A bicycle odometer (which measures distance traveled) is atta tr(l loux+24wched near the wheel hub and is designed for 27-inch wheels. Ww(4oxut+2rl lhat happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a pr/-z3 v09xeg nw :.hab;uegsc oint on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increa-/es:vg;zr3gh 9x. c0newaub ses 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 describe 34sq09+f*d)b4od:bm)qwm jv nggjn cd by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque t mv1yb,5p : r --yffgmokihsl7..o*uehan 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 4(j:d c cz65to 6k ghde24yi1df6gqreto do a sit-up with your hands behind your head than when your arms are stretchedyoe56dr dc:q(21h64dctizf6 gk eg4j out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel an actjbq4dx8gccl+ 97, d three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocke7 9lcbd,c4 gc+jq 8txat? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being jk0ww4g/ b*ghx+qdq, able to run fast have slender lower legs with flesh a /w hwgj,xqbkqg*d40+ nd muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lyt ioew3b s6.6d3x79mjwl4mqa6f a4 cong, narrow beam?

If the net force on a by z823ddr hiki) i/0gsystem is zero, is the net torque also zero? If the net torque on a system is zero, is the net forcerd8) b/i32i yg hd0zki zero?

Two inclines have the sxpcj5:e*me .;z 63ty7nt 1gw p.wekn kame height but make different angles 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 greatern 1k 3z :xwgtkw e7nejcp.6ep*. y5tm;? Explain.

Two solid spheres simultaneyv u --+df,/emwns4hy e 8q/ewously start rolling (from rest) down an incline. One sphere has twice the radius ad -w m y8+nv4wfyeqeshu ,e-//nd 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 bottom?

A sphere and a cylinder have o5q05wh:z mld0xf iu )e2kk 6othe same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater totak5u2 f50:koxe mdo h)z0lwq6il kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum ary 1uvr)rg *02s klkyn6e conserved. Yet most moving or rotat*yyl1ks kr n)2 6ur0gving objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how woulf8rx nonk 2uw2hf ta.31jh-n;d this affect the length of the df22t-3xr huh1fo k8na n; .wjnay?

Can the diver of Fig. 8–29 do a somersault without having any initial rwm*39pi4wc i4whk-* ) epzmgjotation when she leaves te*)z p -* 3cjmiwkhwg9 4wp4mihe board?

The moment of inertia of a ey66fmif8o .y 11wp ,lceg o,brotating solid disk about an axis through its center oeoi ,f,wlb1 omc668g pye1f y.f mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holv,od i)v.icvx nm-a ;k7b4ao :( eh*lwding a 2-kg mass in each outstretched hand. If you suddenly drov,-mdh (b4naio)l; xcvw oe7v. :aki* p the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the s im-do a8 53dm.svlx7jame mass. However, one is hollow and the other is solid. Describe an experiment t-vsml873jdox a5di m. o determine which is which.

The angular velocity of a wheel rotating on a horizontaln9u lcrvr cnw 8+k(/e-g ;9eeh axle points west. In wh r;cv(gnew8heuek9/9 -l+ c nrat 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 decreasing?

Suppose you are standing on the edge of a large freez *qa; gikw7/ily rotating turntable. What happens if you walk qi gzk/7*ai ;wtoward the center?

A shortstop may leap into the air to catch a balpv c 5-cg /yh/w 4rgpd- bv19qfe,ti*ll and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite directiw v bdq ,/-yvc4pgie-9*g1h/l5frctpon (Fig. 8–36). Explain.

On the basis of the law of conservation obktg , f cm9gmfp -4p8ain)9;lf angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or mf8m a9i9cn;kmptgf4l) -p, gbore ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radiaceb(8(f( -j9l bzzrppns: (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 coin8(m 272 uilujfipqe x8cidence. Calculate, using the information inside the Front Cover, th2uqm 8lj(u 72p f8ixiee angular diameters (in radians) 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 fqgw vq02t f7*rrom Earth. The beam diverges at an at 0w7g q2qvr*fngle $\theta$ (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 turn ,3tsc*y,r(j-x eru2o,i e4xaloy ma5ed off during operation, the blades slo a(lj,s3uyo ex* er -4,yrcox2m ia,5tw 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//zsek-x(6s0eph d /5 : bgwr:, v) xgrlpbunw m to another child. If the ball makes 15.0 ren6 v-/ seusb)p zg/exkx::,(bw 5wrh0/ldrp gvolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. H.a4xh 2g9 w0dqhhktgw u h6( 2c5dqeo4ow many revolutions do the wh gc45(.o9qkgdhhu2 h a wq2th604dexw eels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 250:dbbcd4 u4qo.i p: 6dw0 rpm. Calculate its angular :bc d4pu4ddq:bo.6wivelocity 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 rfmphc 21ak,w.evolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a chil21ap.mw,chf kd 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 around the Sun)-kkys( x6 2izyf:p wrfs3hi2    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speedu7f,ei3;nq+lx f/gxy d1e+ pny1m u, )dmwh 5p of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a pmt pd t2.uhu;:o (pqy1iws4.tarticle 7.0 cm from the axis of rotation is ty .p4sqtph1:2 td i;wu.(umtoo experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unifauadl)4y s.i/:k ;p bpormly about its center from 130 rpm to 280 rpm in 4.0 s.: s dli)pybaup4/; ka. 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 radius5 s84l;lo4kjrj,:gh/a*zt f lu ph-dk $R_1$ 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 M+x k/cs sby3ah92t ijw1 xu5887 oibytoon, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribut+h c bs5ty xjyxos8i 729kib8/1at3wu e the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest t42 os19t+ b bms-bhxcco 15,000 rpm in 220 s. Through how many revolutioss t12h 4o- bc+b9xcmbns did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpmr .x 3d:e 7fcbrh21w:j7k nwzi 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 whirlihf((.sf01y qa i o/oxkl9vdc .ng “human centrifuge,” which q1sv(.a (/f l9okf.idyxh 0oc takes 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 r:- i wvmhf,oj,wt xp63pm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled inwx- vfio:,jmw 6hp 3t, this time?    m

  A cooling fan is turned off when it is running at 850rev/b*faw78 pbi9x min It turns 1500 revolutions before it 98*w bb xafp7icomes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformyl awv 3rzr,:s7b)5qo x,e8oxly from 95km/os ,xlxyr)baoeq8 :5 7zw3r,vh to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly fz:qjy j,(w cd1r ,au2vrom 95km/h, udvrw (,a1c y2jq:zj to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all hic(r* nzkjq )d4 r0we/er weight on each pedal when climbing a hill. The pedals rotate in a circle q4ci (jer0rkndw)*/z 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 kf) -n2p (k5fsck vx)ycm wide. What is the magnitude of the torque if n2 )5)pcykk(-x fksfvthe 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 shown in Fig. 8–39. Assu bj jhmeq)/;a3me thathe3)j/ba ; jmq a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass.h) (m3wlc4,wweq r kp m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. I., w3) ql(4prcwwe hkmnitially 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 nks 9).(st-p dwb:g niof 38wp-d.(:9nstnib )gks $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.64i6sr17ebo o 1x c80pnd8 m when the axis of rotation is through ii81s6xbron1dp0c 7oe ts center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diametqz+,nz;ah d 2wxq;y j*er. The rim and tiread+hz ; yx*;zq,n qw2j have a 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 end of a thin, lightp j0cetm+u)(xzx-ys ur-q 4l5 rod is rotated in a horizontal circle of raluq- p(xcyt s -5+mjrx40zu e)dius 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 wheelb; 41qa xns ak0feg/pzr9o1j+ rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay i4 qj9psf+ xno0ge1ra zk/;a1 bs 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Figb ;2optuvn scr 0231lq. 8–43 abou 210lqobr ns;ut32pc vt
(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 tdh n)q oyuy*u:0-jjv, 0 dupy3wo oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct ra0 e )gqdzpuc2w2c10 mocj)(afte, engineers fire four tangential rockets asc w))uf2p10 2a0 c(mzqjec dog shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cyl16-nu (mdbm8ytd1r xhtwf/. jinder with a radius of 8.50 cm and a mass of 0.58ut 68fdn/ x(.y11jdmrb mwht- 0 kg. 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 rest to 3 jpo:/ zx )a5dxy w28sktw4g;x$rev/s$ 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 tangentialvaz/x2c7i1iu /rlh + b hgx+9bly 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 withib7xhvhxi/2bul r c/+ga +z9 1 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 *e4 ;wqlyb e0 o)4qlffudt,:d shut off and is eventually brought uniformly to r e0 w; *byuf,q)ledd 4folt:q4est 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. ka b y :g 3pcfqwhtub/5no/uk+5,t+6f8–45 accelerates 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 thenxkqcrdn-a9 0qqeo e 2.)kf ,3 action of the forearm, which rotates about the elbow joinea, n3r 9)onk xkd-fcqe0qq. 2t 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 eh b7h9 )gtynx+ h) 3dwehclf/6im;b/considered a long thin rod, as shown in Fig. 8–46)hn dchi 9he7/ bh/)mtwxyl 3;+eb g6f.
(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 mayf8x1f2 rwjgy2zb35 j sses, $m_1$ 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 tg9,yk ,boxt6og1gl - full turns (revolutionb6t91,lygoxgo ,k g -ts) and releases 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 moment9/xjk8l ,suon5yc(.3 ww v owm 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 torque o1ps b 5pn,s/pif 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 :i 9onjlh5qz-u 4gf)rcm rolls without slipping down a lane -)5ihl9 z r joqf4:ungat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of to5c 7ea;1s gz(p1*0 je)h0ey nzzjb epwo j(zzo7 pp; s)1nzey*ahe1c 0j0e5e gbterms,
(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 welhz3 xb8)g/ kg and a radius of 7.50 m. How much net work is required to accelerate it from rest txle3 bzhw/)8go 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 rolu h,s ,b*td;kdls without s ,*k tsd,u;bhdlipping 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 veta t2 )o2y / +c7awt+pqapknh0w9dsg/rtically on its tip. It starts to fall and its lower end does not slip. What wo gq)k/ptd9+ytt7 2 w2a0+p/a nachswill 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 the end of a thilz1wqyl.q5ykvk+n o v 6q0awlcf3+w1iw, c7 2n string lcv+5o2w1 71 q.v,wwfc 3w0yia+kk qlnq ylz6 in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whoxf)( j jjx0z-eel of radius 18 cm when rotatin jofxj(-0 jz)xg 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 platformevauh+ djqdpd2 w5oh+78yk h4-.0geb 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 d8+ 4qw0u dehp+h7 do2b 5a y-.gkevjhdecreases 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 reduce hzba0alnw9 lie-b)/i: rd ,:;mlph mwea dx9,7er moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck posit0i:l,mxp9anlbw m9l a7 :ib e)z /;d, rea-dhwion, 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 rat ;tumyqwsq-du8qb e5*,.hi g9e of 1.0 rev every 2.0 s to a final rate of wgm-hue *qds 5t,u yq9i.b8; q3 $rev/s$ 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-m 3ve t4/fusflu,k/e vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of,kef slfmetu-/4 /u3v 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 m .h,b7h* cursaomentum 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 angulaj0moc rzcc** zr+g2.( p .gbk nunr-o1r momentum 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 dr7luvc+7s393zk k9szevp;f7lekbv) d opped onto an identical disk rotating at angulakbfslzu7+7ev;)z l 9k7sdv ec39v3 pkr speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at xn lavm 2g + dc;j01l7t;wyte92.4 $rev/s$ 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 kg7ed0tu mhv 6p/ stands at the center of a rotating merry-go-round platform of radius 3.0 m and momph/etmd7v 6 u0ent 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 velocity of g(ezw bfs0v lq0-s):fi+yrq40.8)wy0q g svi40b -ersfz :lq+(f $rad/s$ 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 eventuallg kjv+6 yl5exgt pw/41y collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Suv gw 1legkt5 +jyx4p/6n’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 wi mxs,unuuwsins6 )v k 1m5or6.s99i7ends 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 l :.w 3u-ldfdwgf11 zq$ 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 m3 h1vjlq9-vdoxfkk7+kvj: 3at rest, that can rotate freely without fric-+v9k3 ovj 3k vh1d7f lqkjmx:tion. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the etlgptfa9d 0.0t6/ u pcdge of a 6.5-m diameter merry-go-round turntable that t0p fp0t 9cda/t.u6 lgis mounted 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 on the ground with the end ofyvkctwp9o s ja/dn r-)* 43-ti the rope lying on the top edge of the spool. A person grabs the end of the rope and walksj*-a r)iwc/ 34n9koyts-tvd p 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. D ntx0 x3h;.p7i as9so2; ifmaqetermine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the lattt;0as7p.aionixh; 9fq3 2 msx er case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist acceleratefm sfb6e./3wytw vj7i+urv*oz 7f2z-s from 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 acqus b h5a3zo.glr(q2qx wc -tsuw*;0k7bires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate theb sk;u0- (tsoh* ra bl 35c7x.qqzw2wg torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of + v;gq6ie. ocahe- j:ltwo solid cylindrical 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 rl+ ag c.vj-eo;:eqi h6eaches 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 angulazmqk -kakd j5,,,wk- sr 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, bmg pld(/ s2*ok/moqg.ut 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*dk.q ( o2mmp//gos gl 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-pojxv/ b8;(og z9j9cwwxllution automobile is for it to use energy stored in a heavy rotating flywheel. Supposecg wjxj/9bzxwo9(v;8 such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without 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 illustratesrf71m nmns)x1 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 h3ba)q)4tl8b4x4j wdj c;sqe bygc 4n(orizontal 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 leng)qvc;ghd6s q 8th 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 force of gravity asdh8;q)c6qvg cts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R.dh2uwqsws0m (m- (d5g It is standing vertically on 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 he0m -hmgu 5 dq2w((swsdight h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is mhzaa69kg e 3o( g07h 2ssmgci)aking a turn with a radius The forces acting on the cyclist and cycl sg9( hm 32seoz6ag0c7ahikg)e 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 z0z0+x /db g+kpa qcwbmui,9(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 9+0wdzq ia+bz,ku cp0/ bmgx(after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms asqjpq2f93 , vhm7u;j *iukxeu 2 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 held tightly against her bouhj27eqp i 2m 3x*9f,jkuqv;udy.   

  You are designing a clutch assembly which consists of two cylindrjwu 2w:7:ma *j9baeadical plates, of 9mj bw:u2*ad 7:aejwamass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rollsax e 1so.bt1gg) r6*ii 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 ist1bga6*rgi se o x).1i to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dbktbd8o ce0v217tq1g o not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as th,8 ox-; dbkg4e:wqrqy e car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the anbexy:8wrqg dk ,o q-4;gular 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