxasfel ja circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 7 i)(lnt:(.znqpp.8u onqx5 jh4 cm wide. What is the magnitude of the torque if the forcunl:((.5h z xpij)qpn8oq.t ne is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39g*wo):kk:o 4yv /cend. Assume that a fr4:w )nky*oe g:c/kdoviction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod which pi u8)t nqm1yvee.ojbf*zij (+.vots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude ymu q*neo tie.8)jbfvzj. +(1 and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require wwhv)2 pdqt111k llw+ tightening to a torque of 38 l1qthd1 w+21wpk)wvl $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere o gkwh xbmq7i59l-k5w; f radius 0.648 m when the axis of rotation is through im hxgk5wkbqlw ;597-its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle w3xx f6fn7z-zo- ,xzdzheel 66.7 cm in diameter. The rim and tire have a combined mas zoz xxf6-f-zz3x,7dn s of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of as7x8 -)og xia j:ptur6 thin, light rod is rotated in a horizontal circle of radius-ou7:jp 6xa)8 sgxirt 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotatinge3+gzaqiw9c qd vqv3/15i +0ssf1kxr at constant angular speed (Fig. 8–42). The friction force between her kvxaqc5q+0z g3fw+ ssr1di /q9v13 ie hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inery1 h z6 2ashlgs hn)k8b,z;6xhtia of the array of point objects shown in Fig. 8–43 abou, hz b6 8n2sg6;xhlzyh) 1khast
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of td+jk0 5ghk,yjq )7 colwo oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cyli bg6jp *rsi3/mndrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satej*i 6pbmg3r s/llite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylwkaa(1vgle )2 t:k- vpinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculateklaevt2)k p:g( v a-1w
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerus2 u *)x(fxhrating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a sma i u s2o;se4j3/mu-zpfedbq*, ll hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting fzm- q ;,psejoe2sd/ui u34*b frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually bxldal(tzu :w48nx+ b7rought uniformly to re4td nlx8a:xl+ w7u( bzst by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–1wmv4qk5 ov( : sp;usq45 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thro5e r:2ueli3d uwn solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscll35:e die2urue, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig.b nsa 9a(/(o,gnxg eg2 8–4go(s 2eg,/a nnb(9axg 6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of tws jt;gig ek07d5ub bmx lq6-dyo/*:.ko masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerateswe*w0 ; zre8- 9 txj6zhqyt0vb the hammer from rest within four full turns (revolutions) and releases it atrxwhwbte8ze;0 6jz 0t*yv -q 9 a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a 24/ freo/uhpyjc1m7u+krs8 m moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of; +cuzn31nrdcw(c x :d 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls witgi3 ch*wd9ti/hout slipping down a lane ith9cdg/ i3 w*at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of t h0cxhh(rure 6.3 lmb29b+ zzuwo terms,hhlur.(+3cm6z9 r hb zux2eb 0
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass ofmtzql-hy43ak-kao(0qg5 2ox 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solidk 0l-a3 xzyo2gq5-qh4to a(km cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 .k;sm,c/ y4ekmfsp5 sl,gb(ykg starts from rest and rolls without slipping down a 30.0kep bfy/mgs m ;, yk5c(4sls., $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall v3cl b0ck ojih l5v79d(+gsc 9and its lower end does not slkvch7 i csc9+ 53bov0l9gl (jdip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of k.n (e0j,qi i 5m0jcrga 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speen(ri 0,gjemjq kic.50d of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding3 lyrk9si)fu( wheel of radiuuk3) rsl9fyi (s 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that i wf4jfv: xg5(vn3s- huq rwdd*m+2qn7 s rotating at a rate of rfm23nds*7h-+ : vn4 w5xj(gquqd vfw1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce herd5-7k6y4+ao j jmx my gr;aix0 moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tu+doa;j7gx56xaj k4iyr -m0m yck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial knvz x wb;h*)sb/jt8 .rate of 1.0 rev ever /txv8b)bs. nh z;*wkjy 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a ve26dg 0rrl3u.b oj6 + pavpm ,ucu*rh:qrtical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter p : l32.jvpbrq06ur6u,d+ m*u arh ocgthen throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure sm*wiwtn +q+bsx9; a y:kater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Earth-i9sx;s r2xjl
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inerti:(ew *wgxw9u da I is dropped onto an identical disk rotating at an9w:u(x wd*gw egular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at t zi4*ecm86rmvg.7m1 x) vjxo2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go4 nwert93bw/ ums nhsu9f, 5m7-round platform of radius 3.0 m and moment of inertimwet9b 3 wuhf4,m /5u7srnns9a 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-goqv-d1hup n.u o3 ,tlg+-round is rotating freely with an angular velocity of 0.u3 u.v -hntpg 1l,dq+o8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white hmyu *,)+h 2rrn w(hfvl1rzv* dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carriehvvyf1n+(hw* rhu r)2z *mlr,s away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in exce qjnm+f)w;yhjl;8d2 1rf wg*uss of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass , */kwd cj5uiaj qnm5r/9o9oc$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platf b-vu0f uphl019ccnmaz8 ;q1lorm, initially at rest, that can rotate freely without friction. The moment of ine ul81 nz0;-1pcbuclfqm 9avh0rtia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at o:f.( /cu j;c5 qxfvazthe edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and h(q5fjc:o v;xa fuc./ zas a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground withf- s:l0f+ dj7zw7j t(s4ywpbh the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the sw 47j pd(-0 +s ylbthj:sf7zfwpool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earthd-v /d gb.gd r4u14il+m7i yfe. Determine the ratio of the Moon’s spin angular moment/f.ud1 i gd4i-eb mv74ld yr+gum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1-xor+ 3.8yq 9wecb dygkif y:+ m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uiw/mhfe)kvyhl8m .)61uayt3 niform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at conv k3y/ 1hmwa.)l ) hmueify68tstant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050gc(j ut b4* 8begxtt58v99gc1xdm l1v kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Uvtb4je9m5 x8t(1cu1c gggv*9 td8l bx se conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheel2/ium -4.zltvudsso g ,/n1ao ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun,ekffsy 2o:u6 + but with mass 8.0 times as great, were rotating at a spe fof6uk: 2+yseed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution aut .4g,k lhh0e hmu2,+isgtsgx*omobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg,h mu g, esglxt40.2gis* h,h+k uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates anibjn 21hrb9: zl l1/rw/ mjs2r $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizo wwczlatiw4v+x7z 4u +pu(t36vqi:+ ,ndj te3 ntal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and lend 2b f4ary,c4gc(s(mogth L can pivot freely (i.e., we ignore friction) about4(cs(myrb42 ,ogcfda a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radiu 0jgr7l1glel62 jl 2cqx3o; cqs R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, jlq l0g3127c6;2jroc gqelx lwhere h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a0+s7:p ak-jcwzb;gv;ufj6:wdp qmc*i oqd79 turn with aq:9-vpmwaj c;o: +cbzpdg7skwu*q ; jd 07f6i radius The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling ob 5ux2l ncc-1u4 qik0lf length 1.5 m and begins whirling the rock in a nex lnkuc15 lq0u2-c i4barly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, makw.u 5)mhlqirn : z6)na(fwr/ting reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arwr/m). f:)z n5iqrla(6wtn u hms, and with arms held tightly against her body.   

  You are designing a clutch asgx3j3zix.;4bmhhp5 x sembly which consists of two cylindrical plates, of xh33 g; zpx5bxmi. jh4mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. 8–dspz9i.kvd.gjv 9 rd p4ycw)ati 1v/f,*e/ 0 x58. What is the minimum value of the vertic4*d epr0v9dizs,t.xj c fg pawk/iv).9/ v1ydal height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not a z,ke-t:aoh kp 3.:iavgek)aqw9s)e-ssume $r\ll R$ h =    (R-r)

  The tires of a car make 85*j1h cttpw1r :j iy,;b 2yzv6r revolutions as the car reduces its speed uniformly from 90km/h to 60km/h1z1*6r 2rh jct: ij,vywt;ybp The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s