A bicycle odometer (which measures distance traveled))snoxt*bubq)x xw(e*lc) d bhn, 9g9 2 is attached near the wheel hub and is designed * e n9hd*(g )b,2bxxo qslnwb)c9 uxt)for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point +pw1blz 1:yxak;nvii3 ) m 5vton the rim have radial and/or tangential acceleration? If the disk’s angular velociivl+iam :ky31 ;nzbtp 5xv)w1ty increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be desc +gpf: idw m.4ijwds.7ribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever c (9mquf)whjg 37i,xiexert a greater torque than a larger force? Explain.

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

Why is it more difficult to dac9w bg2uufz -z e70xs /kl0.jo a sit-up with your hands behind your head than when your arms are stretched ozb9 g cw.kel/f-72z usj00 uxaut in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thnmfe8ig)u y o 2fx67b)ree 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 fron)yg i2b 8nmxu6f 7)oeft sprocket? Why?

Mammals that depend on 6)h49 ndzpsrotp: ls;being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–3t )69rl h ;onsp:z4pds4). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35),.d xf;gipj 4qv 2bu*bqz/5 ij carry a long, narrow beam?

If the net force on a s5y: zew q wv8ze2u9k yy:iv5.zystem is zero, is the net torque also zero? If the net torque on a system is zero, is the net fou95 e 5iv.::yyyq8kw zv2ezzw rce zero?

Two inclines have the same height but mak5n w1g+z7mzt6.fh0kmoc9cyh e different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball m+zo .5kycc61hfhz w7mn0t g9at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an/ dif5(fe6hme jtagzv j r+p)jmz;7yl4-:e+c incline. One sphere has twice the radius and twice the masj);c+d 6/jmhvt :m-+z 4iz pejy7afrf (eg le5s 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 magyx::s4(;. hj f b.bxhy k9tlcss. They start from rest at the top of an incline. Which reaches the bottom fi:9xhh (4f.tyb.sjlkby; :x gcrst? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserv: hmgs. )k5j h-fl3gurw)ayt0ed. Yet most moving or rotating objects eventually slow down andu)ljhfa)h.wt:5 3 0myrg- sgk stop. Explain.

If there were a great migration )hkrab84+b duv sofh .y0f43aof people toward the Earth’s equator, how would this affect the lengthu) of3vdh bfy4 8rhk+a 4s0a.b of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatioq zxmf(b j -a+4n ;(z,z;mf9opt.rax cn when shezr49+nfjbz;az, cx-(tm.ax;o q mfp( leaves the board?

The moment of inertia of a rotating solid disk about an axis through h 1n0;liqld0wxch, 4cy trg2- its center 1xt gqnrwh0c2dhil,c -;ly40 of mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool 6 7ylynw: 7ytmc 7ft;pl,s 8ehholding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase,y7ly ns7wht;cm76e :,f p8ly t decrease, or stay the same? Explain.

Two spheres look identical and havefkilzxjr m t )(do 67 -p5j(bdww8j.;q the same mass. However, one is hollow and the other is solid. Describe an exkdxbp rj(wj t q )z7w-8.lfi5(jdo; m6periment to determine which is which.

The angular velocity of a wheel rotating on a hori:e7 nmlq:kn4mzontal axle points west. In what direction is the qkm:7nln4 :melinear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. What nl: q4u:dvtl3)vk4wo1ir0 kb happens i1 4lt)w d3 k0ivo:unrqkbv:4lf you walk toward the center?

A shortstop may leap into the air to catch a ba+ne9 voi 3gs34y go)jpll and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly yo3p ) +yi9g3e jnsvog4ou will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss whywz2 uxhl92afskno6 qhy*2p4gnoe 808kr6g n( a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can opernkprnx6gg26sa fy2 qkzwl848h 0o hneuo*(2 9ate to keep the helicopter stable.

  Express the following abf3gp: 0-)tc (qazdh5a1y cic ngles 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. Calculate, using th2c toz)(e5d 9:igxm cse information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moo ced t:mz)i9gcs5(x 2on, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. Tyzl 3r7 roy- h,2xo9wyhe beam diverges a o,3lyxo7r zy2y-9whr t an 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 a ll 7: +z)om6hjlr4ktof 6500 rpm. When the motor is turned off 7rkt al4 6hj+:z mlo)lduring operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a le ,g44ydo*u /smb*av.bqpx/z mvel floor 3.5 m to another child. If the ball makes 15.0 revolutions, bx/ zyuq v gd,p/m.*4osam*b4 what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutionu7ga un zu66wbqs+f9;s do the wheefbq;uug 7uz as+6n9 6wls make?    $rev$

  (a) A grinding wheel 0uw.66dyev k;qt vn ,)j.35 m in diameter rotates at 2500 rpm. Calculate its angular 6)w e6ud v;v.nyjq,t kvelocity 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-rv g9r;yn0+pdj lxhd r,+r78mmound makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child p h+xr8 dmrn9l,+0vgmyjdr7; seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Su5 c5wa8: y27xgbas mnfn    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a gjotllaty47t a6lj4:(39uy g3c-zp h 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 tw mrzuqh79et: e*ar (9he axis of rt(r*z 9rem97aqu h:ewotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abodudr x bc8wh +bnmz n*zr(av3l:h/5*0ut its center from 130 rpm to 280 rpm in 4.0 s. Dete8nxr:b3*u0adc zv h5mhlwrbz+ (/ nd*rmine
(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 radius -e.fkh i 4r sqk)ws6e0$R_1$ 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*w:vvuzl 6 f(uauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute * uvwlf6( vzu:the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000q58k1 pe w7/ismhcqz) rpm in 220 s. Through how many revolutions did it turn qzcs wi51e/)h87kq mpin this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. C75a7m.euwe k7ngq o v1alculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed sjide k i*,.;aw le3+kjets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.j3ks.a +ewil d,*iek;
(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 24q*cfvb 2p(7rjxh tx/;0 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in th*/vrhf(q 7t2c ;bjxx pis time?    m

  A cooling fan is turned off when it is running at 850rev/min It o55dat ne okv,i-12tu turns 1500 revolutio k5u-t5o,td nia 1ove2ns 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 its speed uniforml0 hvh+e6vi i dy dmmys5u77u m1;;t4wu8n8xooy fr mm;e8wudu7nm6 ovu0hd iy5oy+x8iv1 7sth ;4om 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 reduces its speed uniformlyl 8wzkz9pwp.m4m2v2 g from 95mwg4.mpz2w 9kzv 2pl8km/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 pedal when cl7g-ulrunm e hr2705mtfb7 y3u imbing a hill. The pedals rotate in a circle of r7 f730ulhm7 t5ery2 u ubrmg-nadius 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 5 hjj 3n)4b(uvtw;bg0r5 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exer0 ;tjgun)wbh(j v4br3ted
(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. Asya *zc2mk/ve u33djckf2 p 9:osume that a friction torq:u kay*9fe3zjmpv 23d o/2kccue of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a mass ksm3gr:,; 2leat kd:qless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitum, e:t;a 2l:skkqg d3rde and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a t sutyodm*z x4as+ww 42i-p*1xorque of 38+ts*p*odzywua4- x w4ims1x2 $m \cdot N$ 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 10.85 ul gwj ) v6l0lw dxq4juks.l5 .t4v.cn9pi5a-kg sphere of radius 0.648 m when the axis of rotatio9ljs lvvqltpukad5jw46l) 05iuwx4 cn5.g ..n is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim f rjph* /k qhg859xzy(and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (whyfpx h/j h *r895y(qzgk?).    $kg \cdot m^2$

  A small 650-gram ball on the end os+ vkok+v6f4u, 4c 4t tpwvb2of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculo pcvut4b ,4wk of+vv6t2s+k 4ate
(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 rop 1r ac:s+6d,9/hh;y*rah pv;it efy gtating at constant angular speed (Fig. 8–42). The friction forcehacv ,;yf6h+ *rrp ;hpg1dse/i 9aty: 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 theoe j1hrcb2,,23pq 7hvqu/bhe array of point objects shown in Fig. 8–43 aboutqe/ h3c1 ,,jvbhh27qr 2oepbu
(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 conx- o8lfl gc5 fu3utvy g/ux40.sists 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 cylindrical satellite spinning at the cq :6snc4 yq4/ vq),jy4s/ ysjvop:jh pzbe1a0 orrect rate, engineers fire four tange4yqj,bq av j) pe0y: jsposh4z:/ nqc 6y14/svntial 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 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a rwq .e;hnm-owgf 4-g nny:6r* (iino9 madius of 8.50 cm and a mass of 0.586;-fon wi -i9hr:g4o.m( qg wnn*ne my0 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 toj:ad jal8u2 d6g7i1 du 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-f upy,p+x jq25h.my*zround and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass f.pjy5 mpu q yzh2x,*+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 rolvx;m2+1 v 3jfbfy6mc tating at 10,300 rpm is shut off and is eventually brought uni1 6vfm2+yclj3b;x fmv formly 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 nnh i))8n;u89 crwpk, iu:nwt3.6-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 ..)xo9s83k) tjx;djf)y xnt h 6d tll,ds/p muby the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is anj) xxdt8u l,fxlk9smtosd )j3 6 ;d./t.)p yhccelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long th)zko;by y cm1 /bz3pc.in rod, as shown in Fig. 8–46)3. bz /y1ckz;mpbyco .
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses,nr (4gk+ny;wwn pa6k- $m_1$ 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 34 j8xxyr r)ds from rest within four full turns (revolutions) and releases r38dr) y 4jxsxit 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 mom4es4jv -kb3+y cbg*w hent 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 torqueyewz 7q aol6 nr*+ ju018ufk)w 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 slipping down awacnd )3q 2-mb lane at d-3q bwa)2cnm3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of thqei*3(p /ss,rzpy. 7zi1tco me Earth with respect to the Sun as the sum of tw(yss1pqz . z7eii*c om,r p3t/o terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius yuvi + ip++b0 s)1pbp vq+wqn8r5*cy lof 7.50 m. How much net work is required to accelerate it from rest to a rotation rat y1cvli+q+u 0yv)* 8iprnbp+b +sqw p5e 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 witbna6*kv9 nw/2;oqjn eh(6qm mhout sli2v e;m6nkjo9(aw nmb6h /n*qq pping 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 tij2f h+5nt rgx(p. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole justf+r( tj gn2x5h before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular j cq7/-o,g4cgang6ph z,7z v,so:xlhmomentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of , / zxjcq vgs7z-nc6a, g7 :hh,ogolp410.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg un) 1+nhqqc /e) /sp uujkpv9a6oiform cylindrical grinding wheel of radius 18 cm when rotating at 150pusq qn /)h/v9cu6ke)1apoj+0 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 tha eh+311igfxk kfi8s yop-2ui6t is rotating at a rate of 1.3f3k6 psx1yfe2h1i k8u- iiog+ rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her momenm;w 4d1t l5 *,ejhq6x:qisc htt 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,c e jls;t6:wqt im14hdq* 5hx 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 incr dkdy)r;xruw*4;d 8paease her spin rotation rate from an initial rate of 1.0 rev every 2.0 drydk uxr 4*8)d;;awp 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 threi -yd)m7mrnn0hi7q o7st:4r ough its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximatelyn- oi q7 d7hysr:)nemr07i4tm shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3.5 jqr:jp (71bflcrrjf /j6x92d $rev/s$ 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 mom-jdc*f pkb -8ri bf/t,3dhqk0entum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an 2f, *z 5gkg;ik;xnviq identical disk ro zk5kx; qf ;nv2g*ii,gtating 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. ) 9+s3 btjpqqbj,u+yz4 $rev/s$ 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-roun zsk/wml;*2j3w mlh3 od platform of radius / km*hojs ;2ll zm33ww3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity o17aanjgk9w9g 0oed 7h.a olw39 zr1nxf 0.8 kwlaj99 zdaeorg onw7 1gh 190 n37.ax$rad/s$ 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 dwarf, losing about half . cxenh8;4)t5v r6ea/gor idt e k8;vqits mass in the procrke 5;eq tn ; a 4/)xr6i8cvtodeh.8gvess, 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 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*1,jh dhd:n4/7g y h*l5gaokktq :ugq 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 dgizac6rz:13ird *c :$ 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 c 8t1gir ghp9hn(.ng m9c7f5hjan rotate freely without friction. The moment of inertia of the person plus tpjhhg 7(nfgn95m81cgtri 9 . hhe 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 3ed60dxd4l 2- o6u h+a czb)ia9xjxsh stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frix ca-bjl63ao9 4zx 6dh+2d)shx 0diuectionless 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 qen,j15)iqe5r (hs ,eyhokwd:y om3- ound 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 ontq):eyd3k r(,qj - ws51 mniehoo5 ,ehyo 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 tkm (/ntte(f6fhat 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 l( fk6(m tnt/efatter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/lsqcbqmex8h 5 c.;s3* *a (zxs$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.2:(acqr1 aap nge;x6l4 0 m acquires a rotational rate of from rest over a 6.0-s intervalpa1g4racn :q;6(aex l at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made +/o:bltv3ph,p m7f o wof two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of masbpw3t/+: fp7,hovol ms 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 speedx3)gkt jbax2 bepik-gv 8mgl/tt/i 8.+ 6vk /q 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 sizl d.c/ sx1-boz9jz t*9h enes/e of our Sun, but with mass 8.0 times as great, were rotatingh1-9 .ot/e 9bszldsjx */ zcen 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 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 eq5 x l:j7xcu;(qd dc)o2 adzw)nergy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses xl5u7dadxcj:z)o(q w2cd q ;)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 illustratesg 1uc7p;jg2vl8+k pm o 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 horizontal suroy s9h jybhyw2.u .59dface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frictt7ad4 t gofa2*m* x (z/tr6 nw*lb2qgsion) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acc*7g(romzdg/ ttabw nqtl6*a sx* 242feleration 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 ral ktltw0 .)macbc5o ifo .aw4,6-d6qwdius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will cliqalk-tmf ,6 tbcdwa0col.w) i5o4.6w mb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with szk: 9- c (qhwn vbv;fqna7+q1ppeed v=4.2m/s on a flat road is making a turn with a radius c7a9nq +zqbw ;-(vvqkhf1:pn 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 0.50-kg rock into a sling of length 1.5 m and begins whir1hqwppmm44cbue: zhxe- l ,9 8ling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm ,bm 1um8w 4zeexh- q9p:cph4 lafter 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 ar- ihh r.k; ncc mnmp0bj9)69vsms 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 body.n.9 n-6bmvm;0h jpcsk9 r)ihc   

  You are designing a clutch assemnqj(d 5m:jx*a3m4j jp bly which consists of two cylindrical plates, of mjq j:5dm*3pmn ajj4x (ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alonvr5r9:5,prvwt unum3nbfm ++ g the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point r vrnt9v rbmp+:+w 5umu,fn35of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assu*:9v8; walx nwppihl2me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car redua*y fjxucxa3p7gh, b.80in,g ces its speed uniformly from 90km/h to 60km/h T7 pxjbg, ufa3x8,0aig h.ny*che 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