A bicycle odometer (whke p4:oe8ucu.jugc+0 ich measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inchuupkoc. j ec4g+u 8:e0 wheels?

Suppose a disk rotates at constant angular , lq9 7u4.qf0th0oxz :oyb bhdvelocity. Does a point on the rim have radial and/or tangentialyoh4db tx0 : .,qfohq0l7bzu 9 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 velocjau-n jgcys wh:5u4 i)id9d5 e/ q3c;vity $\omega$ Explain.

Can a small force ever exert a greater torque thanurewmxt2 hknsi0 9y9r)7l y8 p g.qq.1 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 do a sit-up wilm/my *rs6g9 sth your hands behind your head than when your arms are stretched out in front of you? A diagram may h9r6gs*lm/ symelp you to answer this.

A 21-speed bicycle has s (teanco:vvzg :w: px7 *4qhe6even sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is i:*tez4o n6ea :q7hcx: pw(vv gt harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slenwkme21 bx8; /w9ihtzoder lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On ewob1k9w;/itmh 2x 8zthe basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a8dl,w9rih+ hd long, narrow beam?

If the net force on a system is zero, is the net torque alsmwy2m1 g,t*s5pyt c :a uaj42oo zero? If the net torque on a system is zero, is the j4ypmuo2 t5asa , cgt21 *y:mwnet force zero?

Two inclines have the same height but make diffv. kae )o q,gqng34ts (e9hc4eerent angles with the horizontal. The same steel ball is rolled down each inclines3k,qg oeh9cqvg.4( neta)4e. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously startbp .f8gb-ufu9:w vx/8rox3 sm 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 the -8v/fxuoxg8m wb f9uspr.:3bgreater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start fy3;w1r) fd lllrom rest at the top of an incline. Which reaches the bottom first? Which has the greater 1 )f;yllw3d lrspeed 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 conserved. Yethl70,nykh*r.cw -sjgw na (;t most moving or rotat7k j(nyg.l;h,n tarh s-cww*0 ing objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s +q 7ub/+xh3 pxob8xyr equator, how would thx3u hrxq++ ypb 87oxb/is affect the length of the day?

Can the diver of Fig. 8–g +16kck4 7qxo qcv-xs29 do a somersault without having any initial rotation when she leaves thgkq s61ox c4vc7k -q+xe board?

The moment of inertia of a rotap:sb8k up/c: j b1+q /wy/axolting solid disk about an axis through its center of mas:pc8spabbxw//:1+jq lou /yks 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 knb we2f j6y;jn+:n2c rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain. ;wjje+nk yf2:2n6bcn

Two spheres look identical and have the same mass. Howev,y)1y oqs3w.o(ms u mder, one is hollow and the other is solid. Describe an experiment to determine which s1wmoyu(odq .s)m 3y, is which.

The angular velocity of a wheel rotating on a horizontal axle pohpvhj; r8xs3 g5q0t )dints west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points eastp8)r3jvhq h;dgx5 0st, 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 rotagc:wl gxgb1f:jt ./a/ ting turntable. What happens if you walk towardfx j1:w gl/tb:g a/cg. the center?

A shortstop may leap g;)zi 8qwgr( b9 v2 veel (7alqyps.bnc4r8 y(into 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 hips and legs rotate in the opposite directio qr 2asvcr89(y eb giz4;.wq(gl7ben8)( lpyvn (Fig. 8–36). Explain.

On the basis of the law of conservati- (p zf8;vaekson of angular momentum, discuss why a helicopter must have more than one rotor (o z( aksefp-v;8r propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following kwxl -.alz72d (b+z cqangles 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 co0tp ptwx3 pgdq9 mzuv66- /-+tz+zpnfincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Eartdtz/vwp6g+- +qm6zn0p-tzpp 3 u 9fxth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km frow6+k5 ( hk5gmxvfuxl(0pe t;qm Earth. The beam diverges at an angwt5mkepl f5xhg x(;v6qu0k+ ( le $\theta$ (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.7gfjwnr 43pu rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angulawj u3f n.7p4grr acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another kcj odp9b6p *mf -cu*;child. If the ball makes 15.0 revolutions, what is its difjmb p;dk*96c op-*uc ameter?    m

  A bicycle with tires 68 cm in diameter travels 8.: dy dleic13+ut)*sjocug//v ef h i-10 km. How many revolutions do the wheels ltdu*v13 e )icg dc/+o:- uysj/f1iehmake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500d2 ioyic29qzf, rcmqt 63q)k9 rpm. Calculate its angular velocityqm cr,qdi o9c9kz)t2 i63qyf2 in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig.gwl zp opacia+ jsmh-z,9g316/ r vwaog6 3.p1 8–38). (a) What is the linear speed of a child seated 1.2 m wzwar9gv-6j lmg/, ic zh s131+o opa.pp 6ag3from 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 ta h 5z90 s+ (xpiqd4 )m2pnfjshf-bo/hhe Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of1y-k+ejnvd 9a 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 tj 9wq- sa.dat2he axis of rotation is to experientq-92a .a jdwsce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center0p z4j 8qzvc 5ywc9fr0r*l7ht from 130 rpm to 280 rpm in 4.0 s. D zw84fzyt cq09 07rlvhcp* r5jetermine
(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 bm2,ozcj,3cf $R_1$ 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 put themselveg)t 96pqf))iqb:qy*yowcnzq /4s6 fx s into a slow rotation to distribute the Sun’s energy evenly. At the start of their 6 :6zp cy)qisw9 q4/*yogqt)xnffbq ) 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 22dmgpmt 4,p z: q271qj9ojoj d80 s. Through how many revolutions did it turn in this timt1924 7d oj gjp ojmmq,d:pqz8e?    $rev$

  An automobile engine slows down frou8xswx4zcg. : c :.,dvqgcmd o3on52lm 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 highspeed jets inebm( ,jmob rvh7k e*))n-8 gog a whirling “human centrifuge,” which takesbvgmn8 k-rmob( oj7eg ) *h,e) 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5 vsv8 s1m;sg2f hw9iwl8 -up. fs. How far will a point on the edge of the wheel have traveled in this -s9;g1wu v28s8slvmf ih. wfp time?    m

  A cooling fan is turned off when it is running at 850r fzcs8nu-4s m: kc7z:tev/min It turns 1500 revoluti:tzu4: fcmnssk 78- zcons 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 makem- ; hh m;u(;.dkpzj e6rlnlh5 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tireslnlmmk.pj5;zr ;-;( u dehhh 6 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:o .cfinm-q20ok,sj )2yqn u m speed uniformly from 95k u02y msq)k injn ,f2qo-cm:o.m/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 my *xt*pd:48gruuftg*t2 : mqj;m 8rwall her weight on each pedal when climbing a hill. The pedals r 8ud:r4mym2;r m gqttx:pf*ju **gwt8otate 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 of 55b 7 qce-9mdld 6v 98+lrejsv9m N on the end of a door 74 cm wide. What is the magnitude of the torque if the9-d9mvbr 9m edc8 j6slvel+q7 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.gyfi5 *p1 2xg7m,bht2 0mpt wi 8–39. Assume that xyti*mp7 ht1w0g,gbf5p2 i2m 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 pd5rgfzrs8 pkc8 (q9ro d*:e2attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque o(28dkedz*5crs: 8 q rpr fpog9n this system.

  The bolts on the cylinder head of an engine require tightening to a torque of u uf9yap3;q ng+ k+:xr38 ;a+fun9rup+gx3qk y: $m \cdot N$ 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 x7h/fdu9a1f uz98i xjm when the axis of rotaujxhfai8 f9d7 x1/z9u tion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 6e45vni- upcu a;)gw7u6.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub c; )wpuv47e u-una5icgan be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on awvk1;/.jzk8km n(x nqcs:q:q x8ud8the end of a thin, light rod is rotated in a horizontal circle of rad:8 x;zc1jdk /:kmq skq(unanq8v x.8wius 1.2 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 wheel rotating at constant ang(fjvgd)+3d6 z;o cwmvular speed (Fig. 8–42). The friction force bvgdf63+ o zmjw)(cd ;vetween 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 p38rcxv:.nte loint objects shown in Fig. 8–43 abov8:er cnl.x3tut
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total ma8jh0iooa (**vqo htm wr8a l91ss 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 satellw pfw. .msio q8u65jibe6o4 1dite 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 ro566bo1m 8isowpu.i4djq.few cket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is egs5 + n kqo01h.0touqa uniform cylinder with a radius of 8.50 cm and a mass of 0.580hot+g ue0so1q50. qnk 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 to2o*/ev/pyy9-vp icn b 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and is abl *n6*em5(3 5j jsxczz*afrpq se to accelerate it from rest to a frequency of rsp 5zfj* 3a*(zmqecx *5sj6n15 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 ropt90o7s5uw9 lvoyx*a 7 (ixphtating at 10,300 rpm is shut off and is eventually brought uniformly to ru ypos*a 5pvhx9lxw 7o79i0t( 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 3.6wbx 8mwy6j ( k6a5roaw*r xj+3-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 thell9j 372 6bvdwrg.ay dvld0p5 action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerat3lddwd50v7 lalrb962. pvgj y ed 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 mh4l; dia*:;mrcc9w jt9 mc5m8nc s y0long thin rod, as shown in Fig. 8–46jrl99ac m:ycsw;8;5m 0inh mm *cc4dt.
(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 q dtxje2w sji70g4jops7b;/ ,consists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four fuq s:7ucqzs 1f,bl zafa) 4o)r ehl:m5nd8 s/;lll turns (revolutions) and r); z r:f8mqnf 5sdlsozau /1,sql4a 7el:ch b)eleases 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 moment ofp()9 gxci0 nam 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 t )ao1l pqp /b1a,o3kldevelops 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 radius 9.0 cm rolls without slipping down aq(c;,vj/m nwb bb.t.a lane at 3.3vc.n,bb;ba. jwqm/ ( t $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of d ghs : vv8nnu 93zf(b(x)tlq*pz .6yvthe Earth with respect to the Sun as the sum of two tervbv*(tg.y)dhqunx :nplsf 93 (vz6z 8ms,
(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 of 7.50 m. How much net +/vlopuy1al;:1t, h5ydimdkl dm( 7 lwork is required to accelerate it hot+ylu;m(iyd517 1 :,lml/l dpka dvfrom rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80hpm xn*kie4l )mt,:3 uz.qz q0 kg starts from rest and rolls without slipping down i0tm:4x*)zqzlmnhu .q e ,3pk 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 vertio/ jd5xit6obkb6 z-hb07g:np cally 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 before itdhbj: i6-p5gb/t7 6x zobo0kn hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0o1 /p8 v5;vz)znilxjuc6d j- m.210-kg ball rotating on the end of a thin string in a circle of radius 1.10n j/5 x1 l-zv8opj6;idvu)mzc m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momenttiy r1avpqj+sns. :h1 ul1p;1 um of a 2.8-kg uniform cylindrical grinding wheel s :+arji1.p1tyuphvn 1ql ;1sof 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 side2rj-kca+ o eu3, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decra +rjo ck32u-eeases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shownlqpk(+t*qqi nw n2(a3 in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight pos3t+ qia*qk q2pln (w(nition to the tuck position. If 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 initu unbs71b13ltial rate of 1.0 rev every 2.0 s to a final rat1 7t 3nlsbuu1be 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 r/ 5qzo tdhu2724u +plby0uvbbotating 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 80hobuv y2/bd4 p5lzt7bqu+2u .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;fy smw ).0uiwy *st)n*rja)x 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 mnp: 21 g(5cppqrf in3nomentum 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 momentafi61lwg k9 4m,rm) ft of inertia I is dropped onto an identical disk rotating at angular speed fmtl 146 ,rkgimw)af 9$\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.44*(0,zc,aj mn/kvbg4rymwe q $rev/s$ 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 kl vf5mo ump012kx nj1r,0ik2rg stands at the center of a rotating merry-go-round platform of radius 3.0 120umjnf,xkrkv51 po l02mri 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 wixy, (01bm psqymf); cx3 avpe6xs82 jkth an angular velocity of 0.8m;)sey 2bxfxx,sj pv 1pcma0(68k yq3 $rad/s$ 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 i8 pbq.5qwxo,fyhvq 7 s7qbs*7ts mass in the process, and winding up with a radius 1.0% b5yq787xf qsbqpv.swo,h * q7of 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 in exces(q -dywzfn-q v di04i+s 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 ffkhyb 54x/or0r z: 81ea,onv$ 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, p ) 1d1w*ev x stra5qc3itc(s.initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform is 3 1)ivdt p *qsc1rw5tse.cxa($I_P$ 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 merry-go-rof61bvc* ti69dh0owaj 2j 5xv ik6pt -rund turntable that is mounted on frici1ht 5vikx6dv 9fbj0j2oaw c*-p6rt 6tionless 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 ground with the end of the rope lyino20(yi 2a8pnu3/lrwqr n tk;pg 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 spy8(inu2p2k /q wr aol3r0;tnpool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always facesw0qm us-ix 66f the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, i-w0xq f6s m6utreat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a raevcjq7g:v pd s;.wuqx:9x32 nte 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 radius 0.20 mi8uefwa88- b z4gr )of acquires a rotational rate of from rest over a 6.0-s in 8 w48fa )urgei-ozfb8terval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made o7k/30 5mc pemtvd wk0)zgqq3xf two solid cylindrical disks, 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 3/t 0w7k305m vzx)ckpgedmq q 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 angulabnr 4nxv ,41oqr 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 witfd,h9da0r 0u7 ;pm rggh mass 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 thatgf;hd ga9r mr udp700, 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.-lp6d -.* i9sjjkg3refv etbfu te5 9fym97a 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 diam 9bpjs *j g-9 7 katf3.9yu6.-ltevifd5frmeeeter 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 illustra-u 2jlkp 5n2xdc fl5-ates 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 surface n)*p g2qb:lglo8z 8gmat 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 4 r5fmy9t(ls8t kwvc 9M and length 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 angular acceleration of the rod, and (b) the l9lr5fw vyt4ksm c89t (inear 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 standh8hi+ 29gjdjn ing vertically on the floor, and we want to exert a horizontal force F hn9ji8d +j g2hat its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road ;)os4jcgbiovwt) .wpi d5h2+is making a turn with a radius The forces acting on thd.sw+c ob4wjio ) ;) gp2thv5ie 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 slingn12(gbu3oc fx of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque 1nbg2fuc 3ox( come from?    $m \cdot N$

  Model a figure skater’s body as a enl.a8 +yfum-e0so,fqd e8 x)b 1d. 3hpej(adsolid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculat emd1. s-neaq.d(8lx,uf fe+d8yb3) p jo0eha e 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 clutch assembly which consists of two cylindrical plates, unr**+a h )kchz 1b.uhof mas* +) n*kcu1r.hz hhaubs $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 a q utqa mly8/b*2f.ay+m j00xynd radius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vmmq0ajtuly y2abq. 0+8yxf/* ertical 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, bugg sggr,,f afu3lwaj-);ph5., yfq; i q4jj(ht do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed le n:vsj)jjl90ma-h(ky pz7xd :k-i2bv; t0z uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the an0j:m- kh2xld9sjivy(k0tj zenb7z- a;:v l )pgular 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