A bicycle odometer (which measures distance kizl6m a ++y*rtraveled) is attached near the wheel hub and is designed for 27-inch wheels. What zya6r*li+km + happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates atn3eay mji65: )*w hky4f 9ghlq constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity f4)jk 36hqgw*ne9li: amy5 hyincreases 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 spg uja6mzf4 0bwkz4 b3. 8z+pgingle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a 6 i bi ;6(6rbinq/l*pf sv5kbplarger 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 diffi 9yw )2f1lzqlreqh--d cult to do a sit-up with your hands behind your head than when your arms are st q-r 2d-1ez9q hly)flwretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel/cidj:3ju v ejr)l c6qg*tc b5 uh2y4* 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 c d3iucrevc*5:b/lhjyjq g4*u t6 2) jwhich gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slend8j ya)1bcft 3her lower legs with flesh and muscle concentrated high, close to the body a 1y)h8 cbtjf3(Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkersa lf t.iwi5v okko)*22 (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? I3sm kf m,tbp-zp 5vv1-f the net torque on a syss mkp3t-,pbz1-m 5vv ftem is zero, is the net force zero?

Two inclines have the samaewy ka646xh n)lj : ol:q)a0ee height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will6n a y):aqjxkeh ael0o)64w :l the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously4o 8jgp1 n9b b5e;6cnl0ogubd start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has jd6 pb4bcnnbl98;u1 oo gg0e5 the greater total kinetic energy at the bottom?

A sphere and a cylindeq8.is (viogr 95 dd2uor have the same radius and the same mass. They start from rest at the top of an incline. Which9d ui2vr(5sgi oq8. do 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 m a;l7e4cvtf:c7hih:d x4ajr(g;da( c omentum are conserved. Yet most moving or rotating objects evcceha:ax; (gc javt;id4l:7 rd(7f4h entually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s/9luuli+eu na w8/vlpd3dw9hrm3w/: equator, how would this affe39v/:eld ri3 uu 8h/alww/d9uln+ pmwct the length of the day?

Can the diver of Fig. 8–29 do a somersault witc b,ly7(i1f3i-zqys; ) yfvfihout having any initial rotation when shef1by, 7fis (zii yv l)c;-yfq3 leaves the board?

The moment of inertia of a rotating solid disk about an axis through its cnlp zzs :5c/y7enter of l5p y nc:7/zszmass 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 eac 1r ztmyv;y81dh outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, ory;t1vym z8d1r stay the same? Explain.

Two spheres look identical andjj+6zjhd eag p2-gs19 have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is which.z 9-je6s1+jjggd h a2p

The angular velocity of a wheel rotating on a h+ : f:rur4ch ab orpp5aw(d,.corizontal 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 this point at the top of the wheel. Is the angular speed increasing or decreasinop+fpar:w5b au hr:.c dr,4(cg?

Suppose you are standing(ijzw3x0 pw,y9 p d4yy on the edge of a large freely rotating turntable. What happens if you walk toward typ djyy4 wip,x w93(0zhe center?

A shortstop may leap into the air to catch a ball s adutj( rr;xl.;(-a6s m4ajs and throw it quickly. As he throws the ball, the upper pajd-a.alxum(4 rs; r taj;s 6s(rt of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conse wa-2ebb06 3yo,zkbene:ed4ju l2 +rzrvation of angular momentum, discuss why a helicopter musyd2l 3ez bk2j en-bzuob w6,er:ae4+0t 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 angles in radians: b oo ;aq1 open r/u74jhhrwnj9*8 +s53f azzj:(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. +dylmb i a(+w87(xt0 nu vl3pzCalculate, using the information inside the Front Cover, the angular diameters (in radians) of l ux avz+lm078wb(3ind+ pty(the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moos)5jl.tpb+ s9n kgj/tn, 380,000 km from Earth. The beam diverges at an ang sk)9 bgt5 .j/+pntlsjle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is tujmic 2xzs7y m(m8y(c 4* mia3qrned off during operation, the blades slow to rest in 3.02 j3ca7(xyzmcy(im84s *qim m 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 chil2/n01i+ug3wij c4 xuwfurn 0 gd. If the ball makes 15.0 revolutions, what is its diamegj03/u0uw rni1cw2fig+4 unx ter?    m

  A bicycle with tires 68 ryyjx.f.q) l6iavr 7,+p ;keh0ldj )bcm in diameter travels 8.0 km. How many revolutions do the wheels m.0eahb y.lr jx)fyqp+ r ;k7 d,)6lijvake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 24i3hoidn1/m 6qkc pu-up b*g 7500 rpm. Calculate its angular veloccn g/hupb3i -1k* m7op6iu4d qity 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 rw:6n* qy n12nq d:gkxck+6kqi2e 0 stmakes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a childnrw e kg2 di2:s0tckn:6yxqq q61 k+n* 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 itsi/5/zieu:t07w w2)9ifbzuwg u9egpc orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of) 3x q7abwycx1 a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axis oun5 5hh r.+b5fw9wajmdl1 mx.f rotation is to experieh1 .55rm 5wa 9x.dfuhmnw+lj bnce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acchhzx.mfpy*.7fuya4 ,)*k cpms,f q*l elerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s.h* ,pyf q m*mf*ck). shfz,la7x uy4p. 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 radiu5j2jpfu oms 0/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 th7h0uu57bio( p)hhfd b nz)km1e Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their th715)hp di7ub)k bmf 0o(hunz rip, 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 uniformlhfbv28a b*9suf5bv , yy from rest to 15,000 rpm in 220 s. Through how many revolutions dib*5 u8ab9,fyhsfvv2 bd it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Cal* b 7a6nbb+ ms28wcuecculate
(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 fon69x.j cnqz2dr the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before q9dc2 nnzj x.6reaching 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 3p .h )maltk(4l60 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in ll)(h mka4 p.tthis time?    m

  A cooling fan is turned off when it is running at 850rev/min It turnfs5. pv:ma,; s)zsllo +6 ldk*y/uajus 1500 revolutions before it comes to a sf +u6ym oksajpdv*u.l:s) ls/ , z5al;top.
(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 revolutiou:e zx+tw2erca3ql4h d5x t1 7ns as the car reduces its speed uniformly fromh:tq wrc3a7z xdl52t4 1+x eue 95km/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

  The tires of a car make 65 revolutions as the car r/o mra3 v1-zxoarw. hc1g g4m b6*3zhjeduces its speed uniformly from 95km/h to 45km/h The tires have a diameterh1j6o 31 x.3zzmrwa*4 gh/cmv r-abgo 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 gyfxoj ei6 s:2o9z79)phwf keted7;6 her weight on each pedal when climbing a hill. eh66j7x9k esgffo2:z yip)o t7e d;9w The 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) g fkhu9+as/ glww+z,f 55 N on the end of a door 74 cm wide. What is the magni/uhg zw k) 9sla++fw,gtude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assume; v(8lnohv ae/ that a fri vno;e(8vl a/hction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attacq f -n7. )gto j-sm+m)uuakhk3hed to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially7m3ag km juuhq+ts)--o. fn)k the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of aouvcvc5q x n3 k4)zz5y1b62f hn engine require tightening to a torque of 38qkh3y6 z1uc)4fo5 cv 5z vbx2n $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the lv 1li).emqr:; ke)hm l8wzx 8axis of rotation is thriwqlx 8;zl vh:1 e.) 8ml)kemrough its center.    $kg \cdot m^2$

  Calculate the moment of inp s2 jqcx,(rd 47y7)vvxh y-lzertia of a bicycle wheel 66.7 cm in diameter. The rim and tire y),2hx j7yr -vpxq4scv7lzd( have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a tu,ng53smh+mj euiqhv2 0d:7n ,bexl)hin, light rod is rotated in a horizontal circle of radius 1.25,+sl j7n3dihuemxu:eq )2 v ,mbg n0h m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wes50 gdeoy(x0 (;ln oyheel rotating at constant angular speed (Fig. 8ny 0 gxdyo(e0eo sl;(5–42). The friction 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 point oqeo g7: ne(1ggbjects shown in Fig. 8–43 agne :e(o71 qggbout
(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 consisesmg 0+0 fy+mhts of two 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 cylia:sx7f(( o lqs0.hqb5fqiek. ndrical satellite spinning 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, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 milfkiqhxq(o(5.7:f s abs eq.0n? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder witjv-4f463kg m /ugpq iwh a radius of 8.50 cm and a mass of 0.580 kg. g -ivkm/qu fwp4 gj436Calculate
(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 swingrwf.;jfizt/(grs5 ak:2 k gy5s 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 s2 sj d;a++z(s6hny42u zllmv wd zl-*xmall hand-driven merry-go-round and is able to accelerate it from rest to a fl 6sd( l2asnxlj zh;d+*wv+z4ymz - 2urequency 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 rotating at 10,300 rpm is shut off and is evenk /ark3*;om5d fhm;gf8hbj; p/1mmzltually brought uniformly to rj /ka 13m/ph; drzm ;ffko8ml5;mbhg *est 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 3.6-kg ball at dt 729.zgq)pk g8vyq,tgii +h7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the,xv-fues*o89 a 4zcdf forearm, which rotatesa 9f u,doc exvsz-f*48 about the 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 considered afshg9wrclg 80 4:7(ci n:fprs long thin rod, as shown in Fig. 8–46 s:pfc wg04 hnl(igcf7:s98 rr.
(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 consistsza ,o9xqtu(g (j-n.b f 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 hammplg22 s)3ybower from rest within four full turns (revolutions) and releases it at a speed o spyb l3wo2)g2f 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 indit0mp2+ 6gv tertia 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 z:tjgxg0 7b1dzho pe93)l7 -p7iz xfqdevelops a torque 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 raj cez/*jtsy56 dius 9.0 cm rolls without slipping down a lane a*jt s/6jy5zcet 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as t b unvi8gf8:6rhe sum of two tervb8ru6n 8f:igms,
(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 a(yvu ,wa wv5 .9c)mi osmkbvw4b/q 2a1nd a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of vw)u(/ayq1wm42ba v mv c9wb,.ki5so 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm ahupr*ib. bb+ 4nd mass 1.80 kg starts from rest and rolls without slipping down a4u pib rb.+h*b 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. Ix n 4)+2q,ofsei7t bwu*.ntw at starts to fall and its lower end does not slip. What will be the speed of the upper end of t.xfn a t ,iswou+e24*)n7 wqbthe pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the en4o+w/d,/ -/tecyi h brlx5s end of a thin string in a circle of radius 1.10 m at an ang/lixw n/ d/c+obr heyse5,t- 4ular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniforz)7qe-yug xa8lr8c 4um cylindrical grinding wheel ofqrc ay488u -u7zleg)x radius 18 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 isaai b+ bkmi)0dz+qv pax-* 2*m rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the sa0zd2vi a-)bm+kbp *iq* +am xpeed 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) ca4mk l9,jp3kgm n reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2 9l3g,kpmjkm4.0 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 increhupyi4 m,6q)4y3n hscase her spin rotation rate from an initial rate of 1.0 rev m3h sqh y, 4i4un)cp6yevery 2.0 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 through z t1yc.+uogrmzl61, yits center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. Whoyr+z. yg l u6m,11zctat is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum : 4evw9apkhow k1p.-m of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the9kee1n:y nk0k wx:k:m hr w50y Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an i)w 4nocsc c2o3dentical disk roto4cc )2w3n oscating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4x b;qir/ lo3qcg-ktches-9( -8)ggah $rev/s$ 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 mmf2c h8tasn-81zjo .44 ot aukg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 8 8h4.fzoat4-a1 t sjcmmun2o920 $kg \cdot m^2$ 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-roun 8dzr7nithg:1+jw3p md is rotating freely with an angular velocity of 0.8 n8:pgh+t rmwd 7ij3z1$rad/s$ 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:qafb5 si: pl.cu.e.g collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new c fu5:pil.ebgq .:a .srotation 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 0hpwpk38s(k,zhs)v;s m u:q wof 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 f z/yjy;kc 1ae6qh 66g$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can r fj28twz5n5rnm ix v91otate freely without friction. The moment of inertia of the persowfmv5tzj 8i9n x1n 5r2n plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of 0rpm sf3ri4:i4qx(.sz s4 jmha 6.5-m diameter merry-go-round turntable that is mounted on fricti3(izm rxrh4 qs .f404:psmi sjonless 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 gr* tydswcs56 qqq u::+qound 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 length of rope unwinds from the spool:qqqqs*65 t: ydw s+cu? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always facdlme4r3zk z: y3d;mribz,) v+q3 o 7bkes the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentuz7rz3d+:;l3m veok b i,43 bdk ymrq)zm. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates frsfmatm x+2 upi7 ;*r3vom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cy70hl:wz6u p 5 63rqel7 w zj+lktwgly5linder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant 6p3w zy5z +lewl 5jq:lrgwkul06ht 77angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diskelfrw.p9e j6d ,0g x7c7y1xh ts, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed odeh0.cyp 9t6jf7rew l 1xg7x,f 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 sisk; v yvt1:)speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with maexd *zkizyo0f; be/(1ss 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and t ;y*zd b/ex 10oikz(fehat the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for itb 02/xu1p8rc jmpw ubcs7(x,c to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of 7bcwp8 u/c0 jc2 bxprmu,(s1x 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 illustratemx+u7 heb:f9 ws an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizon03r g0xn p7ggbtal 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 l). 5d epnxirc44rhew0ength L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angd5 h prx.40 wr4cein)eular 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 stand wzpwg 4;aav.2pn7tpdf6 rf:) ing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has heig;v f7)rwpan26 p pwd.atz:4 fght h, where h

  A bicyclist traveling with speed v=4.2m/sr:zaywud2k;9 on a flat road is making a turn with a radius The forces awz29u:daky r;cting 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 0.50-kg rock f y(8wbg k33:hrt0tt2glb.tf into a sling of length 1.5 m and begins whirling the rock in a nea: .rfyg23 38b (fbgtt tw0lkthrly 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 cylinder and her arms4eoof/c2 ob(x yb u(,v as thin rods, making reasonable estimates u( xyocovb/ 4(e,o2bf 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.   

  You are designing a clua:phdxn6gj832v or2w o6emr0 tch assembly which consists of two cylindrical plates, of m36wgo j0dan2r ph:ox 28mr6 veass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls alonetk adyt3 9c8vv-iz7ynu ; 4(wg the looped rough track of Fig. 8–58. What is t-e8ty v;iy3vdkaz9( 4tu7nw che minimum value of the vertical height h that the marble must 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, but do not ar (*bkike8x 26afs9 mossume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutiw5.-; eurogvwix.3c7 lqi g/dwg+ qk) ons as the car reduces its speed uniformly from 90km/h to 60km/h w.o/ive kdqg-c3 )r7+x iu;ql5gw .wgThe 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