A bicycle odometer (w nf*zg(-bx;x uxpz9 t(;z.y orhich measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What hnxox*xf bp ( g ;-zrtzz(9;uy.appens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Do* kmgd3(b3 tobv58mg t2sn.9 )kkx zurp+z9jxes a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radian3(*xgz)x2p3m9jzsbk o.+9b vtmgd58 tkuk r l 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 5fkz k:d9)(;tmov b:j :he ipiangular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? c3*sd6kdzqdvo8 a 2o;Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head than whebtt+tv buc ;m t+3:h9en your arms are stretched out in front of you? A diagram may help you to answer t:mbcv u;3t+b t th9te+his.

A 21-speed bicycle has seven0avj0co(+h ow sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front ov0( w+0j ocahsprocket? Why?

Mammals that depend on being able to run fast have slekgu+cw) :w 4ctioqytq0 q/l d1 3l,jr,nder lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, expl:1 /)ql3grwl k+dw cut0t4jqq,c,o y iain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow bea/ i6-lmmzkb;cm?

If the net force on a system is zero, is tck7hz9sm: 2k the net torque also zero? If the net torque on a system is c2:zkkt msh97 zero, is the net force zero?

Two inclines have the same height but make diff1cy l -g1-fozserent angles with the horizontal. The same steel ball is rolled down each incline. On which incl1-z lcofs -g1yine will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start roll h 1w9ki5tu 1o.u)kj7nnk xi7 .r2cugjing (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater spe21h )wntc.ig9 x n5ujo ukuik7rk1.j7 ed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass4ot yu px /-+ 2dcm+)uqp1xsf8m -rjuy. They start from rest o- 21y8txpduusjcur x/y-m+pqm f+ 4)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 the greater rotational KE?

We claim that momentum and angular-qvji hz.nt1lk r,-kx,- wb2e momentum are conserved. Yet most moving or rotating objects eventually slow down anr.k,qeb2 wi -zj1x-lh tv-n,kd stop. Explain.

If there were a great migration of people toward the Earth’s equator, howc23c )fddwthbu)x8i3 f a .9jr would this affect 9ca3fr)xicdw f8tjh 2u3b. d) the length of the day?

Can the diver of Fig. 8–29 do a someree neg 95(jpas 2cu9guad:h6 2sault without having any initial rotation when j(:e e6 2sagga9hpd9uec nu52she leaves the board?

The moment of inertia of a rotating solid disk about an axis th j2k 4:q8qj0 y8tf8kemf2hms9ynb g/j r0ei3erough its center of 3482g980jr ftfmjyqk0se/ ni km8eeq2 j:y bhmass 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 ond b qs8z 6skzuu2i3+4lu mm mlfg+9*n4 a rotating 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 sa2kmuzl+ m 9z3q+d s insg8*m6luu44b fme? Explain.

Two spheres look identical 5m)d4*/pcs gshmpvtb r +q8v4 and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is which.m*pg d / t48p+cvsq4ms5vbh r)

The angular velocity h- l2yu,l)dr*,oa lh3 4a)oodbph, pbof a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, descrdh bh bdp 2*,oro4a,h)-) olaul3, plyibe 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 la/h-tlu umwcvy x:: jjtvx -500rge freely rotating turntable. What happens if you walk toward the ceut5 0tmwyl-- /:xjj 0chv:xuvnter?

A shortstop may leap into the air to catch a ball and throw it q3.,dmrurkzd4 f i3oc6uickly. 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 direc.3m iu,dk6dr34rof zction (Fig. 8–36). Explain.

On the basis of the law7-mc+urgwe lm,7ox t. 5x*mxo of conservation of angular momentum, discuss why a helicoptw m+5mtu- x.mloo,c* re xxg77er must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angle s4j )4pz0*vjv*nelawhec ,* hs in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an/1cuus w/uy,(+c tkrzr x e1s+kexv5+d5u 6zw amazing coincidence. Calculate, using the information inside r5(x/kwsuk6 rcvs c,ezt1+yu+ w/5 ue uz 1+xdthe 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 s mz qu)9oz o(:m kf5(+q6 2bveai*qwcEarth. The beam diverges at an fovzi(*wq6zo +k9 c(maqs m)5 :e2buq 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 yd) t w)s2sjq-g:-caz1 ys.hz6g+jz wof 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. Wyj1s-ww j z+qzs)t)g.2 ac-hszdyg:6hat is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makes ((qwdq.+dsq1 lj bpoo8cq +2,t v8ul j15.0 revolutions, whatldqovt(s+uj88+. cw1,q do lqp2q(b j is its diameter?    m

  A bicycle with tires 68 cm in e- m/l/q 7vztydiameter travels 8.0 km. How many revolutions do the wheels me/t-zl/ym7q v ake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its cna*lf/ tp 2v,angularl,v/npf c2 a*t velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one comple a p7rdw(53wqmte revolution in 4.0 s (Fig. 8–38). (a)(dmr p73wwa5q What is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) i 50l)gsg6 *5jijfrw+zku oo.jn its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of av:o)0gzz/pvbohp ( 5y 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 centm7xb8m9cjcz9q t ;n/f/ plcc*rifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration/t;pzj8 9c* xfccb97m/nlqcm of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unifg5m7 ,sm tykkutdxb shb a*k6rj:,-+(ormly about its center from 130 rpm to 280 rpm in 4.0 s. Deterg-(,b7jhbt*sm xkm +dyk k65:urat s,mine
(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 radiuy+cc5 sgqu/bui(g csf+24z8ds $R_1$ 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, astron nl.zz -e3pdp:auts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minud3ez: z .-plnpte 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 uk+k zy9 c)glz4.m: hminiformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in thi+mhzkk .:)y4cgim9 lzs time?    $rev$

  An automobile engine slows down from 4509jc8j 1fg5hbw 0 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 flyinn :nn5- iafr(kg highspeed jets in a whirling “human centrifuge,” which takes nna:-5 in(rfk1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360,u uetw ch wq yy;(/sd2.8s6pw rpm in 6.5 s. How far will a point on the edge of th wwwhtq;uy.(e6,sudp8s/2 yce wheel have traveled in this time?    m

  A cooling fan is turne ( qrno tabaoxsm( h8(b87((eud off when it is running at 850rev/min It turns 1500 revolutions beu8 sox at7bhqor((en((m a8(b fore it comes 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 revol50y hut-(g6gjag ,al 8k*lsdeutions as the car reduces its speed uniformly from 95km/h to 45km/h The8y 6g tse*uh-a0l(l 5gk d,gaj 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 xaq 9x wypn(es)0q+/lpt-z+m reduces its speed uniformly from 95km/h to 45km/h Thepm/te) zxl pnq +q+x-0 yswa9( tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when cqabu q .m ;l.f; jti5:d3o1nvtlimbing a hill. The pedals rotate in a cir.;31; o.5t qntjmuvdqa:bifl cle 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 bq 1xw qj9tlvci) c;gw b;s1 i3k)-vc*a door 74 cm wide. What is the magnitude of the torque if the foqbcw1cwc; ;9ltv-3)i)g qji*1kx bvs rce 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 toro,(ejr:f +-eqt(qlrt wmh;9e que about the axle of the wheel shown in Fig. 8–39. Assume that a friction to(, r9t:q-oqwt (+er;e fmjh lerque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of ma0e,psdc(;maojzb s .2 ss m, are attached to the ends of 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 direction of the net tss;pc.o,0e2(m a bjdzorque on this system.

  The bolts on the cylinder head of an engine require tightening :t; )5pquwa.ajay7qbav n+k ,to a torque of 38 p+q: ).auaw ,;tnjb ayvk75aq$m \cdot N$ 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 spt(dlep z82mq)6;n8 xozvz:u phere of radius 0.648 m when the axis of rotation is through its cente 8;epp8z: v 2 6)xdo(tzqnlmzur.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheelb hb6ymaah b0 1v2u9+d 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub dabh2 y+bh 106b9vaum can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on m1tytn247n, os2ci gj bt;o5cthe end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calcgytntcc, j4n 72 ; m1tbios25oulate
(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 t71f5i ajn3f*uckh9h bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force betweenna 7 ukhth59cf1i*jf3 her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of ineruhlq ,d)gk0m0 tia of the array of point objects shown in Fig. 8–43 aboutu,q)g h0dkm l0
(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 whose total mass z-:mna8*xnh h 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 spinningr,/sr0vt ey -mf +dv(e at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, w, rtdv/eyr- v0(+efm shat 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 8.5;i1jhx m.ajyy wtz: (+0 cm and a mass of 0.580 yijwy (jzha+:1 x.tm; 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 to9 yx)/axdm +0igvi abh*) -kt*ts 5yno 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 and ir;v i xu5hhv 5 )1qoan.qu0w8js able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce th0 nx v.o)u;5wurqhaqji158vhe 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;-p 2m-sbin0w 62qgsiux,g:cm(dlww and is eventually brought uniformly to iwm-bpsw,m qdc 0g iw2(62-;:ug nlxsrest 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 ac9wjwj/fkku*9lb4z-s yzy lm 0(s8b dmc, ti)1celerates 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 actd ,zbp f, (nm0grq -/v,3mkylwion of the forearm, which rotates about the elbow joint und,mwdn0 r l,m3,vf-gyq zp/b (ker 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 bladejtnv6 .m m0--bgtq.;+ suayy y can be considered a long thin rod, as shown in Fig. 8–4yst-0-.v qbnmmty6. u a;j+yg 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 k6djsp20nyrh v9 li+:consists 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 turns (revo0um -e q/td/bal3r;-offjd o 8 3e oq-/lo-j0f u/atdrdbmf;8lutions) 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 mom)7ht/w, uv*zux dc6xzent 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 oue g)wo77.t7pqqhkn4t2a owx 08kc4 bf 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 slipping down a 3h(+id dhe0jnp* ; lqklane at 3.3 dh0+e ;hjqd(p *kn3li $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic e z. v :sbcrr7au1)sw2nnergy of the Earth with respect to the Sun as the sum of two terms 7 r)21uzcwb s.:navrs,
(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 mapewv0 x5a4*jqwmq8 5qss of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate itjqv4qmq5x*a0 ew5w p 8 from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and roog: pkkxz/oicm22n 8e 4q-n1f lls without slq2/k :mg1oikp feonn- x84c z2ipping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It r,,xmx)filf)egs 5yr,zi0 p0starts to fall and its lower end does not slip. What will be the speed of the upper end of the poley),5m0 xre0,lzgsrii ,)fpfx just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotat ndvbse bz- /f8n)j b7 wjt705++efwjling on the end of a thin string in a circle of radius 1.10 /d +nletn7 0-jf7 sw bejf8)bwv5+jbzm 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 whqf9i 8i4f v :o(wolwj2lmw o6;eel of radius 18 cm69qloiljw(wf2 fw:; 48movo i 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 rotatingujvhp-pp 0 ma /oa.9bs4 :bd7sm tgd9, at a rate of 1.3rev/s If he raises his arms to a horizontal posigophmbb.mus0tp , p47 s9a9v:ajd/- d tion, 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 hel ;4p/i nj7jgxk da.2b* hakl0si:gd6r moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2hd.l064i dbjlajgni k;kax*: p/sg27 .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 rotatiyb8/cy+kf y.l2j ll/qon rate from an initial rate of 1.0 rev every 2.0 s to a final rate of y+lyl b/ qy2 kj/8lf.c3 $rev/s$ 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 c.sizk+rzz n1 63cx7w+h bnm2a 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 clay, approximately shape 7 2z1xbcz6nkwh+znric+ 3m .sd 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 764zak 3; osdhert- j/bc9bqk skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momzw2m 0 so*:okexywdlgv+ 2-5xentum 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 i;j1ta q5how cp;0 yiy3s dropped onto an identical disk rotating yj0tp5a hyiq o 3c1;w;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.owd( i9u ::x /by,xvybg 3yj+w4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round platecl.hj:jy( q(+j6d go (lok. cform of radius 3.0 m and moment of iner6 :o(j.(lyg jo kdq(cljh.e+ ctia 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-roobie,- :i gv*cund is rotating freely with an angular velocity of 0e o bi-*i:g,cv.8 $rad/s$ 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 dwl2y:eyo.jg1rl9) xym arf, 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 Sun’s new rotation rm:xyorl1 )y gel.y 29jate 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 involvt 3k3n. r h*1dl4esdsee 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 z 1rv1bo2spcg - , cdf-qy78ga$ 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 hd.(cu g/kj(x rest, that can rotate freely without friction. The moment of inertia of the person plus thd g (.ck(ju/xhe 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 d /-pfb4ibvnne fyh(.4 iameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia o b(nypf44 -bief./n vhf 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 of :wdup-vjbh(kc;bh7q m9k. o /the rope lying on the tok.pd/v km; -bboj 9huc7hqw: (p edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the E+:/1 p wls: ejx3jskfkyk7o, garth. Determine the ratio of the Moon’s spin angu /l+k skj, :fewyjx3o7k gs1:plar 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 f. ezcs 5v3pta*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 z mqb4;kh.2wfn1ini 1ng9 k*jof a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at i km1g.n1w9;fkhn ni4jzq*2b constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 y-a 8p jw+uuv u2lchr4) hm)0jkg and diameter 0.075 m, joined by a (concauh 2ywu mcu4l jh+j8vp)-)0 rentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how, 3zj7+ uzmpll is the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our l*)o3.o80elsaao s lhSun, but with mass 8.0 times as great, were rotating 0lla8oo* eh .o )ss3laat a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pxup,mj-ze.g ;ollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a cae -jxmzu.,pg; r 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 illustrates nt8lzr )wm b geby91j4*:w9gtan $H_2O$ 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 h t2+ 4h lm;bnmhzh0-9ptn zio3m2c 9/jwrly3qorizontal 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 cawxc c+jz0q0,tn pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is x0,cw z+ t0jqcheld 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 acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has raawpuq6sd;4 7 fdius R. 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 (Fif64dsu ; wqp7ag. 8–55). The step has height h, where h

  A bicyclist traveling withm3 /doocaar-3 speed v=4.2m/s on a flat road is making a turn with ardo /a ca3-3mo radius The 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.np ,66mr0kc8kedg t)o50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it fromdk ,t86 e0prkgo) n6cm 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 :bp 57a .ryvgssolid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her bo7 sa5p.bgvr:ydy.   

  You are designing a clutch assembly which consists of two cyljh*o;hab-s0z+ vm9 u lindrical plates, of m+9*m;l z0ohjva-h b suass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ra/anemi (83q colls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble musmqe(38 n /caiat drop if it is 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, b otz qig( f9tlp :q7d4o5e 0d,xi1aktxac48t 7ut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its qj pys y)5ss(p0i3h.0 qo(cfw +u-akz speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was zapfsyicqso -) (( 5.wy03 qskju+ hp0the 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