A bicycle odometer (which measures distance traveled) is at a ,qxjnxs p88utc +,lohmm4xjwt);f rk),5,jtached near the wheel hub and is designed for 27-i l,4onhx)8m;ftx w jpxj 5),uj ,sm,rct qka8+nch 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 audh: ro6t 5n,0foe*u jdh(*xrim have radial and/or tangential acceleration? If the disk’s angular velocityot*,djox dfh n a6u* h:(5u0er 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 a8,x:ayhiw3p1 /xvv j be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain.zkal 8s53f +lm

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,ixjq *5gg0brp9f 2mp do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help0 xm*pqi rf52p bj,g9g you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at zh h9ep9gn/ .fe 1)45zmymw sfthe 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 front sprocket or a large/fg pznf zm hh)m4ey1e9s5w9. front sprocket? Why?

Mammals that depend on being able to run 5bxpix (sg2jkt v;19- upwa/rfast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageouswb/- sv ;5ti1 kxr ujppg9ax(2.

Why do tightrope walkers (Fig. 8–35) carry a long, narrowvs.e.(wuc (vt1m jzs0 beam?

If the net force on a system is 4 )s-cxxhnf /hzero, is the net torque also zero? If the net torques)xx n ch-4/fh on a system is zero, is the net force zero?

Two inclines have the same height but make different angl) 1jmgio n60kqes with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the b) qmgo1 6jink0all at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. On,g-8t4men nzu; l*hmbe sphere has twice the radius and twice the mass 4e hbgn;l-8 * ,zmnutmof 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 bottom?

A sphere and a cylinder have the)n ss9.gp17h b (zxyyx same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottonsh sy (7xzb)y9xgp1 .m 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 moviuidip(q*lruwk8: t xf 5qa6h p)j11 c9ng or rotating objects xc65tra uji9k p:hf)lud i*81w p( qq1eventually slow down and stop. Explain.

If there were a great migration of people ib)48f4 v feiiz;b,mo toward the Earth’s equator, how would this affecv; ifoz,)m b if8i4e4bt the length of the day?

Can the diver of Fig. 8–29 m ;yxd)3x,.f wuufqm pph:,*w do a somersault without having any initial rotation when she leaves the u :wxpmfmq3u ,.;w ,d )y*xphfboard?

The moment of inertia of a rotating solid disk aba 3z- u4m:b t7j)esm* rmkd0d-x 7pkchout an axis through its center of masdzh s bxm d)j0kck 3*-mu7- mte7rp4a:s is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg m/r. hw3c: ndk7i8yb fqass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Expr 38ncfkd 7yq: ibw.h/lain.

Two spheres look identical and ) rlxvmztdgw. 2v)4;lhave the same mass. However, one is hollow and the other is solid. Describe an experiment to detem42 r);w)zvlxvld t.g rmine which is which.

The angular velocity of a whee-uomi9poak0a g+)e9ix 8sp/ el 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 decreasia+eip 9gki)m e/9 - p8ousa0oxng?

Suppose you are standing on the edge of a large freely rotating turntabl1(m *b.krlw dme. What happensbk*1(.m m wrdl if you walk toward the center?

A shortstop may leap into the air to catch a ball and 7j4u(yvew4,bipa m 4f f 7+aszthrow it quickly. As he throws the ball, the upper part of his body f w7ay7j(p +f44z4v u,imasebrotates. 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 o 5xahqz2b)gb7f conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propelle gax2 q5bh)z7br). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the followini ,8os/,xm/zyqzuuz6 2 hc)vzg angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, usidb 5 nfq2(fb:lym4 b-fng the information inside the Front Cover, the angular diameters (in radiabm-: f(4 qfldfy b5nb2ns) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam di3isur)r6qv r3( 2rhep*f2z auverges at an anra6qh2zir 3f2)sup erv*r3(u gle $\theta$ (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 motor is turned /v jzauag6o( .dc h; 3taep+k4off during operation, the blja/evc.ap(gk +3 ;t4 uoz6ha dades slow 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 another53lymci : wf8yrbh j85 child. If the ball makes 15.0 revolutions, wh8h i3w r5fjl8y5m:bc yat is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many issrd0+o4afure 3f4x-uaz2 n4 k49g crevolutions do the whudx4rf2sf9 era 0oc u3i4 g-nz 4sak4+eels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Ca.(gudo po(z3q lculate its angula3z( d(po .oqugr 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 m8 rcb/klp*qj1r09 ic* cr+lj oakes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the center? r*co+1rlik/ c98lbpq*0jjcr    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velokdh p-0h)+eq2 tql2bt d,b1i rcity 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 speed of aq d3 greh:iyl, .r w) ;p66f7phq1wczh 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 cm from the axix/voj;0uuyfjn 2t 820h 9njqws of rotation is to experience an acceleration of 10jy2n9h2 jn0f xuq8j 0uw/o vt;0,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acziph*v onh5 pq;o;l 4os(03kecelerates uniformly about its center from 130 rpm to 280 koqs ih ;pn5he o ;v3l(*04opzrpm in 4.0 s. Determine
(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 ja 5,matvpge z:u**-fq4 r tknt5e+x7$R_1$ 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 Mob8uxkgd:qq2pv +z33hon, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be g bq uq2+:3xdk3h p8zvthought 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. Tkj.bsaz-fwf6tk3 ,us+5, k3jff9*fs d fmk ,ihrough how manyfsb9-ja6tz kk53fds,,f im kjw*sf.fk,f +u3 revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to dhao: 4o. obbfvpj 4415:yphw:7ja3s mflj(d1200 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 flying highspeed jets in a whirlix9, x+z5hy j9pvv9e kvng “human centrifuge,” which tak5 9xyvx,pv9 hz9ve+ kjes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in e:;eb8whsw q96.5 s. How far will 8: ;ebsw9qehwa point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is rer6wde yz 4qg,m 0gzn59nc1(lt -;peounning at 850rev/min It turns 1500 revolutions before it comes to a zzyem0ntg9( -ed6p oqlg;ewr51c, 4nstop.
(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 caromqfia9-z0 / ja y74a) vg(pxe reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m. 4(ape0jao z7vx9amy) -gqif /
(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 spee.mknwh-ua (m)o 0 j,oqb:f /eqd uniformly from 95km/h to 45. omh 0n -waqmej/,:ufqbk(o)km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedalnb5ouh;(q1 f v o.2pip when climbing a hill. The pedals rotate in a circi1phofn.up 2bo v; (5qle 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 twvno ng 0n4gx;w(t +g:+f 29i2r zgfvmhe end of a door 74 cm wide. What is the magnitude of the torque if twontin gvvfmw9gzg: 0 2+nrfgx 2(4;+he 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 axle of the wheel smdg y14prq7gf2x u +.9p gl-cqhown in Fig. 8–39. Assume that a 7lrpuyg m+ dp9qgg2 1qx f4-c.friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, am q 1mmb+le v/a/;t6vdre attached to the ends of a massless rod which pivots as shown in Fig. 8–40. I;v/ +t6/ mmldqab1m venitially the 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 engine require tightening to a torque ofwc 4tfm02iugu;.di 2rj72x 5 y o;flks 38 2 dsgl2io ;xci2tw; 04 u j5mkfy.urf7$m \cdot N$ 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-to+ba w7/+ qrskg sphere of radius 0.648 m when the axis of rotation is throug+ rw7t aqsob/+h its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamett 5du -rzw-8p ,70ir wpobsc6oer. The rim and tire have a0zwo 8pocswi5d 6 tp7ru,r --b combined 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 ezpxm/k; htv6+gf x7c (nd of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Cal/6m;fgt(c ph+ zxvk7xculate
(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 aeqczrza0q a**3* j a3q bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands *3aq * z3a*qcae0qz rjand 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 o30ox c,hd w ,.t8hk3l wpt+ewktyc:k)f point objects shown in Fig. 8–43 abh:eo tx3 kht,p8lkc )wwy 0wd+k3.c ,tout
(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 o kjb4s7 hse9zl ;+q7batoms 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,ie;6/ jsmk72rr2 tqvgy i3( (vo dsdo9 engineers fire four tangential rockets as shown in Fig7o k/vvr (tqs;2jr6gd(2yd9 esmio 3i. 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 uniform cylinder with a radius of 8.50 )46 lymn4fb/vsolze2 ;i: up pcm and a mass of 0.580 kg. luv ms6 p:nl b/zyi2)ef; 4op4Calculate
(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 rest toe7ova+ d1oxe d,yr ))zl4l( ia 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 ske8q8ec nud3 2mall 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 produd3 8c8ne ek2quce 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 is shut off and92+ wpr 7gbxwn is eventually brought uniformly to rest by a frictional torque of 1.22bw7 rn xg9wp+ $m \cdot N$ 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 b7y* g 74m slv(7glfkxnall 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 sol )p jqn)vy r;pt,;gkcd,oq(x: ely by the action of the forearm, which rotates about the elbow rnp:o);j,qyt qvc dx (g,)kp; joint under 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 blade can be considered a long thin rod, ass, kb5 d3 .kvpr3arpo/ shown in Fig. 8–46v3p ar/3bd oskpr,.5k.
(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 consiac:a gx e/ n),qx1j/:qj 7yvljsts 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 acceleratezje +msa,e jm0o+dv6 7s the hammer from rest within four full turns (revolutions) and releases itdjms jm+eo 0+ev7 ,za6 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 hjk4u 6t(rv;m(o t+t0 srkn s;aas 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 develops a torquv5sf; -svcyn:3 qzf5f e of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mast4l ql 2vl3fh.w2dz ,ms 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3. q32l 4h, wdlz2mftvl.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with,a)cml ozj7 r1cx81ylxu .-zu respect to the Sun as the sum of tcl8 yx1 u7,1c) zramo-zl.ju xwo 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 wnvg 0;3(y vuesa;.t vof 7.50 m. How much net work is required to accelerate it from rest wvvea;tu0 vy(s;n3g .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;nv0c c,bj c7k kg starts from rest and rolls without slippi7 0cj,vcbkc ;nng down 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 vertically on its tip. It startbuk0:n/qc r2i s to fall and its lower end does not slip. Whatu2i0q:/bckn r will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on iqb0z*7.2b g*ymv6yge ub82 p7vqhlgthe end of a thin string in a circle of radius 1.10 m atqbm2 z 877.*gyyh2eb bg6qiu0 vlv* gp 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 grindinghhi *xo6i)km+pkea d8 3 7clr, wheel of radius 18 cm whpia7kk +xh rdi8o*3,)ml e6chen 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 n zvb -a.-4wy4 loi:ova rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48,i:4 -vzvlo4nw-b.aoy the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–293;dq ;zfe/xfj;h:awrg cs plua/ ,/,t k+z2ti) 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 il2 3 /q ;:trf pweh u;+jca,/ /,za;dgxikftszn 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 zxsh4)* e)dr/ ojep y0rate of 1.0 rev every 2.0 s to a fina/ odree p)4jz s0*)xyhl 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 it. fa ;hi/styf6s 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 chunth;yf6.s /iaf k 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 momenltvrz0)m 8am 5tum of a figure 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 of the E +*std(1fiphx4 qzm-r co1 ns6arth
(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 o-rr25 ff9a1tp,l+u wg6sldeu nto an identical disk rotating at angufwr9r dt1u +u,6pg2s ae5-llflar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns y)gi 5, g1ytojat 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 fz mge st:eh8y3,45iwka x-p-ter of a rotating merry-go-round platform ofpewfg: -,34 z -5 hxy8etsakmi radius 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 angular veloc)apsr / tw*am2ity of 0.a a*/2prst w)m8 $rad/s$ 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 eventuem.uistx0,m 8ally collapses into a white dwarf, losing about half its mass in the process, and w x 0ism,8.temuinding up with a radius 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 excov;,hd-oafz ,hd: esf-16 hd yess 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 ktpx m8mc/;w):1 tu jw$ 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, initpkgm0gd(): gkially at rest, that can rotate freely without friction. The moment of inertia of the person plus the plat gp m:d0kggk()form 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 ac7z bykox qu :-oli93v *vp,;et the edge of a 6.5-m diameter merry-go-round turntable that is m oikveb 7c 9*uoql;z3vp-xy:,ounted 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 roll -mqdd)p/qzn o0ib2k; s on the ground with the end of the rope lying on the top edge of the spool. A ppqqk-)zb/ idd;2om0 nerson 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 the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth 6, oy7pf.iti nx12qqg. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Eary qxi6oqni,721t. pf gth.)    $\times10^{ -6}$

  A cyclist accelerates from rxpau3z;*x yvkx d8*o, ph z97jest 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 radius 0.20 m avx;j1l* ,:ur hn(cxcvx85t ducquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered jn1 ;u xxvr,t8:* vcdh(u5cxlby the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyli( ,p.t fwtks.y+ ,6k xqsv0yftrz+1il ndrical disks, 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 string, if it is rele(6.frt,+k,ytxq ys .wzpl+f01kv ist ased from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular bqm -fa95j4v 8b bce0wspeed 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 ou(yju5us b8 nkec )k:5pr Sun, but with mass 8.0 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-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 c 5 5(jbp8)n:u sukekcyarries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it /0smk fqlvp 10to 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 k 1q 0lpksf0/vmm 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 illustraik bzp7lauypkd)b+1t 6s ods*45m -b0tes 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 v=3.3 s l4o bxg,q66;ivxsd 1$m/s$ 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 lenbm-0h)/jd ; o;/xz mf ( jev5qghy(6vcyyb+mlgth 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 (xf qyhg;yjo0v/my 5m +-(( );cmbz velhb/6 djb) 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. Ixofh2pwmuam r ;76c)(t is standing vertically on the floor, and we want to exert a horizontal force F at its axle sfcphm;o7(a uw)2 mx6 ro 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/s on a flat road is make2gu8nzd)u, ving a turn with a radius The forces acting on the cyclist and cy2dzu,e u)nv 8gcle 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. 7 8yw5uziu8(j .crefa5 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 required to achieve this feuuy afrj. c8 w(z587ieat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinx* zha, 5q . jqi3yzfzwsl9dh+eo .9h+der and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms he5+. hf3z9w x.lhqso*aji yq9z +h ,dezld tightly against her body.   

  You are designing a clutch assembly whic axl),lva ih8(h consists of two cylindrical plates, of mass(xaa,v l8)h li $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rol,3:p+ j3lhjk)pey1xg fb vuwz jd 8r0*ls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reac)de 83rg bujjf*01 wl j+pkx,:y3 hpvzh 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 9nf4gyi pi g1kg;u a9 hg7jrm/47gxe;not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the xq/cv/-t3w . pttvf/b car reduces its speed uniformly from 90km/h to 60km/h The ti x //b t-fctvqwp3v/t.res 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