A bicycle odometer (which measures distance traveled) is attache, 4zs,txhk3kx w0edcy i0 )9 i3 o+a9:nmetxpwd near the wheel hub and is designed for 27-inch wheels. What happens iexn hct0dotx9z, ,3w:+wk x)y4mi ike9s 3a0p f you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. D4yyqsgj;2, xi oes a point on the rim have radial and/or 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 li;s4 y yg,qjxi2near acceleration change?

Could a nonrigid body be descr14m(ebmj dk8 cmnvu)9 ibed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Exple odrp6m8ys0;ain.

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

Why is it more difficult to do a sit-up witsx w8)*tfq djob-qe17h your hands behind your head than when your arms are stretched out in front of you? A diagram may help you t)7 8jqws*otde q1fbx-o answer this.

A 21-speed bicycle has seven sprockets at tpyw.lmzy ut3 k+gxjnp2 o)+;8he 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 sprock otl pg2yzpnu+ 38kyjxm;+)w.et? Why?

Mammals that depend on being able to run fast :jebbqn .5qhxv2zmo/x /1 m8vhave slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On . :v hxn/x1mv8e2j boq5zb/qmthe basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walcdx1fw3j -m z+kers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, id6g mz30lhqphx64x m -s the net torque also zero? If the net torque on a systxd-g3lh xmh pm 64qz60em is zero, is the net force zero?

Two inclines have the same height but make diff q38ui4vgpn l.erent 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 greaterp3iv un8ql4g .? Explain.

Two solid spheres simultaneously start rolling (from rest) dxeq ot,lx:p8lc+m1ajd5h s: 0own an incline. One sphere has twice the radius and twiex xlms5 0o,dp:ql h8 aj1tc:+ce 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 and the same mass. They start from noqs* f4uz.u, ruqf1u*b, qz5 rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE 1 .fq*,qrqu*b zuuz4u ns5,of?

We claim that momentum and angular momentum ara :zuv hp9j7wlt4 ,p9u5l5acke conserved. Yet most moving or rotating objects eventually slow down and stop. Explailtk aaup95 jp74lzu,w9hc:5v n.

If there were a great migration of q7azl zcfem/5. :z k z84e s:0lobvn:vpeople toward the Earth’s equator, how would this affect the length of the da:4.: v /fbz5zlmzlz vo n7asck0e8 :eqy?

Can the diver of Fig. 8–29 do a somersault witsn+rz 6w9lb :4qm/ l81lutu6i(n kju shout having any initial rotation when she leaves the bostw9z k1bml ju8n:+s rul6u6 q4ln( i/ard?

The moment of inertia of a rotatcpgumu. i9u,+q o k3z9ing solid disk about an axis through its center of mass +uquipu9gmcko9z ,3 . is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each +8hv,u -v7uj f sodl+foutstretched hand. 8,sul-o+ fh+jf uvd7vIf you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical an.csufs/g a6 -gd have the same mass. However, one is hollow and the other is solid. Describe an experiment to determas gusc /fg.6-ine which is which.

The angular velocity of a wheel rl5g+yh3n2c9qy :qddp otating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of thisq53dgc hn+yylq92 :dp point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable.z6 wi, ak0ome)wl9x(rk2ua*t What happens if you walk x9okw rti ( 6*uza2)k0 ,lemawtoward the center?

A shortstop may leap into the ai 9pw,e g )mp87nlg d kfgds:pyu:c3-0er to catch a ball 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 hipsfk3 0u)myw,ceg8 l ps:-np:p7gd9 e gd and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law 1vhfkyai3;) v of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep thef 3a;1y)vhi kv helicopter stable.

  Express the followinu(ds-ty .4r/d6qbol sel.;y g g angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Cal:6 cwpjauvl)ys+*qw0culate, using the informationp:q lu0wja +sywc*)6v inside 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,xwh 0 8/,lobr/gvtqe8 000 km from Earth. The beam diverges at a/lx08tqehbv/, rw go8 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 6500 rpm. When the motor xp)c xr43umw8is turned off during operation, the blau4rcp w)xm 38xdes slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the bdqg2(a5w*jk 86 sohs ball makes 15.0 revolutions, what ibkog8sj(2d qh5s 6aw *s its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions d(*.dh 7jq-i5( mz(xsn h wp,-fuaqrzxo the whees(qqhz5, f-nzpw.xmu- (rx (jh*i7d als make?    $rev$

  (a) A grinding wheel 0.35 m i( dm-l8 ihav4vn diameter rotates at 2500 rpm. Calculate its angular velocity in vl8d a(imh v4-$rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38). (agcyv1;8 zs g9 oz)pq7v-iv y+.l 3mve)fgeyc+) What is the linear speed of a ch.8 sm-g yqy y;of3)c p+vgivzz l)1+ce9ev vg7ild 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 a+v)5gyk9 euv lx8a 1s)geleo0round the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of anmhkl r99c5w 6yo7u; /,5unf+jqo tay point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axis ou(bko 4 mgh84y/g)h4+5p koehs8s bokf rotation is to experience an acceleration of 100,0 8kb hmpoh 8bo(oyg5/4)hue 4k+sgsk400 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates t0s8w( 1po )ld ulhfw(1c wpc2uniformly about its center from 130 rpm to 280 rpm l0fcsp2ww1wl htc(do1p )8( uin 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiu9*eapug. 3z9zlov.c a s $R_1$ 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 spaceczv*5v33 poehpadp d/-raft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder vp3*5va d3zehd -pop/ 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. Througuahgvor4d6e i*- k:n- h how many revolutions did orh-ui e g* 4nk-ad6v:it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 120+fv:oo5h;i pe0 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 e j;fpl83je.( t51hszsnwf b- highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn thrpe -15 hb(ezss8.nf; jw ft3jlough 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5 z)p1:u fn 72lmkxglma um:6i0s. How far w:k 7 6iun0pfgam2u mxl: )1lzmill a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is runnin32kfjs, e9gpct +c 5lkg at 850rev/min It turns 1500 revolutig3,l5+kf9c2 jc etpksons before 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 revolutions as the car reduces i.l ,18mzgsgvjy1)2ui-kbxi d ts speed uniformly from 95km/h to 45km/h The tires have a ,i1y-klijgs1xm z du. v) gb28diameter 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 uniformmi;su0g, /4e mwlt xg9ly from 95k;slx4 0ueg9gwm/m t, im/h 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 her weight on each -l ++/6qk0fjsix yhsl8y jy 1k4i o:pqpedal when climbing a hill. The pedals rotate in a circle of radius 17 cmi:6 ji0 qky sf- lp4oyj8hl1k/sy+q+ x.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. Whatk kjyi*(-z m(fm5a9w lsto* .qqz s9i; is the magnitude of the torque if the force is ex k( s l-mmqzi(foy.ti zjw5;a9kqs9* *erted
(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. Assumew+ sr,gh2.bph0p3 kma tws.pg b2r ,3hk+0amphhat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are 1k:gfb76hsmrn,m. yk 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. ys, 7mk b1: fh.r6mngkCalculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tighteningy),s t*u*6t4dry ubew c1s fc- to a torque of 38w1 sc,s-t)d4uyebyfc*rt 6*u $m \cdot N$ 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 tndlcd l ba c4/f8(5f :zh(rgye o06e(of a 10.8-kg sphere of radius 0.648 m when the axis of rotation i/e d e(ffy4alh6nb dgc(t 5lr(0: ocz8s through its center.    $kg \cdot m^2$

  Calculate the moment of grmi)+q9p 1 1flb dhbwou9i1)inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a comb+9q oui ip )1df1b)9lrg1h wmbined 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, light rod is rotatedkynsw 8;4;-via kfl7x in a horizontal circle of radius 1.2 m. 7sn ;a-ik; 4lxvw8fkyCalculate
(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 rotatinheam1z flavv.q4(p3 (g at constant angular speed (Fig. 8–42). The frict4q3emp1va(a(lvh zf .ion force between 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 inertia of the array of poin)vjiqm+i rh.7t objects shown in Fig. 8–43 about 7).jrqihmiv+
(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 tf ;iypge l z/*ln;,r z;m1o9vwwo 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 * u )n:(qghjl n-i m9slbrn*7fcylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the hln f* ) -9ji*7 mngbslu(nqr:satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass 7wb i 6d/5cxnrof 0.580 kg. Cx n7/ c5irdb6walculate
(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 swing0q qje0npn2o:) lq8b os 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 tany83 znd)ko tsqsdy(17gentially on a small hand-driven merry-go-round and is able to acceler)y3(o s18n q ksz7yddtate 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 the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rx-ge jrt6c i3zq4al:2otating at 10,300 rpm is shut off and is eventually brought unifo e3a:2t4xzq r-ljcgi6rmly to rest 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 accelerates a m un 13d0hcs hs53bxq53.6-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 if 4z- /icsxoiz2 9oncm,9w ;mzs thrown solely by the action of the forearm, which rotates about tho-x czcmi sm ,z;zof42in99w /e elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be consid mua2/bqc/okgn-j9l :4k*wh mered a long thin rod, as shown in Fig. 8–4/q 9kk4mbghj2a cn:mol- uw */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 consists of two masse2.fn 3;+y1ufbjqdrec s, $m_1$ 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 s51m82 9icz4gvr rsbq from rest within four full turns (revolutions) and releases 8q9 sgrzs1r4bi m25cvit 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 momen*m msc ,,a4nsdt 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 g/bazz5 ,d( )ifku+t jtorque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping9:vdwfg.2/h+3qj lhs.6f ;g nrwnw7gr husp2 down a la h r6us9.pf.qn:l/ 2gf2+3nrwdg7j whgvs; hw ne 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 sum ofxk 70)d v-xu,zk 5dr s-fd5aif two termsafxzk vkfu-7s5 ddir0d -x, )5,
(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 radwl)0r/v0thng2ia a,ynerl*, ius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rat,)*lyvnhe l0a/ ,ga2nrr it0w e 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 rolls withr5/jtx vc6jy)tv)-i vout slipping down )x56y-ritvc) jjt/ vv 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. Itgqzp ) dk.u7u1 starts to fall and its lower end does not slip. What will be the speed o ukpd71zq. u)gf the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular mometq5pdnc,ho ai9f- ptw 4r*w+e5k ge6/ntum of a 0.210-kg ball rotating on the end of a thin string in a circt4r 5anq o +itfwce*kd,gh 9-we5p6 /ple of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momen.qv5h:*mrp z sz mqukx rfycf ;,zwe np)9;9/(tum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm ;; z 9pf9rv(x)n.esf/wuc hrzqq:kz*my p5m,?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hand/mx/v *gyvlsjo)0) yqki+t9juwq +5z u 6jbj6s at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed6u j+ v+/yjisw6 yu xzov9mt ) kqgj/ljb)5q*0 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 heujzegyao2vmg7 5 r:ts)gagru j/* *w*s 2l/e.r 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 wh**s*vr:e)rw/g5yala2 u7g/.2jz jgu g soemten in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase herbri/(a .*ccjt; qdgh*/vmf+ j spin rotation rate from an initial rate of 1.0 rev every rq/*h(*m gb;it cca +j./dj vf2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis tun(m 1akk9 d5xf1y3fchrough its center at a frequency of 1nckk u19xy f5 m3f1a(d.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 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 .8-g5 gapkhl rspinning 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 mab8 b2fch:prb )l8 fh br8x7h1omentum 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 cylindiry6l471x r jmrical disk of moment of inertia I is dropped onto an identical disk rotating at angular speed xymrr4i 1lj76 $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 h1f9wl1u sx4a*g l s;o$rev/s$ 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 pla;kwvg 3qnsmb+ ;:u3 i *vi :lcua)7pcbtform of radius 3.0 m and moment of inertb3n) agim;ubpq v7sc3 *iuw+::lk; c via 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-2fgrdmt( w0s uy+7 sv*round is rotating freely with an angular velocity of 0.8 wds v s72+fygt*u0mr( $rad/s$ 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 cosd6s tttw6/y/n9xb2 wllapses 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 bt6ty/tss/d2w n 9 6xwcarries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess ocq/9 puh b e)f9vt(o,zf 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 mdyx /g7tcz6)$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, tha;t yh5s0 lu4eczv6etc,y o4* rt can rotate freely without friction. The moment of inertia of the person plus the pl;6z* 4e ocv yy trehc0s,tu45latform 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 ofg -s9-6 mouoqs a 6.5-m diameter merry-go-round turntable that is mounted on frictionl6soq -usg9-omess 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 c7ry(y mdrqr07pc i xs,92 mx.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. cxr7y(s7qr9p2mmxi0 ,c y.dr What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth suu)g 5asumoec*/nl /u6 ch that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as )o u/6s5*guelua/ cm na particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a a4v/ h csb/.:dic6ulc.3kl hjrate 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 x:bv+e 57v3o( ;aqqjg aqgg1oof radius 0.20 m acquires a rotational rate of from restxq+og7 vb gev(qaq jag5:;1 o3 over a 6.0-s interval at 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 ofyw(qq+ jaq3h,qv q-le f6na32 mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid 3 qy(-, qaqh wv23e6ql+qnjfacylindrical 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 is the angular speed1at3yb;j-nui,i 3k z50 eh g cgfcy6-n 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, but with mass 8.0 times as great, were r3+h zsc)taua) otating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse h+za)ut3sac) 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 energbypr4b/ a2ip*+h w a2xy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travelxbwhay a4p*/b pi2+r2 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 r(k v0sa b)/53bksm aaan $H_2O$ 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 (hoo6 xzhvmw-y+( c3d9it qz kz.y+p) is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of massxe:zldzr 24d bc;v8-t M and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54zd;r8xe4 c -dzl 2:vbt. 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 acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, andfv2 tc;k1,qpw.34j qndwe4ja we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h,qa, v2 1;.q3p4 4c jjtfnedwwk where h

  A bicyclist traveling with speed v=4.2m/s on a flabfe y,(9 cnbt+4exe(bt road is making a turn with a radius The forces acting on the cyclist and cycle are the normcebbbt+49 yx( (nf, eeal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of lengt(ok1-*ojipd8 kctoi*g 6ne0jh 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 after 5.0 s. What is thojgcd(oj*18 oinp* k-60teike 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 as thin rod+x1 x2z:lw wugs, 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 body.l+1xwgu:z wx2   

  You are designing a clutch as em)2e)na6zlb sembly which consists of two cylindrical plates, of mbeez 6m)n) a2lass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ro wg1d3 hfuvf9c5 mh*8 rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if3c5mo fh9v8d ghw*u1 f 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, but do not a uq/cbh6tg h+yr)7+ yc qg8o2vt5(eobssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutionfn4mf p7fjh2)v ,uc;z s as the car reduces its speed uniformly from 90km/h n7fm4v, h cpuf2fz) ;jto 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s