A bicycle odometer (which measures distancip(s ex7v9dedi3i8g0 e traveled) is attached near the wheel hub and dp(e ii83 7x9eg idv0sis 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 avuc; xesc0b;9 point on the rim have radial and/or tangential acceleration? If the discx e9;uv;0bc sk’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 velocit(1jhqf 4v: obcy $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Expl zzz5ba2ceirt,rw:f /4/0ydz ain.

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 doq m y (7r;nmvd0hzb7,0d ,syrl a sit-up with your hands behind your head than when your arms are stretched outvdshy ,q ry07zm,d;lmnrb70( in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and th )vivritx11gdmt5 g4+ree at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear41t 5igigxv m+1vrt)d 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 slende+y6*7j4 qwm/sanc e (r)k 1 mvb7ehgaer lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain k7j6b wnecg)es7(+vmrqhae* ya/ 41 m why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, nau*ucswr6 9 av:obkw obe; t02:rrow beam?

If the net force on a system is zj/*1uvda pz3(8xs i eaero, is the net torque also zero? If the net torque on a system is zero, is the net force zer3e1i* uvjsa8a x/ p(zdo?

Two inclines have the same height but make different angle ynp ,9v8-xe ataf :uzgnm9a))z3s/g-hrf9wes with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater?y-h,ear 9/a n-xwzfgp99g:zs)f8mu )3v t nae Explain.

Two solid spheres simultaneou akmn + :fw57hj3 xl(p*cm3hupsly 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 greaw3*j:f+mpm uh xk n(7al 35hcpter speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same maw9(:+y ce c39ph9inuqan y;hass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has t9hni ncyyha3 ;(+: 9 aecpuq9whe greater rotational KE?

We claim that momentum and csv3 ;oijm4d .angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Exm;si 43co.vd jplain.

If there were a great migration of people x9rlj.)io(-iwb3yv /a swp.t+ ch7d ltoward the Earth’s equator, how would this affect the length of the day? (yilvd bc /wl.7 jo3hxaw+)i9-s.rtp

Can the diver of Fig. 8–29 do a somersault without having any initi.j (,ugn(v i (h g6yhvkos.r;oal rotation when she l(r j(oy(u 6h.v,;.gsvkiog hn eaves the board?

The moment of inertia of a rotating solid disk about an axis through lg izqi-vm+l s)x1.m:b 8h t3nits center +-:smng t3l1 8zh.xvm biq )ilof 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 sittin8srcw)+k :gr ag on a rotating stool holding a 2-kg mass in each outstretched hand. If you sudd8kcwgr+ :sra)enly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look ide,(hr,m ekrnj4 8y o0kie aw;2mntical and have the same mass. However, one is hollow and the other hoje,ri(rkw40m 8 ;ake2mn,y is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horioz/q 32ap-axsc(w6fv u iv9cf/(zg 8 azontal 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( vpzxvwc(/as f3 aoagqzc2ufi/968- 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 freely rotd75(b5,*ify miuydq b ating turntable. What hbiu yd* ,5m7qi(fby 5dappens if you walk toward the center?

A shortstop may leap intoozi-.iunauv,u 3h 1 b/1*lvn /mes .ki the air to catch a ball and throw it quickly. As he throws the ballimsni/,1/k v u-zuoeiu ah13l.*b.nv, 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 a qq1wi1vna03z+o ank :ngular momentum, discuss why a helicopter must have more thaqak3 n nav o:11wi+0zqn 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: (wn(q6pm/ xk ,7cht 4ida) 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 Earthnifei7h- ek9v 5 80lxr because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and th r9xlvi-58 kfi 7nee0he Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverg+zhl cjj m.dbq):f)d7es at anc: bm)+qdlh jd7jf.z) 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 6500 rpm. stl0 0erlb,wt;i n/ :sWhen the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular accelerationb/r0tswi st: ,el;l 0n as 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 makesgrz;6ha2xg +bii 7ilhr+ )m.v1vmd n* 15.0 revoirgh*ladi.r vh2 m6 )+1x7vgm zn+ ib;lutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutionsf5 g; z0 kgyc2w:yjsgs l,mc0(qz w.6t do the wheelsj, kqzwc 052cywgfgs; m: y(z.g6slt0 make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angude *j 4ogq z0 w4ava15u(nzux3lar velociz(vqx* z51uaaeug3 don4 w4j0ty 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. 8–38). (ayflk3kf+vg5 3 ) What is the linear s35 lfkv yk+fg3peed 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 aron ovjz/:ncg ;f*yo52m+pokz;und the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of ak, f,6+)9 njjzq hqaxa 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 ax3)w,dz9 ls)k;cz gj djis of rotation is to experience an acceleg sckzl j3)d;wd, z9j)ration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center frh. s2-mt; s zi7r srcnb5;s)plom 130 rpm to 280 rpm in 4.0 s. s)sst n 2lrrs; z.b-m5 c7p;hiDetermine
(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 radiuzzy -x ;n y/,o.emrrzu7 -.rbss $R_1$ 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 a;8ukvnh.k,7x sczty9 board the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their triph8.uc,t k 7nxvzs;ky9, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Thyk q/z fg4p :c lhz0zdo*;;y(arough how many revolutions did it turn in z ;(kg0dafzchl;*poq:y/ z y4 this time?    $rev$

  An automobile engine slows down from 4500 rpmf/(q75 yfcg gg bxp op )j(sz*b30.vds 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 stressess3p12wm fbk . 5php- ,+zglhh hz;dnr+ of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 m;h hl .s3g1 h p,p5bf +2dwzz+rhp-knmin 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 accelerate-q9 it ju j) ydp3eagzls98:(os uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the- u3o)yz(8qp9lgiaj des9tj: edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500o24thv t4 a+h(kx ut/j revolutions before it comes to a 4u4ha2t/tk o v jh(t+xstop.
(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 revolug+;8td(p rqb jtions as the car reduces its speed uniformly from 95km/h to +drtqpb8 gj;( 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 a q(fd0b-f+l9hx4mxk0ere 4am s the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.8 9adh+xlbmfmf44xe0 ke0r -(q0 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 climbing 5bgxv gpwrl*5ud 9;8ea hill. The pedals* p5uv ;l g9xwe85bdgr rotate 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 wid vyf+ i6/o ,;a0ir9cxjkske* ce. What is the magnitude of the torque if the fc ,/skck*ri9yf ;x oevija+60orce is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assumzna3f 1a5 w7cse that a frictaazs7 c53 n1wfion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends y rcogbh)1bn4nhl**ur6 d2*oof a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and dir** 2r1uo nhbcn )dh4b*lo6yrgection of the net torque on this system.

  The bolts on the cylinder head of an engine require tighrw)e, akg7uuh v-n48dtening to a torque of 38 un), h7avegu w4dk- 8r$m \cdot N$ 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 radiulfmjfic v/d3mgddh(6 +z w0w,;+ cq5 us 0.648 m when the axcq,5+ uzg wmfhd f3v( dd;+ wmi/j6c0lis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66n )4 a, 5uvwkm1fmje3mf)t .sf.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. Thmfvk e1)t wafj)4 nm.5us,mf 3e mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated inxueh.+ r6vg y e5wr;+mat 4c/ g7ipct).*sxlq a horizontal circle of radiuxstu ggy. qwxicc hm ;r*r74ve 5 /l+a6.et+p)s 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constantaw .* m)zs1bvy wmy-.u angular speed (Fig. 8–42). The friction force between her hands and the claymuswya) *.y -wm1vzb. 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 oqvuo-c;o+(n g) uw 9)xhe8gf-icwzl-9jz9a g bjects shown in Fig. 8–43 aboufe9ocjwl )h9xuigcv zz u-an+-o(-9;q w )g8 gt
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whos 84ynimqn eb *h3nq/t8e 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 satellx(my(zaa kfz 5(t,v/zite spinning at the correct rate, engineers fire four tangential rockets as f/kzz(, xzym(va (5 tashown 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 uniform cylinder with a radius of 8.50 cm and a mass of 0m49fgraidt 9 -.580 kggdma-i 9r9t4 f. 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 4(i7 o9bkmqte a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round ifsm( 4i1 hn9oand 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 i(1ifno9 4 msh760 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 is shut off and is eventually br kp qyvdz4i50(c -d9xwought uniformly to restvp50d c-q wdy(4x9ikz by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 acceairv ;f 015itllerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the fk; ( /eo d)cf.la( a9*.pcngievovdo7orearm, which rotates about the elbow joint under the actilv.*o( (/efn7 .;iav9eo dgcka o)dcp on 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 ddu9at5b(cnjmg- j 7269i x mprod, as shown in Fig. 8–46.d mj9i x -a5bu7p mcjgndt29(6
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consiszb+g- -6t gv4s yhju3ots 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 four full tu0/b urbeqw 57irns (revolutions) and releasebrq ui/ 70e5bws 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 h+drizrs5if o r5oae ;u0bi(5 3as 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 torqund;eiw81t)dirl i g 2(e 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 slipemjx e:0m x w,zec*.pja d0 y,pc-r72pping down a lane a ze2xym:ec* 7 jx0dpm .p0j p-,,rawect 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic eni 3*;uozlqu (tergy of the Earth with respect to the Sun as the sum of two terms(tu3z* u;l ioq,
(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 andzprlhpgfj2tum2 3 *- / a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it hr -flm22z*g ujpp/ 3tis a solid cylinder.    J

  A sphere of radius 20.0 cm an5 -xcj1to,b d( 5xq/w nz;emr(rgz+h-p bbhj;d mass 1.80 kg starts from rest and rolls without slipping down a ejr; qz,5dpn(h/rx+ o;xt51m-b hjzb -wbg c(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 startskg *azj m .cob:w.p /a:wlc1y* to fall and its lower end does not slip. What will be the speed of the upper end of thpojc1 b *.ayk:g/*:awclwm z.e pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-k( ah .asl(yeekrf 9,3yg ball rotating on the end of a thin string in a circle of rad9yar(a(,3s.fleeh yk ius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angula t8+ pt)caq8hl6.euh *u9hhyar momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 150yeht8 a6atqhc*h 8+9 luph) u.0 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 ap/aj8wyqvxjmq./kf m a i; qn(xb/+657 0cn jct a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotatioaq; 7 q 6qj0wymp b. mc/(xk/nafni8xcv/5+jjn decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can ree*ddbi xttfo8s p+;1,vhqm7nf,)u u.duce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If shux)*ftu n. 1hm 7 ,;s,qiodt8pef+dbve 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 initiajjd( z -q qzidv1w5s6)l rate of 1.0 rev every 2.0 sdijz1wj( dv)6-zs q5q 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 fr/70fz4pyp l xsequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius x4pz7/l0yfs p8.0 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 slj3u h; d8 y3.,fy(rlzd4kk gx1 kb:jskater 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 mokd .psvab7+v8 6l(e6la x cmy;mentum 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 id1q , b/kepf9nltw ol6 97vzoy9enticaq/b 9,l v9et n7fw1po6o9 klzyl 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 at 2.9t;1 k 6.bvck6 yxdvqq4 $rev/s$ 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 cent /dmzke)yu;6x)g+5scamzw- ner of a rotating merry-go-round platform of radius 3;z 6 cwmeudxs)/k-+n5)y azmg .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 angularshbx35/ *s q puirg+,tb9 ykq) velocity of 0.8 9ys *s g/ ikb)+qpurh3q,x tb5$rad/s$ 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 5r :rlkpcqb4;into a white dwarf, losing about half its mass in the pc5kl4;pr:brq rocess, and winding 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 wi2 meecziwl8 po 90csv,-6(iexnds in excess 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 u wfqgppwfg)6/c) 7 sh4-gs5 m$ 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 can rotate frees*i/ao3l2m m* t (rj ud5sqkh6ly without friction. The moment of inertia of the 6hl/2j su 3k ar(*5otq *mimdsperson 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,j umy1 2s ly/:fi;dda diameter merry-go-round turntable that is mounted on frictionless /af sm, l2dydy;1iu:j 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 enjp ve .emi ,bq9e:-qh6d of the rope lying on the top edge of the spool. A person grabs the end of the rope and: hpmq9eqebi6 .jv -,e 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 8ofkw rhu2(xfmc:n: *mw7/vxEarth. Determine the ratio of the Moon’s spivr*:uhfcmfn xx7 2w: (k8mo w/n 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 fzjr3dl0b-pk ah *+hotk i0 m4;rom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquirgu*z nm:(btl 7i*f iu9es a rotational rate of from rest over a 6.0-s interval at c*(b 7u iutng9i*l:mzfonstant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each*3dq,r b+p rngny 3r7: .xyxvc 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 g,q7v3:rxrnbp rnd .3 yc+x* yenergy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of th *tix6 s8cwm ct8lgt3 +ye8n+ze 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 8.0 times aeani9a8 o f7*zs 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 thati zae7afn*98o the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it1gn bwvrlsij3--a m9 kbai 6 4i9pwy./ 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 togi-kia4-bl 9av/wns m.b ryw j3 pi196 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 illustratekyd+17p mlr-rxt/0t) yi( tmj* fnv6 gs 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 (hr8* hud*lf k+noop) is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length 9lm5mb+oe1 iy,o+ixx 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 horizon om1 xo m5i+ibe,ly+x9tally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically x-tv: c07s kci+ d *bx+mn*bngu9v-blon the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where vx0n-k+tvbis :ndb *+g c7c 9*lx-mubh

  A bicyclist traveling with speed v=4.2m/s on./z 0v/ qf:s7pexggr a a flat road is making a turn with a radius The forces acting on the cyclist and cfvzaqep: xg s./gr70/ycle 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 and begin6z*d cm3 rq -npg;,keqs 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 requig m qpeqdr3z ,*nk6;c-red 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 thixd1 vo5 r .48xm7imnqjn rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds fm8o7xqrd jn 54vim1 x.or a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrici1whe1 *s iy2l 2pw6vial plates, of mie h11vpli2w2i yws* 6ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. 8–58p1ji- em-vxn5. What is the minimum value of the vertical height h that the marble must drop if it is to i5p-xe1 vnm-jreach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assu op;h ctuk**n/8-spd ume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as thu5ux mop 4gm+,e car reduces its speed uniformly from 90km/h to 60km/h The tires have a diam45mg,u oxu pm+eter 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