A bicycle odometer (which meauti9jbms if6 hu*3pwr 18.0jjsures distance traveled) is attached near the wheel0ru*jpih ti13.w9fjsm j8 6 ub hub and is designed 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 on )kj/d/ep nkh 6the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the poinh 6e)n/jpd/ kkt 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 velocitak /sd*5vtf/p y $\omega$ Explain.

Can a small force ever exert a greater torque/er,5x0yj3 f9khhbni 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 do a sit-up with your hands bmgnt za64kf5 rwo43g5 ehind your head than when your arms are stretched out in front of you? A diagram may help you to anw6545gg tn3rzko 4 mfaswer this.

A 21-speed bicycle has seven v g,27m+aub go ztj48dsprockets 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 peau oj7d4g,8bmgtz v+2 dal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slen+e 2q8r(f3 og5fzc kbqder lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is a k8gq3borf(z5e c2+qf dvantageous.

Why do tightrope walkers (Fig. 8–35) carry a lq l)f7.l910 o zzezrcsong, narrow beam?

If the net force on a somi( g/;jbc9 rystem is zero, is the net torque also zero? If the net torque /ocij;g(9rm b on a system is zero, is the net force zero?

Two inclines have the same height but make different angles with 1pzcjrrmizf14 izv:nhoix)9 w8z7+t/f8 7ec the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom h4n8ij8x z7c97 e )pifi+rvtzzf/cwzr m1 :1obe greater? Explain.

Two solid spheres simultah(lxajl w9a48 neously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater totaj(xalha4l w 98l kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They startwzfu;/mk 3731 ymgd q)w ows-v from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational Kd 3/;vwwf uwo)y qzmm71k3- sgE?

We claim that momentum andz8z1mwkqh t+4: :: 4apwvag(ykpiy/i angular momentum are conserved. Yet most moving or rotatin8gak 4wwz:qtykiva i/ my41(z+:php:g objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how woubvb9)1ul h) ydld this affecyu9vl) bh1 )dbt the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any init*(ojrh* 0ejc j8:c8pwiia + tiial rotation when she leavihtco cj0rw :88iaj* +(pi *ejes the board?

The moment of inertia of a rotating solid disk about an axis tqs9ghj bl h,,9hrough its center of mab9,q ,lsjghh9 ss is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2v2ygtch 2 1 k9hw82cwo-kg mass in each outstretched hand. If you suddenly g1o 2cvc2w8whkh 2yt9 drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass.1sg,-*s kv,z okert52 w6qaxi However, one is hollow and the othe 5i6r zxt*2wk k,-sae,govsq 1r is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. I0h sa0hkl -8epq6aj ;en what directio0jhk l6 eeq-sa;a80hpn is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a laswh/ p; c nmz4m:hb1ngbp)a/2rge freely rotating turntable. What happensm gc nzn/a)/4 m2p1w:bbp;hsh if you walk toward the center?

A shortstop may leap into the air to c gughk*e) -cv*atch 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 hv*kg)g hc*e-uis hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momenxd*/9uebr8l4 yt ;goqwbn j.7 tum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second pwu lybqoje/dg 9 7.x;tr 8n*4bropeller can operate to keep the helicopter stable.

  Express the following angles in radians:hbr kxtb o 6t 2.:x:zoof3y0t) (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 .zgl sj12g0w 0f,aduoof an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and 100 z,sgdf awj ulog.2the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The n7jmil zx4ps c3+x*-l beam diverges at4m p*n7l3jz icx ls+x- 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 rotatej5t,odc wdf jyqk+7l+n5( (ru at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to resncjkt(dr5wf +ud,ly oqj 5 7+(t in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If theu g75lk +zv,uv ball makes +5,zg7 kvluu v15.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many rm8 cm34qx-i isevolutions do the wheels maxq 4m -c38msiike?    $rev$

  (a) A grinding wheel 0.35 m ,. t f6rte0fthin diameter rotates at 2500 rpm. Calculate its angular veloet6tht0f. r,fcity 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.c92h cbhs)2 2mf xt6vo 8–38). (a) What is the linear speed of a child s h2bxs)2fc2 mvoht c96eated 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 tx8(- i0blj vj:ppc/4ebm,k n nhe Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of ou4v2e ig0l4ydre9 ix:-d uv,zd 7 7cba 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 tk*140gg z9ccalnhoyxv; 7i /ehe axis of rotation is to experience an accelerationlyc *x71;ci 9gga/ ohe4nz0kv of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about //l f,hil 2fd,zpobp* its center from 130 rpm to 280 rpm in 4.0 s. Determif,if/*,pz2lbh/ lo dpne
(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 (0vnokb2t+ r4x wgr3w$R_1$ 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, asnql0 0f,d3u pxtronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min timed ,p0unx 3lq0f 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 metfahb4 2viixgljs k1.; 90zo u0/ )z220 s. Through how many revolutions didfgl/ 109 s4zuv.iexizmht2 0ao j;bk) it turn in this time?    $rev$

  An automobile engine slows down * 6adkvov.ozw w27,lvzw43qv 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 nt +vzxkp3:vd ch4r*,of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete r:rc4dz+ n3h*vx vptk,evolutions 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 uniformqe eqyn*s0n6.ba7 cu qt.20bwly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheelqb q*0 qbwe.nuat y.72n6c es0 have traveled in this time?    m

  A cooling fan is turned off when it i 5hbfy)w+yrge 85ha2ts running at 850rev/min It turns 1500 revolutions before it comes t+r8htyyb5hwag5 f e)2o 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 revolutios;m7adr0j d0gax(xroh d3 m6ygp7p40 ns as the car reduces its speed uniformly from 95km/h to 45km/h The tirddr6y4oxa 0rhm37 x (m00pdpsa7gg;jes 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 redha x9/8uf 8q enop6my0uces its speed uniformly from 95km/h to 45km/h The08 8epxquyo9fmh 6n a/ 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 climbinrw)yz-qh g 0o9g a hill. The pedals qwgy)ro 9h -0zrotate 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 55 N on the end ,5s;z1 adel xeof a door 74 cm wide. What is the magn,sed e1 zl5x;aitude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque abs9(by3oxh w n(fu g6n*out the axle of the wheel shown in Fig. 8–39. Assume that a friction t 6*xwy hf(bgn9o(ns3uorque 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 ofxm ykyr/m*b( ymbo1qm-o)o*3u9q9zeut ++ yb a massless rod which pivots as shown in Fig. 8–y9-yzmx)qr mbtu/om(o+*m q y*3uyk9be1 bo+40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder heifqp,* p5a 8dga +pw3 u1yws9bad of an engine require tightening to a torque of 38 p8p3d 9fsa i u1+*abgypwwq,5$m \cdot N$ 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 av; h+ g+iq yma: 5 f4ln: v77ggypqc*y-yvvv+inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation ismc5y v yvgq++ -*ynlph:g7avvyg:fv 7 q4;i+a through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycl+ r2drcfr2yw.yj/ u*w jlx:p50s ) kafe wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg.ld/rwpx5)0r jwyafy+kr s ju:c. *22f 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 roif7,9 oh8om5kwp;mn 5: ycwrfd is rotated in a horizontal circle ohmok7 5;ri:ocf,mw5 n8wp9y ff radius 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 bdfv )l*w; 2mspo7dx6powl on a potter’s wheel rotating at constant angular speed (Fig. 8–m opp*w6dl 2df)vs;x7 42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown innravq1q/8d, kfqp 6jp1 g4n/ z Fig. 8–43 aav nrq/p8d jq4qk,/ p1z f61ngbout
(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 consistsfs+hrc1f:85vai o +v k 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 sabh gl +injih9n8jn 7mo,6+ b p8u;4wqptellite 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 4,7o;gn+j qbn hlp m6n98j4wphi8+bui.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius ofzn*tn(l-f/g 75e 3wjxu;k tvek0cga1 8.50 cm and a mass of 0.5nga/ nz t1-f5jg3eu vle; wk7*tkc0 x(80 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 batjjk);k dqd 2l(, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a sma am kdeihv :6,jr(7d8yl9qj i4ll hand-driven merry-go-round 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 of 25 kg)4hlj y,jka796( eddiv8:qmr i sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually *b/yw ):pm*cyjf s p+tbrought uniformly to rest by a fr:fb*)+sp /y*mt jp wcyictional 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 accelert: qqfq4s brn0u-/ ey+ates 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 forear ;zr3xf llq:x5rwrwfd;g-:xrz1 g5 z4m, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8gg 1xzf:l5lz5:x r 4rdzr3;q ;-rwxfw–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 shown in Fi+qt9z :a;/sk dvxf*td g. 8–46. +: ;tq t fd9avxkz*d/s
(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 consa8dix - wov4 ki)ia1,vists 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 acc z1o3jg 3gv ;zf-u 8a2dr4pbvuelerates the hammer from rest within four full turns (revolutions) and releases it at a speed of 28-g;ur 3v bzz d8p f24ujv13oag $m/s$ 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 k0igvr o)r.mwl;y9r (a moment 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(em a;*calbi8 develops 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 kgpn69gm)n fl h+,5aqoe and radius 9.0 cm rolls without slipping down a laa f)5plqme,o+g 6n9 hnne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as th;/fwz ++v 5x3kkgrheblqo/f/ e sum of two terms, b/l5e;xfgrzk/ vkfwq h+/+o 3
(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 e;6wpgro;n 80 p6zuo) jkl d9ymuch 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 solid cylind;ryo68pug)ew;jd l on9z p0 6ker.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls witho5hi+c yj 5n4pc5l1z9jg+dnw uut sj hd9c4zlnjg5p 1++n icuw5y5 lipping 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 tip. It st4bf0ecp -by5rbft/ 6earts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conserv-tefyepcrb 0b /f546bation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg(0 l;j u8+qdl+b ;z :ssvxgdoyj+j0fg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed ol+0:jjy+g ov0j lsfb;sgqu; d+8z(xd f 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whr tm8c+g +r82oienp 0veel of radius 18 cm when rotating at 1500 rpm? i 2m oe8rtncgp8+0vr+    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands aqe d-ehz8a3d7t his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, th edh -38zaqd7ee 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. 8–297f14ymi)wd+ d:vm nmg ) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1. )di4:my+1mwn d7vmgf5 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 rate of 1.0 dgbj r .oj0(j8ywh7*goop /0ga6:o sorev every 2.0 s tdhjo8 7sr0:bg0(.*oojw/pgooy ga6 jo 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 nhm0,fww 63ec 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, approximatel,f0n wmwe3hc6 y 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 mom dl y(u)0y*.efehb k6f ew3.-xfdcuo*entum 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 angularlrltcrxx 4,e7c +-f *xmso 6t6 momentum 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 of9 gsr+cg0.* h9h8mk wxrwwku3 moment of inertia I is dropped onto an identical disk rotating at a crs*wx8 ww0.gmh9 +ku39rkhg ngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns awnh39)rre *kk sa6a u*t 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 sh5pl86ci tt, zy ;b:zttands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 6t zci85zhp :b; lt,yt920 $kg \cdot m^2$ 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 rotating6xb.p r(a6z tx freely with an angular velocity of 0.86b6xx.z at(rp $rad/s$ 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 whitem.,mf sqxhj6pc(r o, 2 dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. A ocps,j2x. (qf mmr6,hssuming 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 excess of 120 c056 hras 7/1yhk6xocylae t 2$km/h$ 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 qxpqv;//q2pg2 n flb6 $ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely 0v gzit;/l4jp without friction. The moment of inertia of the person plus the plvti04l z; j/pgatform 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 b e1ym*52ne llof a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearingsl 1e mbn*el52y 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 lying on thb7 mdrc xo.ya8ki53sz/ k.) uog00rwye 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 fro0w 3o.dsicmoyb)x ar8guk0 7/z.rk 5 ym the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that th51)s0/g:yi6 wr qjlmpbnyc 1p e 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. (0 pclq51ygmpwrb/jni61 y s:)In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates f3d ctmnnr4s8:5axkfro75( wn 2vau - mrom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a unifg)kb- kn: zoic6pb.c7c9x2toorm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleratt.b6)koncc9 2zck :b7 g p-ioxion. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and di7 xj,gk1/*c za:hglphameter 0.075 m, joined by a (concentric) thin solid cylindrica /zhxp7kgg l, 1:ha*cjl hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the anguhgkql :s suv4 1+l*+x xq/4kzjlar 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 sizqp g)jo3(e 4lo:p rm:be of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 dj:( o4 )pqlbmgop3e:rays. 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 aug..+h3trv;cs:zd ,,one dhqc tomobile is 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 chv;og.er:n c,+td q,.zd s 3hof 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 illustrates azhd8ce8.o;w3w ekv t. n $H_2O$ 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) mi2v*c q jqp.bh7/4cf7mjt v 0is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore friction(d as/+4yzs ri) 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, a4/ id srsz+ay(nd (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 radijoy sxx8e cn) .u0-z:xus 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 step has height o)y z -ne8xjx cx0us:.h, where h

  A bicyclist traveling with speed v=4.2m/s on -r . um; :zbvwi5m+akwczzyz6( -q su0a flat road is making a turn with a radius z-bmw.ysc z(-6a;iz:qk ur5 uw z v0m+ 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 rh wxebw 9d/+,qzrsm+ko /q,e*ock into a sling 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 t*d,we mw h +9orb +sx/qe,zq/korque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skatermuch+d y-u*.s j a6ml0’s body as a solid cylinder and her arms as thin rods, making reasonable e l6 y0cuamj-s*.mdhu+stimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two c7vs-,efk3w dj ylindrical plates, of mafksj, 73-edvw 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 radiusi 1 wz55p(gt w ,to k59qob4ie59doizf r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving tqb1kgt5i(i4o55zo 5w o9pw ,f 9zied the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do wjm2uirq 6xj:;gw8-ja/0yl 2l b,kat not assume $r\ll R$ h =    (R-r)

  The tires of a car mb5hjjw3b4 v:((t 3exfsfb -hk ake 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accelerf bw(jvk33b 5e4h:xf(tj-sb hation 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