A bicycle odometer (which measures distancddfo lc,)+bk 9p*jc )ne traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle wid,l+jn)bpc c k f*d9)oth 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on tjqxo ntjf) w5xi7zt16 0q*/alhe rim have radial and/or tangential acceleration? If the disk’s angul* 51z6o qwnt7fai lj j0q)/xxtar 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 0mzzajwrn ta+5cf4r;l+w,r / described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Ex3esr 65.;i ti3 cmqpwpplain.

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+hv1 zboq)8 fl do a sit-up with your hands behind your head than when your arms are stretched out in front bo zl1q+) h8fvof you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the ot+9j/(e a3n:o1yc2 mv t6qqfxoks5kpedal 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 smaltscno1ko6ok(vqxt:3f am q +9jy52e /l front sprocket or a large front sprocket? Why?

Mammals that depend on being able tfvni1qp q6cbmmf3ik9)n0o 0w, s .3tao run fast have slender lower legs with flesh and muscle copn.qn fmw ka o1i fqimcb033 6v9,ts0)ncentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fiqa)j)h posj7 dz,(hnpr+ do1eh :4/yug. 8–35) carry a long, narrow beam?

If the net force on a system is 9x)l wtl4s71 icpvl 51q*xjkkzero, is the net torque also zero? If the net toi q11p9*ls7 5xvlw) l xj4cktkrque on a system is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizontalenhm*)rk c,2o68b x pd. The same steel ball is rolled down each incline. On which incline will the s*ebxcmnd6, r 2h8)opkpeed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) dow)jgtycsel p1.c,l x/z4i3e2 in an incline. One sphere has twice the radius and 13lte .)/, 2ye4sxiljc pgczitwice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. Tht-2 kzk81:xfyc tj7 ohey start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which ha x2 hftk7ck1j8t :-zoys the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and z1*g yunc oxq 7.1aii3angular momentum are conserved. Yet most moving or rotating objects event 3zu7qaic1x gyo*.1inually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s eqbb/h)y)rxg axb9p v j7w549n vuator, how would this affect the length)g5 97vba9nvwx/ )ybxb j 4prh of the day?

Can the diver of Fig. 8–29 do a somers4eva 0(2xir7c32mhi weyp*atault without having any initial rotation when she leaves the boar7iwi aexc2t*y0a4 r32evpm( hd?

The moment of inertia of a rotatiy,zy)pl. , tw 8vqhj:sng solid disk about an axis through its center py ql,.:vswtj) yzh,8of mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holdis9ofw-iwm0o8 ke72 wc ng a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, dim0oc87k-29fes ww owecrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one isrd/i d3qc.u ij3r xi)3 hollow and the other is solid. Describe an experiment to determi .drxiu/d rj333ci)qine which is which.

The angular velocity of a whee3,ql3 d1a5 e*hsu yhnbl rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tan5l s hhqey31n3d ba*,ugential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on t iaf*ovzc2z, n)q2 k 346ac4fyeq2yjd he edge of a large freely rotating turntable. What happens if you walk tow4yj6e222,4dyq)qvf cf zanko i *3zacard the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As filzc:2:l,.v, mornh 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 direction (Fig. 8–36). Exp, riov:hlc: m,2nlz. flain.

On the basis of the law of conservation of angular momentum, disc*ar-9kkv hyo5,:of a auss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second p:h ykfo aa5-kvo9,r*aropeller can operate to keep the helicopter stable.

  Express the following angles in radians: (axy2uy6go. 8h d) 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 coincidenceqz;) / 8l ed a- pbk58-hrcub0t*btrgu:t av,m. Calculate, using the information inside the Front Cover, the angular d-vhpd - maqzg) b c,kt:au08te b t/;ur5r8*lbiameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Eag6 ;q8qxwc/hmrth. The beam diverges amw gxh;6 qc/8qt an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turne4t91yuc;nhgi zk 0:otd off during operation, the blades slow too0t 9;zi:4hnky tucg 1 rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on afcy)c t yt,hcw6t-k(/,i8 k jo level floor 3.5 m to another child. If the ball makes 15t w()k-,t8 ct ifh ykc6oj,/yc.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolux*jez(r:3 ee p +swj,htions do the wheejxp h,:3( +j*s erezewls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculaqcr.9l 1 lu4+h u)xlkrte its angula1c4 r+ q)uk9l.xuhl rlr velocity 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.s 7xs23d(f q wegphi;90 s (Fig. 8–38). (a) What is the linear spee( 9hws7spf3id g2x eq;d of a child 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 o:ra8*tjl rc(e tb 3+7l8kf5p leh8elp rbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of-v9 lrdw;6nj ux)42uaw/ uozn a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the ax 7xi)xbpbxila. b*;3t u66tkwis of rotation is to experience an ; ipi763 bk.bl6)x b uxtxw*taacceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 0c:2:hu4 uig rlysz t1130 rpm to 280 rpm in 4.0 s. Determ0g:hcrztl s1:u2u4y iine
(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/x yuz(5ejv1 y $R_1$ 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 ab,.qov, x np6 w, f.7og:twlphboard the Apollo spacecraft put themselves into a ,vnw 6otqx woh: pp7f,. .g,blslow 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 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, 4oedzmb7- ;6hzum stzza8g 97000 rpm in 220 s. Through how many revolutions did it turn itm7u a 8z9oz;4zghe z-b7s6 dmn this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcu(8xs 0yqab vx-late
(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 stressesq-80+baj03eyc ecz/vx1ruku of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its f01/ccyu+83jxz0qeb k-veua r inal 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 rpca3s(lb */sgmp0 /nf pm in 6.5 s. How far will a point on the edge of the wheel have traveled in this time? g3c0m*p(slbap s/ /n f    m

  A cooling fan is turned off when 4-efb 7-x6j bt5pqskpit is running at 850rev/min It turns 1500 revolutiotq7 fbp-x bsp4jk56-ens before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revo61 hmntdnt*)(zsoy7,jr: zrhlutions as the car reduces its speed uniformly from 95km/h to 45km/h Them 6ro,znht sy1:*h )z t7rd(nj tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions asv.z w/x dfd)8;e3rjh t the car reduces its speed uniformly from 95km/vj /h etx;8df.)zwr3dh 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: bnlqquw1in3aj16u; a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of bi1lw3; a n qq16u:njuradius 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 enewgh+tq2q q )kuyo5b25-: wgdd of a door 74 cm wide. What is the magnitude of the torque if the force iet-d+yugqo2w kw2h55bqqg: ) s 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.43cyox(vqmc w,tlmy)4 1;nyv 8–39. Assume that a friction torque (xt cwv3c1) yynv,; 4 mmlqyo4of 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 mass1 w;gu.wjk5yg less rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal pos gykw15u ;jwg.ition and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require ti8us r xz9qpn r)c/-ts7fkao. 1ghtening to a torque of 38 rtf8 osr1qkx)nu9pasz 7 /.c- $m \cdot N$ 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 1.1 f nbgf2 is0zxk51jczdoi/ 00.8-kg sphere of radius 0.648 m when the axis of rotation is throughxskbo2 1f5z.f jgiid zc1/00n its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamete5rs27 bjzdp(fr. The rim and tire have a combined mass of 1.25 k7sz2dbjp5r f (g. 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 ro(ri v;bm(7 otizkn75z j:7 w hfnv7/dkd is rotated in a horizontal circle of radius 1.2 m.m kw jdhi(kr7zvt ni/ 7;(bo:f77nv5z 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 a19 ix-cn)i 05jx3tgdnv sy9; +fnp lvlt constant angular speed (Fig. 8–42). The friction force between her l1 t ;3x jxpf9gvn)+-cinl9d50y svin 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 obje hjlb;dwgw3c.1n*;q eoo p0a,cts shown in Fig. 8–43 e wdlgb; wo1o.p,chqa0 j*3n; about
(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 5i:0-z)sgtrxazv9h uwnwjpq0t;6 l(whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at ty m4v0m nbkcv5 ;uuuv2sh45c4bf 6f,khe correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite i h4vk6cummbs;45u20 uf ckyb4 fvv 5n,s 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 ossb* d--w c)axay *lllsp17, cf 8.50 cm and a mass of 0.580 kc-7y*ds)sw1*lb,cxs a p- lal g. 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, accel 0 kw nklgkqufk ;w3x0+:)sis+erating 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 small hand-driven merry-go-round and is able- tzl c3ef7po2 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, and7 ecopl t3f2-z 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 g*4ofwht 0pm,10,300 rpm is shut off and is eventually brought uniformly to rest by a frictp04tgfwm ho ,*ional 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.+. kmdh9rqv9l 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 actioucoyxla0)3bbr 0/f a,2 fqzrs)zur75n of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is fx)cr5l/ a 0szufbyo,2zar)u r730qb 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 Fig. 8x6yk d7 m8 6fo1qydum,1eyh.f–46. y6 m f1om8y,dq67ukxe d.yf1h
(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 ofp+kv a6ia ugq(-)5zhz4h djfviz j,-7 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 wiz9di0 mu fnj),z+/vgv thin four full turns (revolutions) and releases it at ,9vv z+zm 0jdui)n fg/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 of inertd0mi;o (.( 3eolvfqni lt 62oyia 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 torque q*yor5e9m;jiwo pektk+ p )( qea,s+(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 wakar;mhod4ww+u-ep 9-0po/pd jj .p* ithout slipping down a lane at 30/rjhd4w-uo.ppppa;9 adk o e-*w mj+.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy +kxj0ryl:jc +) bppk9of the Earth with respect to the Sun as the sum of t)jpky0 +kc jlrxpb+9:wo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.v q*wezda8)kdbkw7z ( b3xt.(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 solid cyk* v)x( ezdbw w(a7z.3d kb8qtlinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts g* jjwa*ew i:+ r2+yzofrom rest and rolls without slipping down a 30.0oj r *+:e +izjaw2g*yw $^{\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 balancmp pv(fi 5p-z.(hl yvst535jww4rq c +ed vertically on its tip. It starts to fall and its lower end does not slip. What wilsl4jyq(z3v5mp5-pw 5w(v+ f prc t h.il be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kgfks3pfd+- t (g ball rotating on the end of a thin string in a ffp d(g3s-t+k circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.hfz726jl :(nhah .gxa 8-kg uniform cylindrical grinding wheel lxah6 hf n:g7h2j (za.of radius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side*ixb blx1j uw88t,v/,hr2z*p+sx ad( h+a ztj, 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 rotatiau, lbiphjr8ttwbxz1+xaz d v8* +x * 2s/(hj,on 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 ./6cjst*c (dobp rc zq p(atu3d/)q a*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 1.5 s when in the tuck posip bjr)3q**d.tca dosu6tca/z( (q/ pction, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from k wi),.naeuw7td1 ny/an initial rate of 1.0 rev )ye1n,d w.tn7kaiw u/every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axisnzsyna6kvjq ,r5v-fq 7i21 x* through its center at a frequency of 1.5rev/s The wheel can be considered a uniform 5if- jrvyskaq1nvzxq 6,n2*7disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinn3sc91wfopu 7 x3sdpxuj4,k5aing at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Ear q +ku)auku8mhkm b,34th
(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 ide.ahej e0 ;),ylrxco)ontical ;,).al)erjh co0yoxe disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsf7vbcx; *g/ozovpt5.hi k0xit4:s2 o at 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 cented,5 cyg /hmz:tyy2)fl r of a rotating merry-go-round platform of radius, h2tym)5yycd gf/lz: 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular/w o7(h-snrs 2y4gg1ta*glqj velocity of 0.a2g/qywj1 lg4 ( ghsnts-o *r78 $rad/s$ 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 its.9k3uw 1nv oqk- 3wbzz mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentw q k-u3v3.bkz 1noz9wum, 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;i 9x pr7ww0ekierd+4s 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 isj7zzd4ud5b uol ((v/0+sidy 2mci 9du5r* k$ 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 freely6cvu /;7qb0foiokje 1 without friction. The m16j ;0bvfc o7q/iouke oment of inertia of the 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 merry-gofl0b hk vw5n5:.iubh--round turntabl5v i5wk bfl:0u b-h.hne that is mounted on 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 the groum;u p459sxprvmvf*y3)v wx r4 nd 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 from3*yv pmw 5v;9)xuvrfmx4r s p4 the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side 7zw35 ffbk q bjxg7s,1always faces 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, tr3 g ,fkb7 wqb71fzxsj5eat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates mj: +4h; oq4vadka 5doyj* da7yw-sx-from 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 idr4o d4v.6-u/sc v,sfkvs5pt n the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a ,f v 4svk5/c-sdp vro6tus.4d6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two sol 0 ql,.mqb9knqjt;8u lid 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 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. q;8nb0., 9mtlkj l qqu   $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear whe*gsx.i5y .t) ehgy cl0el ($\omega_R$) 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 ovkkylpd+6i jhte 33;q l7 mr7.f our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution evervh36 +qlj7pk .dtm;i ky7 e3rly 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-pollut8 ,mqiolwxz 79ion automobile 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 f7lwx,8zi 9 oqmlywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates autski /r t uk:gl)3gyx /+l9;cn $H_2O$ 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 rolsul ;j9w9b* bbv9gus- ling 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 pijhl o;0lh6d 5em38 doxa8,anavot 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 show0h 68x o5eajahldo,3n8mld; an. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on thk iuf qnm1-h7v/x1f2qe 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) xkf- nfqmi1/7h21vqu. The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn wi9om6qrto0 8 / :qsz9n wfeex(wth aq9q9o nem80xezft sow:(r/w 6 radius The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins whirliapnealjs23-e-/8 rka r p3 q8ing the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rp3q 2/paspl-i naaej3-r8k8erm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as th 7g zqh12ax7wjin rods, making reasonable estimates for the dimensio7zjqg17 w x2hans. 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 cylindricaexpx+,+b d8z2+h nlxml plates, of x8zxh+b 2 ++nd,lexpmmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 thedqeelt;rf .(rz0 ;7l 8d:f8v k+zu dos35htbb looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that5 rdodle;+uvrdbl88f;t0qz 3z seb.7f(k t:h the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do nofyz ,19um;0htt-t tmst assume $r\ll R$ h =    (R-r)

  The tires of a car makeqyj/p+t buw*rd9gz3p 9s4nd7g 3ru j1 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 acceleration of each tire?/79p yw n1ggtj+r4 *9j3rub3udzpdq s $\approx$    $rad/s^2$(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