A bicycle odometer (which measures distance traveled) is attached j/bsnfrl e*- 6 eaps63wr3q t .0fn3ygnear the wheel hub and is designed for 27-inch wheels. What happens if you use it r3l te 0asnb f3p*g3f jrn- y.6q/6sewon a bicycle with 24-inch wheels?

Suppose a disk rotates at constant aneu l+jr2 i9ax0o2: :,jknrp osgular velocity. Does a point on the rim have radial and/or tangential acceleration? Ifr+k2o :rjap,29sn ox ui0lj:e 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 the angular velol(nl)fi8 5 x(q0vnotecity $\omega$ Explain.

Can a small force ever exert a grp0jg8ox(asfon j2 c.7 eater 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-upup m.fx8tz2 d*ba yoh()5nu )d with your hands behind your head than when your arms are stretched out in front of you? Axutazp 5)(by )nuo2df.dmh* 8 diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel1vr1k4 e y5pd9s p:jptz77wdw 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 wh9 ywp:p5rk d77vzp dsw1e 1jt4ich gear 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 slender lower legs with flesh;hcmofk+6,k h and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explahm ;+kc ,kf6hoin why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carxm2 j ,emen6jv9.( pghry a long, narrow beam?

If the net force on a system is zero, is the net torque also zerofhk o*allw1 zh3bdvv5 mt 5vu -)y(e(:? If the net torque on *1l)vl:3-ybhve vdamtk5hw(uf o5z (a system is zero, is the net force zero?

Two inclines have the same htzu wi(+ 3z2djpzt2+r wv 0iw,eight but make different angles with the horizontal. The same steel ball is rolled down each i(wzrvutiw0+z2i3dzw+ j t 2,p ncline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start ro4g k)17qbv nbik ram.8lling (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 the bonbk81bgk4 ar7q.m i)vttom?

A sphere and a cylinder have the same radius and the rdtzg/ v8n-.inb1ul 9same mass. They start from rest at the top of an incline. Which reaches the bodi .rln/b8g-9 tnzu 1vttom 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 and angular momentum are conserved. Yet most moving orkrk l(p2d34pghsq z, , rotating objects eventually slow down and stop. Explaih(3dp,qr 2 lgpk 4z,skn.

If there were a great migration of people toward the Earth’s e /e ,tkxs6ie .ahtgzini 98s5xyn-j22quator, how would this affect the length .i8i2,x/ez9 -e2 hi6y j5n tksgast xnof the day?

Can the diver of Fig. 8–rv;ub9(* bcvxp-ve:v l 6mr 5m:rjs:j 29 do a somersault without having any initial rotation when she leavj:be(:vr;vmjr*- :svur m 69v x5bclpes the board?

The moment of inertia of a rotating solid disk a* oa3*8y j;yrm v)rqvlbout an axis through its center of ;qvravm* l3ry*o8 yj)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 stoolyy0h7 y9gv 9bf holding a 2-kg mass in each outstretched hand. If you suddenly db7yy9v 09g yfhrop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass9pxnr .2ddm )gz2d fq/. However, one is hollow and the other is solid. Describe an experiment to determine which is which.zmdf. )2/ dqxpnr d92g

The angular velocity of a wheel rotating on a hadkxwd3z/i(e0 h8 1vwvb ; cy-orizontal 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 kv;b 8y( xwh/ddzv01ew3i-a cthe 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 edge of a large frees+t 1gw0d(5jkhc dc9fly rotating turntable. What happw cf+s5(h tgkjd019 dcens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he thh34-uo++,sk r:vzx zzyg;k eyrows the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and lh:zuyzkrgs- o4e +k;yv,3zx +egs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the l1, p 1p njddifp.gz+m6aw of conservation of angular momentum, discuss why a helicopter must have mop ,1 16.jdpnfpgmd i+zre 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 radians: (a) 3t/ 4+-u p2egtjtzz5zbmm**q d0 $^{\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 be:4 qdxr;ysw)g cause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as xwgs;q:4ryd) seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed p kb/j;a -alh3uv83gyzxw:k 5at the Moon, 380,000 km from Earth. The beam diverges at x -abkk 8/yhu3wj3g ;:pvlaz5 an 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 650ktca 4 *zap0aio m;ze83cw7dk g* ,)ph0 rpm. When the motor is turned off during operation, the blades slowzk** ta;w3 h78,a 4cc zidp kmae)o0pg to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If m)xw0v4rptc9 the ball makes 15.0 revolutions, what 0 rpw c4)x9tmvis its diameter?    m

  A bicycle with tires 68 cm in diameter travelpxreorw*c14pih u66 3s 8.0 km. How many revolutions do the cw4h6i prox1u*e rp 63wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates 90 wa4jgf4: n19tioaj y4dby udloi:7at 2500 rpm. Calculate its angular w4 7fui o49b9 ddn oy:ija1t4y:l0gjavelocity 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-r5is,9s wukbfgp5 c 8;cound makes one complete revolution in 4.0 s (Fig. 8–38). (a) Whas c,p55fwig98u s;c kbt 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 velocity of the Earth (a) in its orbit ar;*g.7joc 0d wgei0x psound the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poink0bp7 zkf,z6qd7kq w6t
(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 cm from ty+ m/3je rpk. tgu9b:0q;sahh he axis of rotation is to experience an acceleration of 100,003m/uayeh9+ bh;ks0prq. g:j t0 $g’s$?    $rpm$

  A 70-cm-diameter whevxg kphv. 1q37el accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. D3xp7 gvh1kvq.etermine
(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 radiusq4y7j4mftsuduu2tw/3 m,da5 ,l nh (e $R_1$ 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 aboard the Apolly zf-f )p7 4qs 7wh,5stl4wkgqo 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-ms7lf-)s, k4 wq 7y gtzhp54qwfin 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 unifor7tuaih :o8amui*p ,j+mly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in thiuim+jt,a :8 7 pih*auos time?    $rev$

  An automobile engine slows dowh+d8(sekq az+n 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 flhg +zru m.n;/or the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 compzhr lm;g/n.+ulete 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 accelerates uniformly from 240 rpm to 36.bcim4;).db(s zc ayc0 rpm in 6.5 s. How far will a poic4ym.zb);(s b a.di ccnt on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it iuf f(6,axk;;u8ck go vs running at 850rev/min It turns 1500 revolutions before it comes to a stop,8;f6kvkuu c( fax;og.
(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 cajfn m4xerrj/a/+.,vs r reduces its speed uniformly from 95km/h to 45km/h The tiresj r.m+a,xs /vfr4nej / 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 revolut:8xf*c v)9g.wwwjq n+xti-9lbsa z 1 qg 5fpm-ions as the car reduces its speed uniformly from 95km/h to 45km/h The t a:+)zn8w t.gsg b9qqfi-x9-5 p1vfxcwj* wlmires 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 weight +4gov)mwcz8bmq+8l wf oa ;4 non each pedal when climbing a hill. The pedals rotata+fb8 lcnzmo4wo4g )m ;q 8+wve 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 end of a door 74 cm wide. What is th 1rmbk2cmar-4d,n u j0cm9u5g e magnitude of the torque if the forceuk 10mrgr2 4c9mun-mb j c5,ad 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/g64fp*9 hmoc cjt 1l76ozcfi)w md2q axle of the wheel shown in Fig. 8–39. Assume thatf74cqp6zc6j1hg/ ci fdl2 *wom9m)t o a 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 e enicyt9:4gz/a0jtw1 ( tt3kjnds of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizont(ji9yn03at/: 1wtj etz kcgt4al position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinderft . pce*/c-(dlmi; ij head of an engine require tightening to a torque of 38 jflic(c/ d. t;p*e-im $m \cdot N$ 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 1nr eeaqi946d of inertia of a 10.8-kg sphere of radius 0.648 m when the ax 4rqe6da 9ei1nis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in 0y2x y5zu3dk wexz8 o6diameter. The rim and tire 02zx8w3 dx5ko6 yy uezhave a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram balh mp. q-zfik/9 gu;up dff(t4-l on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Caluf9f;mi/g -dqpf4h z(ukp.- tculate
(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 vts* 4rv/:--yttfqa pjj*vg+constant angular speed (Fig. 8–42). The friction force between her hands and v raqs**v-t 4 j v/fp-yttjg+: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 2 i74rnitap*oeiv x un 7fmh-;i-i7,s objects shown in Fig. 8–43 ahot 47m2-pix7 iusi f* i,7a-rvn ne;ibout
(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 consfl s-twq f1(4iists of two 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 corre u6, tx:nkyj9oyud, s+jhy95lct 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 satellite is ys,j9kx,yj d6: 5t l+9no yuuhto reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform 6b*8jt a*d wtwcylinder with a radius of 8.50 cm and a mass of 0.580 w aw*t6*8jtdbkg. 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 bat, accelerating it from res ,iom w7g3yx+y av;cr+t 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 tangentiall-d / 9c4: wgfc5qlvehr/shavav:.is6y on a small 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 frictii9r6wv-:/vevhs 5gcf cd aq. hs4a/:l onal torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor ro16h5tsh :t:df nmfb z;tating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frict5fs :;:1fd h zhbttn6mional 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 7;1zwr*1*e(k lj suhs s $m/s^2$ 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 the f b3n8duj hj3 d0+81v6uv wogdforearm, which rotates about the elbow joint 86 vjud 8v uhg+1f0wnojdbd3 3under the action of 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 bla9twk i,2yrseolvh/ b j6-.bp:de can be considered a long thin rod, as shown in Fig. 8–46.k vr-bitysj6l 2eow:/p bh9,.
(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 masc 5i;ldx5f f, 5gjafhinrty58c m778nses, $m_1$ 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 accelern+3id*y lb16ro 7 vr)o.cfq goates the hammer from rest within four full turns (revolutions) and reif3 )go.l+*voqc rn6 o7y1bd rleases 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 moment of ineu*vo 4k:nc x/n-hiyx0rtia 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 jt.i 1p b:j2mt280 $m \cdot N$ 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 wit,j u/wh9evrt . vpsw6erf4)l(hout slipping down a lan,ru s (rvljv pe96he.4ft w/w)e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun/8kjkmf;(e 2 xv-nqv s as the sum of tv(kens v2;/fxkm q-8jwo terms,
(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. 6;xi+nffnac bac:,j,,tx tjsy7 q6m(kg.9ps How much net work is reipff ,am jjsxky:( s 66g,an; x 9q+ct7ct,n.bquired to accelerate 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 rest and rolls withox *z qq0k,r,ghut slipping down a 30 xz k*0qh,qr,g.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 vertically on its tip. It st-vx6 7+msdhfu arts 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? [f-6 mhd v7x+usHint: Use conservation of energy.]    $m/s$

  What is the angular momenvrdd jg80 r.j k lkq3-ay a3qq.k9l,z-tum of a 0.210-kg ball rotating on the end of a thin string in a circle of radi-3qqrkll akjg z30k.yq.avd8 , -jr 9dus 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 cylindricyaigm-:6 fk1ogag. ep;gbj/3 al grinding wheel of radius 18 cm when rota6yiega j- 1pkmggo.; /3:f agbting 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 atg05,dcww waxq* f j9,b his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreqgx,awj 0c 59*wbfw,dases 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) g .y:cbes1nssn ) -jmf/ p-wv9can reduce her moment of inertia by a factor ofbnj9e -pn.:) s1sfyg- cswm/v 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 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 iya9 moh6 t4-qg 8rlk4nnitial rate of 1.0 rev every 2.0 skh4t 8q 6-g ya4rm9oln 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 freqj:bnrp)/.o qmuency of 1/ p. obrmq:n)j.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 frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning al++ wbjnkj ht 73zlc- (i5qyc2t 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 thjwv -;abxz;/0( pv oroe 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 i bg( 4jt/dzs9snertia I is dropped onto an identicald4/( 9gbstsjz 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 jt:-4s(ef 9qm dy5ipt 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 center of a rotating merry-go-rounjht7o+uggb2 bqirgrp8l+ r 6/u1fg13 d platform of radius 3.0 m and moment of inerru u g 8l1b +2+ripto 1gg7h6grj/3bfqtia 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 frzx tp856m;qhweely with an angular velocity of 0.8 pmz;6t 5qxw8h$rad/s$ 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 into8oe zjx1bbk, pufga)d9 z 93i* a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming opzuzid9b aek3*x8b9 jgf 1,) 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 bw j4h wrr)+kqkk7/p; 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 1 oauj8xal0jna7iv4pi,b 9t+nol 1 +npgz0;x $ 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 c m4e 7z:letve:o -m vg4x-0hzsan rotate freely without friction. The moment of inertia of the person p o m:hv:z-gevx m7e4zlst4-0elus 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 aip,zpwti,)qlm w gb1* p:5w0 vp,r9jl t the edge of a 6.5-m diameter merry-go-round turntable that is mlvp:,9p1mq ,itw prbi *g zwp,jl0w)5ounted on 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 j,k.lvmakh 4*ul2 :d don the ground with 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 spool? How far does th .ahk 4d v*ukldml,j2:e spool’s center of mass move?

  The Moon orbits the Earth such that the hmj:x 5snn ;ji8 gx,gn,j/ew*same side always faces the Earth. Determine the ratio of the Moon’s s* nj8wig xh,esn ,5g; /:xjmjnpin angular momentum (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 rateem)s8 msz qpt6,o-+s q 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 radivxz x hz,w6;tm)8s2hu56 96lme gx dugus 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.58,l6tx xxzh gzd)sv; hmume692g6w u    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and drp4cvt1v nl 63.xs dq+iameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation oq+ vl t6p3v14nx.crs df 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, h+i/exp47kb;sr zq o,now is the angular 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, g6( s:ezqh r/vbut with mass 8.0 times as great, were rotating at a speed ofh(ge/6q :zs vr 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 autom5(u2m-vp pot8 +c:j 0niitpf mobile 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 1.5 j : v8+u0 -i5nimpftcpptmo(20 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 illustrates0gn m t.5a+nryg8 r-uj vhoc3; muqn3a4a)d,h an $H_2O$ 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 (xvza ft,ayb-5u 0za-u rzdf,p o4p;/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 leng9s05xzi f d 3);zqibcxth 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, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the forzbzxsq3i 90f) cx5;idce 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 sta8qab f5 3j2rb(az44 alxmbxf2nding vertically on the floor, and we want to exert a horizontal force F at its axle sm2x4 b85 b3x rab qfafl(ja4z2o that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with spe o8dv8ai76,g9 5ej w:p*nrd xm ijr1jped v=4.2m/s on a flat road is making a turn with a radius The forces aai7:j*e r68 ddv,rp9jmgpj oi1n5xw 8cting 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 0.50j:4,g wua 0s:9 qqqean-kg rock into a sling of length 1.5 m and 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 re j4gnuqae a9,s0qw:q: quired 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 rodsz2o) waz *g+ce, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning zgae *cz+w)2 oskater with outstretched arms, and with arms held tightly against her body.   

  You are designing a cld5d)* tbdid c8utch assembly which consists of two cylindrical plates, of massdd*)idc5d bt8 $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fi 9xgcyd)bolqrgne /v6ag 312/g. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point rbd 1lv2ogqgn963 gx yac//)eof the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but doe . :jctj,h1wg not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reducesj7y5liioz(2 hx4p*s t 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 ti7ipsj2xz yh5t o*li(4 re? $\approx$    $rad/s^2$(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