A bicycle odometer (woo87 w nud5c (6w+ovghhich measures distance traveled) is attached near the wheel hub and is designed for 2on(o 76cuw gd o+w58vh7-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constaz d1 zn3 1jabg+b-pi2wnt angular velocity. Does a point on the rim have radial and/ z-j 1zbbn132+pdia wgor tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular vz4+j9qk:5x s ihr vq9lelocity $\omega$ Explain.

Can a small force ever 4ei*a,v6( d igv ahf4hexert a greater torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a si4eh5: dqlz0 nx)hli (dt-up with your hands behind your head than when your arms are stretched out in front of yodxi:en4l )5 lz (0qhdhu? A diagram may help you to answer this.

A 21-speed bicycle ha0ccak7oy ;wsug9z qn*+p+u+3y d6r lfs seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rea63c+ w0kzp rc+da g ;nqsuyfylo9+*7ur sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fasue).wn +rp q4i3y8 xjlt have slender lower legs with flesh and muscle concentrated high, close to p8e4ny+liq. j3u)wxrthe 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)9 1i ue6 des,z5yc )c+suqbpg7 carry a long, narrow beam?

If the net force on a system is zero, is t 8 bi,yu(i; gv /w a0c;gquspa*ni6-hfhe net torque also zero? If the net torque on a system is zero,hv suiuf a8,-y*iaw ig6 0q;/cbgn; (p is the net force zero?

Two inclines have the sam d2ec4w8ww:4 2eoug 5(lxwkgue height but make different angles with the horizontal. The same steec w482 g(lu w54wu :gowek2dxel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (fro(vfku(0cwi6lumiu-rqqu4 0 gq:b; d+ m rest) down an incline. One sphere has twice the radius an+ ;uuqu-: igl0q(id0cv(6wrmqb f4kud 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 the same radius ar,g 7tqiwn 7 guarb:8mz/ 9 sa(/,ko(xq- juqwnd the same mass. They start from rest at the t 7b,89a,xto jwqr/nwuuk/(7 s-qg(rm q:i agzop 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 the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving o+l zaqga+zl3uc4-- wa r rotating objects eventually slow down and stop. Explac-+ q+4z llaa-wgzau 3in.

If there were a great migration of pew8f)qj l2nk7p ople toward the Earth’s equator, how would this affect the)pljk 82nfqw7 length of the day?

Can the diver of Fig. 8–w/3 m.wv u yh1 q90wmruka+lx7wapm/ /29 do a somersault without having any initial rotation when she leaves the boardwa9hr mx0 ukpw 1qwvm./7u+/wyl/3am ?

The moment of inertia o x q (w4ty)b4+:kprbq tw8lvjjvf+7 :xf a rotating solid disk about an axis through its center of yf7:w4txbk:qw r t8v)bv jq j(+x4+lp 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 rotat7w+y3p c sxa),l8pfrt 8m8bdring stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Exp8d88 c slr3fp y),wbxt7rp+a mlain.

Two spheres look identical and have the same mass. However, nlww,iio ;y3 vsk9-,)cgq h)s one is hollow and the other is solid. Describe ai)ik9l,wsw-o 3yh)v ;,g csnqn experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In vergxaf.v3pd8.d) k9( 5 *lyw s(pcwnwhat direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangen.39wgw.axcd5vknv(pf e) r* (ly sd p8tial 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 la*0t0v+jw tku nrge freely rotating turntable. What happens if you walk towk* j0n+vwu 0ttard the center?

A shortstop may leap into td(g2xgztx; wk*8s-n+/z/ xjc u +f mnhhe air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. xg th jck*g/-;+w2zzf/nn s8ud+mxx(If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a helicoe:e n )r-op9b ,)datdopter must have more than one rotor (or p))9,aoe-dr ob: etpdnropeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in rajzj,ss0)( j hpdians: (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 becauj8ie6 cqzs,, hyd0 ,qjse of an amazing coincidence. Calculate, using the information inside,j zcyjsedqh 6,i ,q08 the Front Cover, the 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 from Earth.t0/hs p, v0nu(ii5s p,klgd)j The beam diverges at aipik, )v(0d h ssl0g5tjp/u n,n 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 6ay p:b58gqk , eim5a8:f8 :aqomxd4bz500 rpm. When the motor is turned off d 4m::qy b5eao:b 8g5xpifz,amdk8 8qauring operation, the blades 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 ta80zn4d0ec;9vqqj *gbs 7hxco another child. If the ball makes zc d 8cb;qv*9e 4a70xjng0s qh15.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do tia;;i xfeey2 -he whe ;;fiix2 ye-aeels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotateyib8et tm z(vy8 8q rdz/:.ku:s at 2500 rpm. Calculate its angular velt ivd8/ t:bz8k(uqr : ymz.e8yocity 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-roupln,wd44v 5x/e yyxh ch543lvnd makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m fromny3e 4vh4xpl 5x4/d cly5hv, w 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 icjp0v3 go +h dw71uj 5vwqz2l-fs;f)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 cpd)ku 23 c(scy05 dr l,ip0ewa 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 particztuu7 d zk(x06o+;f(;tztgl cle 7.0 cm from the axis of rotuu oz k6dcz+ zltt;((;f70 gtxation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpmprh+6wis/5dcqg hu;1 x+ l8 rj+nz q1n to 280 rpm in 4.0 s.niw1qrj5pqhcx/h +n l; 8d + +16gsruz 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 radiu3f s:p+ *j dpee)j9krz 1-kvocs $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft pun*s guteg.w m( l7kv9*t themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of l k79 *.mg enw*t(usvgtheir 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 to 15,000 rpm in 220 s.,gwxfd gh ;+q/ hhz*b5 Through how xzg,fh +qdwh*h;5/gbmany revolutions did it turn in this time?    $rev$

  An automobile engine slows down sn 66x )uus(2 r)szfjlfrom 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in z:j x;)rtx n9wa whirling “human centrifuge,” which takes 1.0 min to turn jz)x;:w9rn t xthrough 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 accelerxhk9t2l+avwfk /smf9 *17vqrates uniformly from 240 rpm to 360 rpm in 6.5 shl79afxvw/st f kq+r19mv* 2k. How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is runningttfk u/4b)+rn at 850rev/min It turns 1500 revolutions before it comes to a sttunt4f)k r+b/ op.
(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 uniformly ftryr5lz7s2 o8:ame3ts9rqb4 t8 s s7yrom 95km/h 2rro 3 4tty5et8 7m8q9a bsz7: rsssylto 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 speebh86ut+hxp .dw2rtc : xtn2v2d uniformly from 95km/h to 45kmh2 6x. :cw82dt u+2rbnttph vx/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 onpmcq-c20oaf 2o)jl.8+ ebfcw each pedal when climbing a hill. Th f c0o-jeo )2c+.8qb2c mafpwle pedals 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 o(ywxi(w0y1vd f 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exertewwy0dy 1 ((vxid
(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 wheeldf,qe6va0s j+3l0 k ll shown in Fig. 8–39. Assume that a friction torae0 l6jds+l3vq,f 0lkque 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 of a massless rod whiqtml eg90, t sp996dcbch pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then relecsg 96bl,9 0tmdpqt9e ased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder he1 .2s;kg ixfyiad of an engine require tightening to a torque of 38 y.12g; skx fii$m \cdot N$ 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 of2vmq 8, nd7 ct63c;gg3 uzjflj a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its cente62 umq;jg 8ldzngctv,j37c 3f r.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The risv n psod7 .qp3h-)q6pm and tire have a combined mass of 1.25 kg. The ohpnq.pq 63s p)sdv7-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 rotateu l)afl-4pepieyf6zx;f 8r:+ d in a horizontal circle of radius 1.2 mep+)xrp :u; lz fyai e48f-6fl. 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 constaxr rga jls,y/75 pvx1/nt angular speed (Fig. 8–42). The friction force between her hands and the clay ig5/7/a,sx jprvr1lxy s 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,l fguf4u)3 8m:gx8m s*ouvhe of inertia of the array of point objects shown in Fig. 8–43 ,4fof*v)us g 8eluxum:3gh8mabout
(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 whobe(:5) iy8hmqkge 7z .e ,qr,)xpgayxse 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 at0qeh)v3y:lu4 l d af,h the correct rate, engineers fire four tangential rockets as shown inlla 0u h)yf,4dqv3: he 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 massdottlg(n;25w;ysl.y z 4 a 3zc of 0.580 kg. Calculatezgsa3yl o cy;2 .t4lndz5t(;w
(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 swinh8 9r ui/paqgig;dl-ze- ; py/gs 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-d+m 4+4 bairix4lwcp-driven merry-go-round and is able to accelerate it from rest to a frequency ofx cr4m 4plia++ wi4b-d 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually bp - lr3osto9zk)nvi35rought uniformly to rest by a frictionalrn3p-3t zkloo)9v 5is torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-.rf mpq 56nfk9kg 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 o5 li 0wlgki xr)u.2-sjf the forearm, which rotates about the elbow joint under theiwx2. -kr)l 0j gu5sil 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 rok9i s gceoudgn2 (q+23d, as shown in Fig. 8–46qnkid(9g2g e 2+uocs 3.
(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 masses)4d6 zz)nok x 0ug,j8ihuu4 us, $m_1$ 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 resjn+ym-us*p j, t within four full turns (revolutions) anuj-jsm*yn p+ ,d 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 moment of inw(5vl(qcwnypj +st x,h;d) w 4ertia 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 torq: ehy5v,j(**;tigkgf ,cnrvz ue 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 a8izr i8 af33jjnd radius 9.0 cm rolls without slipping down a laij3 8z8ifar3jne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the /v.*.p o w7lss3uxdr vsum of two termsp*worlvv3/ u. d.s7xs,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50;d93/zual 9x zp :jzhd 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 is alxazju9 ;d3phd:9 zz/ solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls wndi-kx *ym(m+nt gi,0v h3:df ithout slipping down a yhtn vfg, ki3(: *dmxmd-n0 +i30.0 $^{\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 vertically4kcf9)h07t 4k mqikzs on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the gro 4)k tsm7h4cq0kfikz 9und? [Hint: Use conservation of energy.]    $m/s$

  What is the angular mommc5j m7m47haa8ikq,4 +2cm ;rityy5 b onmx/uentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.4+m2t,bjy7/r5 q7mk4min ocamhyi a5 x8m;u c 10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a tswd t5 +w z(ycax*j t82+ipo52.8-kg uniform cylindrical grinding wheel of radius 18zcw a*+tjsoy5 +8td2txw ip(5 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate of 6lqa l*;c 4xlo1.3rev/s If*q lll;o64cxa he raises his arms to a horizontal position, Fig. 8–48, 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. 8–29) can reduce her mh48 /ilg-z vejoment 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 posithz4g /vj8-i leion, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate yg5m 8zs9dgm9)p qq8ot t)/) 4xayxpa from an initial rate of 1.0 rev every 2.0 s to a final rate y gqa m 8ax 4sz)otmyt9q/9gdp8p 5))xof 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 vear8*sh2:vj qp up ,x-zrtical axis through its center at a frequency of 1.:ap-p8* hzq 2srv x,uj5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning a:bhzr 1hmigs zce8l5 k 8og jk*0lv),9t 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 Ear, 9lzof 2 vklp6oicq);th
(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 cylindric) mv;qkrv iu.5al disk of moment of inertia I is dropped onto an identical disk rotqv.);imv 5urkating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns755 meg c xfaomdpjv2-uy43i2 at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at t91zs6e iuc6tthe center of a rotating merry-go-round platform6zts9ct 16eui of 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 an1 *txo.1 nuu rb0jcub9gular velocity of 0.8jcbubr1u t *u.91n0xo $rad/s$ 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 into a white dgv7 obg1p7 7 -oo7eizzwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost7o 7ioo1 7bg 7-gvzzpe 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 involcy-j;(bkex2u mx-.rk ve winds 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 r,mptb6ja5knvm(l:8 ,tna- v $ 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 platfo(4o+( dwsi;rjg9o uzfzyz2,w rm, initially at rest, that can rotate freely without friction. The moment of inerz2z4 j;s o(z w+yuw9id(fgr,otia 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 personhi y:r fbopb.;dn.kao;;n s2oa v ;97t stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a momen;bf;s7onr;.i. a:t n ;pkhb od29aovyt 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 rj)c(y na; y;*5cjwldhz/ge dd a3cw/6olls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What len53)hwzya adcy cdnj /j l/dew*;;6(gcgth 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. Deterl.vilvw. *eur gwbr55y54*q f mine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angy lvf. *ugrq5bw 55l.rwvi *4eular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a r xj xwwctfl jg9v0(d0:4 :(wtaate 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 ra- u,+h 0jvxccndius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the t vj-c,uc xh+n0orque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 w 6ig*ag*e:nd.)n k8 pc6ryibdd+ g1z 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 e grgwn1*izc6bk)+i *:ndd.dya6 8 pgenergy 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 o.aafxn- g6 evw9 a2t,zf the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, wertd4q j pil(y1:ro c:x( k2dko1e rotating at a speed of 1.0 revolution every 12 kpdt1 1(4cjo o 2rdk:ylx:(i qdays. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy st v8wd- r(osz2pored 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 d8ro swp( v2-zbe 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 illustratix0k3k2r9 i-uh 8rrhv es 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 rm(k d xd*4ugod7f *s/dolling 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 L can pivot freely (i.e., we ignorejdom;y c( w4v1* d(dne0,seug 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,(; * jd,( 0ywvcog 1umneedds4 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 vv qdt.:fw13j rertically on the floor, and we want to exert a horizontal force F at its at :rfd3v1jwq .xle so 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 +snx*ab8wsy/ww 8x eiq5j+s/ road is making a turn with a radius Thw+/qaw8ibx nwsx s s+ey*5j/8e forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.p(uvhrdolsa47u 5*vz,og w,4 50-kg rock into a sling of length 1.5 m and begins whirling the, v4uurg5 ,*(d4hwpaov sl7oz 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 feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylind ay/70 nd4ur7b0jzwe0ww5r9 qjapn1 xer and her arms as thin rods, making reasonable estimates for 0qba5 7 jwpr4wy/za1j7xn9 ur 0 0enwdthe dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch asse 1xl //v;dfh2sgpxyk 7z- fuo/mbly which consists of two cylindrical plates, of mass ;/2fxv gxu/ldy kf1h o/7-s zp$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 roxk v*, e)bfp5q. fg5/g+ mrgmvlls 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 reach the5) gvm p* ,feqg.fkrmx/+ bg5v 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 assume *-c,n:4nmnw dgih k/i$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car rzgs u+d0 4pykmq*q1 b:lc;;udeduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) Wh zu:m c0y;d d1gu+b sqqlp4k*;at 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