A bicycle odometer (which measures distance traveleb-trdl9fkgi-0 8 zz4 td) is attached near the wheel hub and is designed for 2k 9gf4z0d-8 rtibzlt-7-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular ve*c o 0v )7h)kg0fokhpxlocity. Does a point on the rim have radial and/or tangen *o0kgfxvh7)c 0 k)pohtial acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocityf*p ff4c6y /aj $\omega$ Explain.

Can a small force ever exert a grea 8u; z0ez(v6p9abgshqter 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 r/f7xoysf6(s)ac2r+ohl- 5 kr8qpy rdo a sit-up with your hands behind your head than when youryo 6f5 -r(as8ryro )cs/qfhpk + rl27x arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprocketz: uoji qyr:/; mu+x-ms at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocimqx;+j -ou:y: uz/mrket? Why?

Mammals that depend on being able to run fast have slenderyi1)wj-jb k8r lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotationa8jbw ik y-r)j1l dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lz, oe;4m m2crvong, narrow beam?

If the net force on a system is zerouj:rg,9f/ 3x;w fzljz , is the net torque also zero? If the net torque on a system is zero, is,:f ;u/ fjxlzwj3 zgr9 the net force zero?

Two inclines have the same height but make difft*pclmw4 r)vy f ru) z6(m,av;zoy51i erent angles with the horizontal. The same steel ball(m pwz*,56a14ymfut) v rorv zy);cil is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneouslyne2 vcinw4tzfk;k 8q rkd1n. g* c (bjx7:z5/u 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 grbn;(5j:u4cxzqtrdwk ei82 kzn 1g*kf.7nv/ c eater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and thp7:) z;s ,cbiybppsv 0e same mass. They start from rest at the top of an incline. Which reaches the bottom first? Whic cppszi:;sb0)b, vp7yh 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 con(pf-l2f /ey3vs4 9hmg i jnvn7served. Yet most moving or rotating objects eventually slow down and stop. Explay7v - pv9j ngf2fh/e3is m(4lnin.

If there were a great migration of peoplep0bm( 8exo,qv;9 wvbo toward the Earth’s equator, how would this affect w(vbb9x; 8oo pq0ve,mthe length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotat5iy0i(dkhwlp;yj -1f ypl)e /2xs hi5ion when shekle2jxif y( 1 y)0-ihdhips y ;pl/5w5 leaves the board?

The moment of inertia of a rotating solid disk about an ayh bi.snu6w)wg e : zk2e9)u:axis through its center of massb 6a wh.:z:2 nwe)is9)gyeuku 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 abet vefsat67*9z /6pp 2-kg mass in each outstretched hand. 69*eafbzvp/te 7ps6tIf you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and h(p5q x;vql0qu1 imrx(ave the same mass. However, one is hollow and the other is 5l(q m(u1qrqi;pxvx0 solid. Describe an experiment to determine which is which.

The angular velocity of a wheebv4b.uhwl .)kq wv)o1l rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top . l wk1o4)) hwqbb.vvuof 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 qv,k7lqp+mc+1u qwg vc +9a+6fuqv4phappens if you walk toward the vkqla+6mq9 1g+v7 cqpu fvp4q+c +uw ,center?

A shortstop may leap i 7wd;5 y5pvirqnto the air 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 hipdpq5 5;y7v iwrs and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, dis l p39p1iebbt0cuss why a helicopter must have more than one rotor (or propell9itppl0b31e ber). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in o1: rwc 0kar;hradians: (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 bec -zna(mgd :g8slf3g b(ause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diametenlsgmz3-da: 8(b gf(g rs (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at t.7v.qxd9 kxh0 0cmt lzi8k g8dhe Moon, 380,000 km from Earth. The beam diverges at an ang.8ilg8c 0 9k mqhtvdd7k.x zx0le $\theta$ (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 rat5dmy*ln2 d7 ore of 6500 rpm. When the motor is turned of5dmlron d y72*f during 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 leve1gl vikywr8s8337 bjql floor 3.5 m to another child. If the ball makes 13liy w1kj q83rvbg78s5.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 i,clesv: o, qaw6imhie7n2 4qnb* +a7km. How many revolutions do the wheels make+ia67*cv b2q ,ahe,nis: 7 me4oqlnwi?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angnj+ea*uc o0 w9ular velocity inncujoe+w0 a* 9 $rad/s$ $\omega$ =    $rad/sec$
(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 revort9my+c:wp g(; r 9w( -3s*cq enswoholution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child*9h+gyor(w( rqt owc 9n3psse m-;w:c 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 arounb(6fsx dy-p:zd the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pointv) o(pgrx ro0h*yvq j ,1v.ac4
(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 rotat7 yfzzi:t2gfi)g /4msy32wy te if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,00z i 7gm4:ysf)t2iwgt/zfy2 3y0 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleratem :f5ty kzh(a/s uniformly about its center from 130 rpm to 280t/afhm (5: kyz rpm in 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 radius so+ z8ib (borh: lsshpwt +067$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft p jsbo5 /3 qdlfdmvv,+/ut themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of theiro3d,qbjf sm / v5v+d/l trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Throug)98oq(istlcr0 3vjv wt(t-;*adlspe oc)p q :h how manytp08ecwsi3l)t :v vj9oat(;rs(*oqcd q-l p) revolutions did it turn in this time?    $rev$

  An automobile engine st5 +2drb 8):ox-ezqv3nfgq uqlows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highsp cnj/9rcngzig;,) f *k kp:p7jeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutio ,k)p p*fc :/n7krcg; ijgnj9zns 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 fc8xnko y-xj8(yv*(ay rom 240 rpm to 360 rpm in 6.5 s.n y8y((kc jox-a8y*xv How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is r d83 w a+ p2y4fx:k4pnaysfel*unning at 850rev/min It turns 1500 revolutions befo:py2 ealfpw*nsy k3+8 4afd 4xre 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 uk pr75sywl- *nniformly from 95km/h to 45km/h The tires ha-nsywr75pl *k ve 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 uniformly frxjy.qv2 p 8gf+wqe m hx99f))bom 95km/h to 45+pqb8 fxme)q g9 2 y.f9h)wvxjkm/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 climbing a+ vmyluu)vg 9* hill. The pedals rotate in a circlumy)9vg+v u*le of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a do/wgild) :mlf.davd 7vfj/- b -or 74 cm wide. What is the magnitude of the torque if thel/fdva.g/- )7lfmidv- wbd:j 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. Assumjbr tg4ur*3; s3lsjgda8 ryt2-si wrwrf2+)) e that t4iua gs)3r2 rr+2gbdstjyw lfs*;3)w8rjr- 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 attached to the ends of a massless rod whizv9ff cm:d.t7ch pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the netvf.:c 9ft7m dz torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of 380d6 l0txvvb+ rr0vdtl6v+ b x0 $m \cdot N$ 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 radi7 g-in.mq8jp56 njp rojsm0p. yf z,9ius 0.648 m when the axis of rotation is through its center. p,. io5ip-n0mzmqjy7.jsr gf8n6 9j p   $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diam/5p gd;,lftk j.u mlu u9na4i3eter. The rim jklt l a3pnu gu9 du4,.f/;i5mand tire 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 thin, light rod is rotated in -(smyv z:1:rt hy (mc3doiz0 za horizontal circle of radius 1.2 z( y:yzs-hmr0v zm1t: 3codi( 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 wheelmxt v un:gf89( rotating at constant angular speed (Fig. 8–42). The friction force between t(gmvnu: x89fher 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 inehz6q 1svvo6atarzest .,x 9bo4y ,.1zrtia of the array of point objects shown in Fig. 8–43 abooes,sah,xv1 16az. tyq4t 9vozz b. r6ut
(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 co2kgrg7pq jb+) /rc+hm nsists 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 t 0mtia:w iw35: j(wcyat 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 roct0ji(:a 5i3y :wmwt cwket 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 ofk ltu4,o,v 6lzd0oz)vw+9kfhk6+t ae 0.580 kg. Cakk660+k4, v l,av dtuo+th e olzw9f)zlculate
(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 swib- z5p)+ 7h 0mrdbaf7riz)kqsngs 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 tantzd8nqt7uxhpc5s0; k0 8 gv 6lgentially on a small hand-driven merry-go-round and is able to accelerate it fr zqlt8xvp8;n 7t5gh c0u0k ds6om 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 rotating at 10,300 rpm is sht)zhal3zrf*y*r,j ypt:7 s6mj j)mt 8ut off and is eventually brought uniformly to rest by a frictional torque of6,ttrm*yzarp8 t3jl)ym *:s)h jzjf 7 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 accelexj+lp4n h85my rates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, wf*vzubg, +brhn) .u /ag3ddr ;hich rd)v.bh gn /dr fa*;grbu,+zu3otates 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 a long thin rod, as 2m8n7pcezr82c)gd, z b bulu7shown in Fig. 8–46ub c ,2rl8ug8 em)cn7zpbd27 z.
(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 two2;zy+jfy3sr h5z5che masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest withindr 5xz:fs, w+c four full turns (revolutions) and5zfwc +rxs:d, releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a momq6c2a -yca zyg0-wyfzsel:(bky0 3h0ent 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 torque0,(bxot4yu 5e5ubmu p 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 isgb .1c: lnsuj6x*6py lk/ a.cm rolls without slipping down a lane at 3.3/1n 6.cx.y lsl*k gj u:spi6ba $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic ener n4qukf-reynb 3sr +0+gy of the Earth with respect to the Sun as the sum of two ter n 3f+sey40u-nb+rrqkms,
(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 a6 rd/mmx7oi,:+pc/to1 a4hrrh m5fc p radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a sol1artr,ihc r/6o+p:x ch /df4mm7pom5id cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rrczqmcx8 njud,+5 *k: )q/evxw7 -j neolls without slipping down cj+,cnzvj8 -) k x7qx*ewu:r/e mnd q5a 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 vertical.o9xasb cet x,*;:cw ( cygzp:ly on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just beac;byexo cp gtc:.*zs :,w(9 xfore 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 end of aibaqr; r1,es9 thin string in a circle of radius 1.10 m at an angular spa 1iqs,9r;b ereed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whe-y- ,av9d,ll ybjomh1 el of ra-mod, y,aj1b 9 hlv-yldius 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 is rotating at)m*0n9bsewb /l w*rjhm- cp*v a rate of 1.3rev/sb09lpr-hmcv sb wjm) *w/*ne* 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 e35(fgl 8 ogg bk-6rjxmoment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If8- rg36e5lkf (bg goxj she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial r1qkv 9la2k e207 zjuoyate of 1.0 rev every 2.012kq90zela72ko uv jy 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 rotatin- nst 22mao5sdn2gx5v g around a vertical axis through 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, approximately shaped as a flat disk of radius 8.0 cm, onto the ceng2 n tos2x- 2ndm5s5avter of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular mopaex5 rr/h- wm*;tu ffn,/ /yhmentum 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 momentu jj2no,.i6z ih7u5 j.ccz823qrker pk siw2c,m 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 identic4d6( zf qhsrpq s6ih02qwy15x 4mlq9d al disk rotatinflz054mi 2qys qs1 9xdwq4 rp6(dhhq6g at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atwxin.ebl .23xh b.3 ui 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating me81 ea 7 i8fwxb82 o4edsomcis7rry-go-round platform of radius 3.0 m and moment of ic7xew ss8824e7 ao1 bifio d8mnertia 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 do 2t rqgk:phhs495-yx6 yxd (yisc76with an angular velocity of 0.8qyy4 -o79k6(s56pt hxd xi r:ghycs2d $rad/s$ 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 intw4e5t8 p;-b 5xowplcto 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 5wwl5pt4-;8bp e cxtoSun’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 incjx; y(.tu, os 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 mpak*+l vl68 jtc h/ yi1g.4xs1 sm0ne$ 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 platformj) dl t g7ttd)+8i qe/(ucoux,, initially at rest, that can rotate freely without friction. The moment of inertia of d(q/cu g)ietx)ttd 7j,o8+l uthe person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter c.,q l*nmc /mtsi 5dee2br;0c merry-go-round turntable that is mounted on rnc2/0medil 5 sccm;*e,t q.b frictionless 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 fh-;suh)ybk1 the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass bf;1ush)hky -move?

  The Moon orbits the Earth such that the same side3 szl u odzv,vdi7g,83v:/ krs always faces the Earth. Determine the ratio of s:rkivdvov,d g 3lz3sz 78 ,/uthe Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest a ox7tkodm466m 24ku6 0an pphct 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 cylinder of ra kf lueuc076;zdius 0.20 m acquiresu 7 0kel6uc;zf a rotational rate of from rest 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 of mn2ysd:eri5brlb17.m dw, ) wcass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0b sm)d l5.ir2b,:dw7wrey n1c050 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 angula8* r:+zf0m bbpzij yf8r 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 mass 8.0 times as grearz- k7.ese)qv t, were rotating at a speed of 1.0 revolution eve ke-)7.qvreszry 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 isx ;k smcowny86wr,0)t wby,pylc55 n8 for it 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 diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without nee,yx kwpytosn5bmc86w 8y0ln ;) 5,wrcding 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 anu:9z 5 hdh2nje $H_2O$ 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 horizo (f4eklbl*4j9m a h0yxntal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L agewwv q;t*:*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 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. [:ae vw**q; wtgHint: See Fig. 8–21g.]

A wheel of mass M has rp7eh (ug 9dd2sadius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The stegpd2u(7se9d h p has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a ta3/ir1ymd tu c 2/hj5wurn with a radius The forces acting on the cyclist and cycle are tc u12ary mw3i/d jh/5the 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 mf 47t2:j(idu lcp m 9tx2z(abr 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 the torque requict9(j fb iz:p4 durt2(x27a mlred 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 thindx. 8hn;kam8 9i xvn;u 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 huna vxi;xn.;d mk8h89eld tightly against her body.   

  You are designing a clutch assembly which consists of two cylih-v q1 q3qvl1indrical plates, of mal-qh3 qivvq11 ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 along tht+ri0fs/)doii ;kb9 jjf,gh -q8crb8e 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 toig,9ds j r;if08qkrt c ho8bi+/-j )fb 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 as uyxd./mj (8t)ffd1nnsume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the card8gv ,*u9i i;vp xlc;m reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of8,;vl ugvix m *p9dic; 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