A bicycle odometer (which measures distance travele)v)kzyzvzh)c r:pp6 87 -kcj od) is attached near the wheel hub and is designed for 27-inch wheels. What hap7z):6 k hk vp-zrzpcovy)j)8 cpens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a poiu5eqx2yq8h c6 5w d6jgr* pox3nt on the rim have raj ye6 r5xc*q8gh 5dxwqou632pdial 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 change?

Could a nonrigid body be described by a single value of th( q*tmeloa 17 jhl)ijof2,djyn mo()2e angular velocity $\omega$ Explain.

Can a small force ever exert a g* t(b6f+pgmkf6h vqiin:*u- greater torque 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 difficult to do a sit-up with your hands b* w41 quv1em:6ia sw:zk(py. dkqf/xx ehind your head than when your arms esqazui x (1xdy */k:w qvm p6w:.f41kare stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sproc4uxl+ 0.vqdr qkets at the rear wheel 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 gear is it harder to pedal, a small 0q.q+ldx 4vurfront sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast tasqp/.djy,k0.* dfh ffam s*gj+ k8ffq80o*have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On thekfg qf*0qfpyh ,* .ss 8j+/ktfd*jdfamoa08. basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long;alt/r9k4fx 5x ti g.g, narrow beam?

If the net force on a system is zero, is the net torque also zero?1 +acy7)dcqk k.3u fl:dzeo-rb ks,zm0n 7w w7 If the net torque on a system is zero, is the nonqr :+keu d ad7 73wl)7zcskck0 fm,y.w b-z1et force zero?

Two inclines have the same height but make different angles withd//v+zp z)7gugr1vs(g i:wny the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explaigd/ gnwrv(1:is) g+uyz7p z/vn.

Two solid spheres simultaneously start r*)*k;/n vtih kgkaa c/pc 4qim05ei0nolling (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 first? Which has the greater speed e/i kcgqc k);**pham nk/ai4nv50i t0there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder havdgvn xu7juq:hjm5p zk/) 3z5( e the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the5zn h/upd x7j3z5)qk vgj(u:m bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving savgqgc8j(7wxfl s +k6 0;n yv(bq6f1or rotating objects eventually slgb6( f1+f0kq wn;6qaycs v 7gj8lsv( xow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, ho 9wd23rz t:ot l5vcvce-ds9h5w would this affect 5 tdcd w5-che:vo9rlt3z 29s vthe length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial ro c/(c(,ilg p8u(xd;h2t)ikhilr b7pdtation when s ,h(pkx)2l(cti lcdb;8(gipr/ hu d7i he leaves the board?

The moment of inertia of a rotating solid disk about an axis throu zyo ym,y1td57jx4b*g gh its center of o, d4jy*1g tmb7yzyx5 mass 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 zl ikh- xr2 f iyv:3s,rbc8ldd702op 1holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your ang30- v8lrzbp 2ryh,d lisix okf2 c7d:1ular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howev3elex ln/d2fe2vpjtde )2e0* er, one is hollow and the other is solid. Describe an experiment to determine which e ne2d vflej*/e)x0dte23l2p is which.

The angular velocity f,i1 nxv1p t 4)tgvpi:of a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a poi itgv:fn)4vp1 i1t, pxnt on the top of the wheel? If the angular 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 edgeq;- p*5o : sonvjc2bqs of a large freely rotating turntable. What happens if you walk toward the center?q *bvojoqps scn;:5- 2

A shortstop may leap into the c7u c-9tf:yxh air to catch a ball and throw it quickly. As he throws the ball, the upper part of his bf7yt:c9 xc-hu ody rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservationtw50 ds u5h2gh of angular momentum, discuss why a helicopter must hawd 2 0uhsh5tg5ve more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radi i ipg eoxh1tiwuur/: o15 bu,dho.::3ans: (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 coincidence. Calculate, et/tisxrcdk 6akj+ ) e(8l;-d using the information inside the Front Cover, the angular diameters (in radians)sar-;)kj/ed 8it6 t d(c+klex of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 38qpc;ixy(or;5hx);7fhny :o u 0,000 km from Earth. The beam diverges at an angle 5nho)hfxi (yq y; pxo7;u;:cr $\theta$ (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 6500 rp) kc.fv+ t/yt6y)mt zkm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular /cf z .y)tt6vkyt+m k)acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chhkj08cm +/va ncmw) 7yild. If the ball makes 15.0 revolutions,m7ymhvac k)j+n0/ cw8 what is its diameter?    m

  A bicycle with tires 68 cm in diamiy6tm7vda 2zlm 4tm*4u2i1 cd,jyt:f 8xa.qbeter travels 8.0 km. How many revolutions do the wheels make? zya6c,*4xt 7qmtldbi. 2u fa4 8dm12 jmyitv:   $rev$

  (a) A grinding wheel 0.35 m in ,w)cby i )vr 6k4mizq8diameter rotates at 2500 rpm. Calculate its angular qcy8bi4) ) r6ivkwz m,velocity in $rad/s$ $\omega$ =    $rad/sec$
(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 makes one complete revolution in 4.0 s (Figp*.oi) .xji mm. 8–38). (a) Whato .mixpj.im)* is the linear speed of a child 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 veloekl2v03sbqy2 z/ rc(o city of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sp dh4rouwo)5r k*)d (sseed 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 ojkdl )bvw p/wi(w3) 2cm from the axis of rotation is to experience an accelebwod/ p )wkl3)2(jivw ration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerat 3 nvwl5z76.z y cnea2qmc1y(des uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Dec5ny cq6y7lwaz(.m3z d2e 1vntermine
(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 radiugh lb68n fj,r-s $R_1$ 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 .hednmdeq i/r)* -xb-aboard the Apollo spacecraft put themselves into a sl -)bq .r/e*idnxehm d-ow 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 8.5 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. Throughsd*t/1klo k( 2lg mve6 how many revolutions did it dl*o 6vklstm1g 2 k/e(turn in this time?    $rev$

  An automobile engine swd cs4bbu 74k1 tjs0ycxo)45nlows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(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 fl-mbr8-367 rj msnxfdwa9t.kaying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reacmdxbn68ma f.w7str r-j3k- a9hing 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 accelerates uniformly frowid .05z/gp:u vcm-i hm 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel hav pc-hz:uim/gi vd.05we traveled in this time?    m

  A cooling fan is turned off when it is runtd 2b: 5/y8lsx qjxj2t;zz 3ivning at 850rev/min It turns 1500 revolutions before itiy25:dxtsvq xj j/tb 2;38lz z 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 unif8.f5n 3xjhqk2p v4wj lormly from 95km/h to 45km/h Tk l hpfnj84xw5q3j2 v.he 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

  The tires of a car make 65 revolutions as the car reduces its speed uncn7 qmqejzj/n9 m4z9;iformly from 95km/h to 45km/h The tires have a diam;z7 n 9zq mqe9c4/nmjjeter 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 weig*4rprh5d0kvrh p fti5wk40doln)7v3 ht on each pedal when climbing a hill. The pedals rot7 r0d5 rtknv45*hpf3rd)wovhk 4pi0 late in a 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 ettymrt dq3l+m t59aq: /.sgz:nd of a door 74 cm wide. What is the magnitude of the torque if the force is exerted/trd :3. s 5lt:tz+mqygam9qt
(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. 8–39. Ash.(5gpbz4uen/owc r) t e. u9nsume that ab et)w pue9.(hngrnz u4 5c/o. friction 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 h/7-j0fyshj :4yox) hl q7qbi which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude an7i: xybq 7j4y shjq)-holf0/hd direction of the net torque on this system.

  The bolts on the cylinder head of an engine qvz.41n)m6w5 f up zzurequire tightening to a torque of 38 v.w 6z n qfp1m5uzz)4u$m \cdot N$ 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 of radius 0.6(kcqo5 a h *qlps9*tu-48 m when the axis of rotation is through its ca *ksuq q* (ptoc9-h5lenter.    $kg \cdot m^2$

  Calculate the moment of inermj 75j,zeu alp6 qx c2n)d0ji 2)b4laztia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?)bxli0lcqn2uj5 z6zp)a ad4ej)7m2 ,j.    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated ihk +k/ .(4sj qelz0oieh.k/ fb2f jqk4n a horizontal circle of radius 1.2 m. kk2k4 ilzjj. k(ef/ /q+o bh0 hsqfe4.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 rotating at co4ha:; qiuo- oxnstant angular speed (Fig. 8–42). The friction force between her handsquxaoi:h-;o 4 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 inertia of the array of point objects shown in Fig. 8–415l18n;v zr hzk uoby53 abo51z5yn;hr ovl z 18kbuut
(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 two oxygen atoms who.kiez s1d fl03se 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 cylindrical satellite spinning at the correct rate, ek 47dqxcx; 6r9qshj0cc h lc/6ngineers 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, whach0 d9ck/xq rxs4j6lccq 67;h t is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mas* 4uj;zo: o. mofxxzg(s of 0.580 kg:xo;j*mz4x u(o.f ogz. Calculate
(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 axei2 6gp lvc(, bat, accelerating 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 small hand-drive+/;h n2a; pqdkiqv8z cn merry-go-round and is able to accelerate it h qkc v;pa8qz ;n/di2+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 frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating ath6 tz-zzk 9zl3-5 qi1 g6gmkgf 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque o65zg-zlk f1g k6ithzg3-zm9qf 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. 8–45 accelerate*dyob1 ph ,.mhs 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 thrown solely by the action of eptw xk7 :j2h(the forearm, which rotates about the elbow joint under the action of 2 :xe7ktpwj(h the triceps muscle, 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 lonjn b:(hbg iqzf),):9e 7jcl axg thin rod, as shown in Fig. 8–4xzea7bn:( jfgh:il ,)jcb) q96.
(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 two massesowozp: 6i+epkj:8-l w, $m_1$ 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 accelerates the hammer from rest within four full xsnk)q s8*7z/bu. wh vturns (revolutions) and releases it at a speed of 287s) ukh/xz n*w8v .sbq $m/s$ 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 h 5uril 0q8,aj)mc f*+xtf0pqr as a 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 develop j4s3dzsbrs 3+s a torque of 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 ro*nl)o0ar9quc lls without slipping down a lane at 3 lnaq0 c)ur9o*.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to 8tgm )ie4aoif2lq.* vlh ,2 oethe Sun as the sum of two terti, e a2l*84e q)gilmhvo .of2ms,
(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 of 1640 kg and a radius of 7.50 m. How much net wo7 kn8t.x.d- 76 ozdhs0dqpqy srk is required to acceleraytkds qhs. -7qd06 ox7pzd.8 nte it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from resucg;4 hx7g4dbh j2g, tt and rolls without slipping dowxu4 gj ;h g,gct427dhbn a 30.0 $^{\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 vert y*f2w4n8iv7kz (skfd ically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it n4v28(kyzdw 7 *kf sfihits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of v yzq82u(fg4 ja thin string in a fqv8y4 ujg (2zcircle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical gr wj,x:d 73xwdhinding wheel of radd3hwxj :,d7xw ius 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 is rotating at a r0g)( cm fiwl/gate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decrea0l/ cgim)(f gwses 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) cano )zd,3tf6i)b6fd fq gbc40o tr dls;* reduce her moment of inertia by a factor of about 3.5 when changing from the straight positisl)ff ftdr *6, 0ob b4g )c3i;zodt6dqon to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck 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 rate of s2 b fc5fy/o7s6ckd0 l1.0 rev every 2.0 s to a final rate ofsb5 s6y/ c2kf0cf do7l 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 vertica birmad7mw9n 9- )ieg-evn 2;tl 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 then 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-mn)i; a9egibtw ve-m27n9r d frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure ..h, 0czpa.heg 7b k9yhuq +ffskater 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 momentutkjrr zf7u8y3s )i6)t m of the Earth
(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 inertia I is dropped omsaq.2c wnb)(54 jfvznto an identical4 wfn(5vz)amq.sbc 2j disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns 3hknsam v( + k.y b1ar3r.zrqnvy:-;g at 2.4 $rev/s$ 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 cen lxz (,x;vwab1ter of a rotating merry-go-round platform of ralb1,zx(; v xawdius 3.0 m and moment of inertia 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-go-round is rotating freely with an angul0btiy8u687w ( awnfop2e m*x var velocity of 0. t 6bumx8( f2vo* e7wynawp0i88 $rad/s$ 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 dwarf, losing ab.f6cs2s m 7x/o9gmt drout half its mass in the process, and winding up with a radius .6mdfo/ c 92gmtsrs7x 1.0% of its existing radius. Assuming the lost mass carries 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 excess of osyj;gwk 780u120 $km/h$ 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 *(5kde8u(fph qkjx 3e $ 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 platform, initially at rest, that )pj.pbgr6(h x can rotate freely without friction. The moment of inertia of the persbg h.(rjp6 x)pon 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 the edge a;5 qndzjerc/.vg xn6rl 2 tzpaj6 7,1of a 6.5-m diameter merry-go-round turntable that is mounted on nxqp5.r ja;6l7jvza2t1,/czge6rd n frictionless bearings and has 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 with the enych b..z(5 ,u:m) hde tbxr,bud of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, hol.e 5u.zhx(r y,:bbbthcd u),mding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that7umfkbik4vf 1 8.1ibx the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the1viu b74fkfm8. x kib1 latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates /ho5gr;cxwd-k f2h, bfrom rest at a rate of 1 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 uniform cylinder of 3g7q83dv yktw radius 0.20 m acquires a rotatig ydt3qk73vw8 onal rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disknh + ycg9h4c-bs, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long s g-+ 4hhnc9bcytring, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the aufg qms17:a 3;y 1x8sbiqa w+ungular speed of the rear wheel ($\omega_R$) 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(loe2b,i-+p*yce a3o 4lpp zw of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revoluticpil43opyoe*2 al ,z(pew-+bon 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 lz r-h0;g q(qmuow-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diamezgqm( ;u 0h-qrter 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 an /z3/dwn4: auw(uep k 8u g,ujb$H_2O$ 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 horizontal surface atc*lj.n6.5 tuvc(l i2ee*y cho 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 length L can pivot freely )t .1dvtnmkh/ic3k-t (i.e., we ignore friction) about a hinge attached.th /kv3 tnki-t)md 1c 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 radius R. It is standing vertically on the floor, a/sh6h(4ynk4k zd d(uji3pqj *)gk qy2 nd we wanjd3jh z /(kq6u2s(dpg*k4yhki qn4y )t 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, where h

  A bicyclist traveling with speed v=4.2m/s6ca a(b5 ux z n2)zh2is,ogh1r on a flat road is making a turn with a radius Thz(ug6ox2b,25ah 1 r )ca hzsine 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 of length 1.5 m i:u:rit /l f-2o1l.m ozag nk3and begins whirling the rock in a nearly horizontal circle above his hnagf zi3.lr -l2mku1o t: :oi/ead, 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 ar/0v yrvukfc1 goo9:/ ims as thin rods, making reasonable estimates for theokuyi vg1/f/: r0 co9v dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylyty85n- a ko:sindrical plates, of kns 8a-ty5oy :mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 r 4; p hm8f*y0hxoawei2olls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if2hea08px*yf w;ihom4 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 (.whozels 7my i4 v,uecmqm19 e60hv3do not assume $r\ll R$ h =    (R-r)

  The tires of a car ma*jrsy vwt7c 3zi9* 8lske 85 revolutions as the car reduces its speed uniformly from 90km/h cy8 9izlsjt3 *rvw7s*to 60km/h 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