A bicycle odometer (which /o,h5gy j mdnxm xb1 vo-w.+3omeasures distance traveled) is attached near the wheel hub and is jowoboh1gx .mmx y+nd- /3v5,designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rixn(znd ,x*r) j:jo 0imm have radial and/or nz rx oxm*,) ij0n(j:dtangential 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 bd+lj96xv3 bqrb 7 nwfn08 *j awrzm6q6y a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a largere(2 jmhllsrsqjl-s:4z7 bn+6. 86x zdbi fe.u 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 behind your head than whc a (u2dyf9,bbsb kmz7398 zvxen your arm(v3,fxzcabu sm 7z ykb92 9bd8s are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at th lg v;aa*vcqy uf2 ,x)vqayqs3)3ko6*e pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is )a6qgcqq s,k*)2 3vfau vax yv;ol*3yit harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with fleshe3q5(3ni qolqici04 qxp )x )o and muscle concentrated high, clos3q l)n)oq34ci( eoqx xiq p0i5e to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narr nqtxtvrc 7) 2xfg-tx sb9,/p:ye z;1yow beam?

If the net force on a system is zero, is t* n bpzy1n8 xkij.s0a.he net torque also zero? If the net torque on a system is zeroi1jpna.*. 8z0xksnby, is the net force zero?

Two inclines have the same height but make different angles with the horizontal. 4 9dauf.qpvmqhe,+ 5y The same steel ball is rolled down eac,um 9qhq5e y4dfvp+. ah incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneouvtipo -)5)exe6z4p4 pq a+moasly 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 first? Which has the greater speed there? Which has the greater total kinetic energy at t5pi+- )x4pz e tae4aq6v opm)ohe bottom?

A sphere and a cylinder have the same radir 5.qdrkowv;z 8w/ t9x2gah3cue;:k g us and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater sk; ;d8e9 vzr 5chwo3uqrak/gxt2 w.g :peed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conservlonc(w;)9u vxr1kk o*a (iymc3e4;ek ed. Yet most moving or rota)a(kc9v14e uw;yr3oi xk ke;c* (mnloting objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wo5(-uygl8 nsip3 pbp;c*urx g l6a 9h5culd thgnlpxb cp36 lcryhp-usaig(u8; 5 *95is affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any inich0wf ,uq;an:x j 7ef00esn z)w(xhi3tial rotation when she leaves the boanh7 wf;0 ih0),zc f:wus(3 exnjx q0eard?

The moment of inertia of a rotating solid disk about an1zv*j6kpr8y f axis through its center of mass is yk*p vr81fz6j $\frac{1}{2}WR^2$ (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 s q.x1wu6twhe-z t:l5dtool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explai l6xu-t.:15 wqzw dthen.

Two spheres look identical and have the same mass. zmpz6q6( swe(However, one is hollow and the other is solid. Describe an experiment to determine which is whicm z6e6((pszwq h.

The angular velocity of aiqu73r hq0v 2o wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point 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 de037 q2urqi hvocreasing?

Suppose you are standing on the edge of a large freely rots* 1derc)tn-m:whf k:; /gs(imjtvijg, i6o- ating turntable. What happens if you walk toward the cent*gwr:1 -mkd so h) c,j/nvtiif(imsg6j ;t-e: er?

A shortstop may leap +bsl (3o x)/lvp*tf 8ksd:k ikinto the air to catch a ball and throw it quickly. As he throws the bkksos8ixvlp b+)k *(tdl3 /:f all, the upper part of his body 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 conservation of angular momentum, disc( ( hbt4d3:q d kv1jy0iepf cx)4o*tuuuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep th(t :uf13u)x* 4dbotdcqk4iphye 0( jve helicopter stable.

  Express the following angles in radt mgk. 6ph,u/nhi*,ztians: (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. Calculaij(f7g/w* bx+l ,oh ybte, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen y igw x/,l*o+jhf 7bb(on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed6lx1k hzqb7+h gb7w9s fcj;3 c at the Moon, 380,000 km from Earth. The beam diverges at an angle 3bjhfz c q67+xwskb1h ;7l g9c$\theta$ (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 rpm. When the moto (wo9(yyafx2 .* x+uhtsrv,j br is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration aswyx(*, u2a+bf9oh y.rstx j( v the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball mako4bbs2l dwlz)8qzu9p:)33plqvr lg: es 15.0 revolutions, what is its diamw: gu )3zb) zl ll3pp:do98bq4 lqs2vreter?    m

  A bicycle with tires l+t0e :dm : qsx;vq6xf68 cm in diameter travels 8.0 km. How many revolutions do thde +;0x:qvq6ls tm xf:e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 25k sie;4/m;ear 00 rpm. Calculate its angulareaksi; er4/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 (Fig. 8–37n fg)9 lyb1sf8). (a) What is the linear speed of a child seated 1.2 m from the ce7lfsfg b)1n9 ynter?    $m/s$
(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 o t ;hia,ua zo4tnhctk87:b 31zrbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of/csz3q1guwdnq**,olh cs3 p 3 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 rot6k e nf:xh/kvfzjl)5-ate if a particle 7.0 cm from the axis of rotation is to experience an a 56lhkfevk-f:zj )x n/cceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about i-d82j1 vk 9oydh *xl(dmhk0)y. vwo scd1s8ibts center from 130 rpm to 280 rpm in 4.0 s. Determine(lyyhbd)s c 9sm-20 i hodd.vdw8o*8kkxv1j1
(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 radius;o/oz (acus:2gt wx .nj am,7k $R_1$ 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 aboardg9bsg a (5t3uwb) 9zt)*moa kd 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 min 3aa)(wg*o9 9 b)k5uttzgmdbs ute 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 uniforg;zij yn;6+8w.wp9 05ovartpv)hs ot/s.r a u mly from rest to 15,000 rpm in 220 s. Through how many revow0t s;rs;p yvz ja59 8go)ptu.w o r.+6/inhvalutions did it turn in this time?    $rev$

  An automobile engine slows down zlu1c5,7 4j-eja 1 e-h fzyx 9xolc0ujfrom 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 strilk pu( tko74v8h ,mbwc/o1)r esses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min 8hr(l1)/vt4 wcbiko7m pkuo, 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 acceleraten3kd1bkc gqz+vy5zic9u 3 4 gzw6 tv98s uniformly from 240 rpm to 360 rpm in 6.5 s. Hi +g3 64k kcz9 gy1vt8 dnvz9qb3czw5uow far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850relxwl1t+x h.mzf ; 6a0nv/min It turns 1500 revolutions befonlt1l x zw0x.;m+6h fare 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 ca -4 k,pan12 f7jaqcqzqr reduces its speed uniformly from 9,-4n cpkqqa2faq 1zj75km/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

  The tires of a car make 65 revolutions as the car reducyxkl(pip67i ls6-mjem v q0(/ es its speed uniformly from 95km/h to 45km/h The tires have a diamv -yps07 x(pl (q6ij /6miklemeter 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 weight on each pedal when climbt) g-j 6qswqw ;e50sc6dsy2sbing a hill. The pedals rotate isybjecs6 w;) g2 6wt5q0sqds-n 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 end of a door 74 cm m(36kyq0ox e qwide. What is the magni0qy 3me (okqx6tude of the torque if the force 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 axlrp*3 i:s fe:tke of the wheel shown in Fig. 8–39. Assume that te:*pf:3 ikrsa 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 attachb.;jse.n1 gexta fr r;i) ,:fled to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially ti:lg)b; ;rxa,e.s rjf te.1fnhe rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engiv,96qo6mv 06s zcvza 6b )(tubh 4kchju; mv+fne require tightening to a torque of 38 vfo,6c9bv( 6 sm 4hcz;0jkuq ) uvm zhva+6bt6$m \cdot N$ 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 0.648 m when the ai8t 0c( e ld7bizt;h4gxis of rot hg07e cidbi;tz4(lt8ation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 z0037g2nvt,e myctwt 9 xu5y mcm in diameter. The rim and tire have a combx3 0 ttymg0 m,vzeyc257uwtn9 ined mass 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 ixg5/ csr/ y6trbve7: a thin, light rod is rotated in a horizontal y :b7 i grs/6/5vxectrcircle of radius 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 dl:qvregjfj -j7z42s s0df+8e +l hk 0potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N tota-70j+sjd 2+ g8e qf0z levfldr4ksh :jl.
(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 *c340pmrnv x73ie 1gldnq.dz fc s1q5inertia of the array of point objects shown in Fig. 8–43 about f5qd n *ev1 dn xlr7g43zc 1qpic0s.m3
(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 , onj0fd/tx6.6sa ompe9bcp2two 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 cylindrical satellite spinning at the correct rate, eng6bx ot6)lh mo ;sx6dh)ineers fire four tangentiadhoh)x ot;s 66b6lm )xl 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 satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a unif6x(kd lpqq9qa6(k e7 borm cylinder with a radius of 8.50 cm and a mass of 0.580 e 79qb(6 q6(kkadxqlpkg. 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 a b (8abf w q+fcer;/znh6at, 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 tangentiallym j(d8ikeab+; je:8y c on a small hand-driven merry-go-round and is able to accelerate it from rest to a:ea mkbd8ec+( ;jj 8iy 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 at 10,300 rpm i dejfj9(cmp.b(j2.jj cjeq );s shut off and is eventually brought uniformly to rest by a friccdj2m j;eejb. ( .)jp(9cqjjftional 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. 8–45 accelerates a 3.6-kg ball at 7tma(knk91/rt $m/s^2$ 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 xt,c8 as( elpt)lh4+r ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerl +hx) (e4t8s at,pclrated 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.f*sd461 ,6iwuzd vqvm 8–46. w *6mu ,fd1i svzv6d4q
(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 co(h 1 )n:swfvi;jrlq2 onsists of two 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 accelerates the hammer from rest within fouvks 7u/tr d92j kv7zy0r full turns (revolutions) and releaktskd y2v790uj 7 /rvzses it at 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 50 ,5iw lktasrmoment 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 develo vkj.gav27:wrwx n: j8ps 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 rolls without slip45bc pfkp13/zp(m)ncton) wo1gr ayv) e9 w( gping down a lane at /)cz(wtgcwy pv o5gakmo n91frb3( e41p)p) n 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth withu z8 f;rqib)t9 respect to the Sun as the sum of two te ;i bqt9fur)8zrms,
(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 oq1+fpfdq t4 ltg in 3;-kqa04lx sz;7pf 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it fromgzq7-k;4ttfqla 1 il sx3p+df04 ; qpn 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 stp v:oi 0cm- ,kf0oux1sarts from rest and rolls without slipping down a ouipo0 c k,-0xms1vf: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 bam*laar nu ;g zxpxc39i9:q.6glanced vertically 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 hits the ground? [Hint: Use conser 3aaxig:.gxznm;r9 c9l 6upq*vation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end be:vgj-9 hf a u)qr21 ys 7n,j9vfdg:xof a thin string in a circle of radius 1.10 g1uyn7 a- )qe9:jxb9gs:vr d2jfvh ,fm 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 grinding wo1 5vf-ue/t u8)v+k6fhyh qu lf ;hc/xheel of radius 18 cm whe6qc-t /fe/lokf8y;1hvu +hhuu x)vf5n 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 rate of 33txx iev5rhg/tfw 3/ 50ogcb 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotatobv3cxe /ggxr03w 53thft/i 5ion decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shsd*mb00)abc i 2w3d 4lu8 xjtbown in Fig. 8–29) can reduce her 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 tuck positioul420as) mbdx8cwbbi0dt 3j *n, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation r m+1 cce hqj,lxa yj*cy2,xl01ate from an initial rate of 1.0 rev ev lym1e acych l1qc,j 2,jx*0+xery 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 vertical axis through its center at a freqv.7v)eb1 j9 jg0mg9ri bdl l:euency of 1.5rev/s The wheel can be considerg7)0vb91:eel.ijvj9 bgdm lred 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 frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure qsa6/c a.j8rg)j+*q fima fucy n62c:skater 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 7ca uks8 ubno2 3/du.5syec0d 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 onto an iden e a,x1-dva1/sgk r9zitical disk rotating a 1r-zie1g/9a kd,x avst angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ahu)2vfc6ijj izzim4 8f5 6cj: t 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 center of a rotatin3uutns m:z+();dx asmiao : ,qg merry-go-round platform of radius 3.0 m and mq:nam;,z:mosu3 txdi)+aus( oment 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 roo,1c qz6h(r nt)3fu yvtating freely with an angular velocity of 0.8 )ovyf(z3h rtu,6 1 nqc$rad/s$ 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 eventusi .ly37,dhvially collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its e3his i.,v7dly xisting 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 o iw5fwcug0 c f,:6vn,gf 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 srn2x8t pe kl/vm2 *v(op n.o+$ 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 .r7l3 .eovvwnplatform, initially at rest, that can rotate freely without friction. The momen .3lwve.o rvn7t of inertia 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 the edge of a 6.5-m diw 2vaq58cjfs 1ameter merry-go-round turntable that is mounted 25c1wvs8fjqaon 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 end of the rope lying on foo6)/4 dao7 hzpk.znthe 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 spool? How far)faokodzp7h6o. n /z4 does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always face/e83;tct jtc gs the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In cj 3/tgttc e8;the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fromol iy-z3k;fe0fa) -hb 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 6g2pl7h x 5c2qusjh1ashape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivpqxa h6h2slug5j 1c27ered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid v-usa- ; oh1wxcylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of m1vah-wo ;s x-uass 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 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 a5z xnsf25 o0vangular 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 of our Sun, but with mass 8.0ho -zun0 :;ypk times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarte:pu; h0-kyozn rs 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-pollutioef9uwozc;xv.d 6 gqav yar(52a/r* 9en 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 diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needir dgveyxfaa qc6.uv(z 92 r/e5o9a*;w ng 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 )vs(- s2:hjyr 3tqzy u,ml9gpv-,jqian $H_2O$ 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 at speed egk etc.6z(m nj d;9tvpn:s::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 masl40xxw4 : zvw j9)zbfvs M and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, :)v bxwv40lzzjx4 fw9 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 standik4tz;2bym: hapyb39e ng vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against whze ; 2 yktb4:yapb93hmich it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on hiuf7) :dbg4ya flat road is making a turn with a radius The forcdhgbi f47y) :ues 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 and8pmauwi:av 8u4a ju3(sm, 6md begins whirling the rock in a nearly 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 thws 3mapadm8m 6( jiva84uu:u,is feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her ar7: fvjq 0munc-0: ckdwms as thin rods, making reasonable estima ::muf70cj q nw-d0vkctes for the 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-. iox6q:o5 pa,ilomdih 6bp7 consists of two cylindrical plates, of ,6i-xpmpq h5do7o6i. :a b oilmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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)7 gai vc)hgf+ rough track of Fig. 8–58. What is the minimum value o+h)gc 7fi )vgaf the vertical 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 nxi)x.ef18 rwq d/:g swot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car 8bintn0h*ip tv w kf91 3tj86mreduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? itf1w9h*n t 08btmv 8kji3np6 $\approx$    $rad/s^2$(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