A bicycle odometer (which measures distance traveled) iszh1qz:q1jv-rog y 3f( attached near the wheel hub and is designed for 27-incyqvqr :fh 3- og11(zjzh wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at crc2nn 6s(j dglqum o;m+1) +luonstant angular velocity. Does a point on the rim have radial and/or tanuq6( u;sol)l1d mcj ngn+r2m+gential 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 singleb3x1h3wb 47zhhfv,,ax ft 0az value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Exp,cu/7f 5d5lwaealwp a 7l q/x3lain.

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 k1mk1n **ccrybehind your head than when your arms are stretched out in front o*cky c *nkr11mf you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at thj3 bvm: xuseka*g9y:fs16p a 4e rear wheel and three at the pedal cranks. In which gear is it hap6:g 1a4kj9y: vexsab 3f*musrder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have j2aj*zd90eow fk:5 kqb a6 r.wslender 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 advantage2.*dqkzrwbfo a 6 k9e50:j jawous.

Why do tightrope walke bi+a mce6sa//rs (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, ialkt3rx dm 16. vp+3jns the net torque also zero? If the net torqu 3xn1 v+tl6ma3pjd.kre on a system is zero, is the net force zero?

Two inclines have the same height bu9/ hfcy*i-v1,q wf r6dexk;vw t make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed,rx*ffwdy/i9v6h ;evw cqk1 - of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incli5vi 1o* zn-7g rnwtn1vne. One sphere has twice the r7n1t z-rn1g n*5 woivvadius 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 total kinetic energy at the bottom?

A sphere and a cylinder havdw-+7,gzlne;q y9ob2-h lbt ee the same radius and the same mass. They start 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 rotegdz-9eh 2o-nyw q+llt7b,; bational KE?

We claim that momentum and angular momentum ar9bg 8y-w b;pbse conserved. Yet most moving or rotating objects eventually swb-y8 gbb;p9 slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how woultcn:o8 sxtmzk93* tx)s f)o p.d this affe zk o:t8fx)9 so*).nctmpxt s3ct the length of the day?

Can the diver of Fig. 8–29 do a somersault without having i rfgb60 4g.l *ovr3hlany initial rotation when she.4*0h3grf glo lrv6b i leaves the board?

The moment of inertia of a rotating solid disk about an axis 51dsrex a (cptk+-pf2through its center o dse a1c-2(p+xf5rp tkf 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 holding a 2-kg masm+ug4gm;y 4uwowx: /hs in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay theuwuyhg w4/mo+4 mgx ;: same? Explain.

Two spheres look identical and have the same mass. Howe21 27h/ jcgkg;er muj p(as1e0:n9w iq c-rgrzver, one is hollow and the other is solid. Describe an experiment to deterhpr1r-c0n i1gje2kemqg(r w2z/9jc:ug sa;7mine which is which.

The angular velocity of a wheel rotating on a horu j h pjbfa2/e y: reo,*s(l36yp+6+kyzbr(kdizontal 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 tangential linear acceleration of this point at the top of the wheel. Is the angular speed3ke62sp(e f, rhky(6j: d+py+b/ol u yjabzr* increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. What 1igri+o-7 xsa happens if you walk toward the center?sgx+ai oi7-r 1

A shortstop may leap into the air to catch a ball and pvfc8 vjyla -8/9tth2throw it quickly. As he throws thl8ac2t 8t -fyhvp9/vje 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). Explain.

On the basis of the law of conservation of angi6pk(vq /mko63 g;k91vqwrm6lyum,e(i rl(nular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopt /wm y6q( k(mqp,;9 (6vomklig3 irnr vel16uker stable.

  Express the following angles in radians 2hjp4yslkt nj-bankye-(4ks.5 g3z0: (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 a u qa(re*5km*d66 *mgvm,rhqzmazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun andg 6qrhvk *d( qr*m6 am*z,emu5 the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The ws8:boam*/7i ee okf: beam diverges at ai:/b7* a smk: f8eooewn 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 blenh qddfo6f:85c 7h-mmd der rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades hmdod-5d67qm f :8cfhslow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3fulf1 3tca)bo j)t- /v.5 m to another child. If the ball makes 15.0 r/ fftvoc3 abl)utj-)1 evolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions doc) aht 3loob.2/sds)je w* +gc the whee3 2a+)gsd/cehcs) .o ol*wb tjls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angn(3c oq-qwb/ kular velocity in wn (-qbk3q/ oc$rad/s$ $\omega$ =    $rad/sec$
(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 comp8v3qbv)l,ac i 6rhjh9uc) *b plete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the center cbp9l *r hu)j aibq63hvv)c8,?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity+6hih xa*o)4,v ktpt c of the 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 of a /5dw,z-n zot x c jqhffnaj,v 0vy6 2+pa/i+i8point
(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 centrifugqk 4- (;9hgtwnibxi id18 *ybre rotate if a particle 7.0 cm from the axis of rotation is to experien*y4 1hwk (q xirtb;di8bn9i-g ce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about it r2pk m)b : q91u0txqdoozvlu,+cg(:g s center from 130 rpm to 280 rpm in 4.0 s. Deg bg d(tkqoz0cmr2,lu)xqv1o + 9u:p:termine
(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 radiushf4 h0akqf o:9 $R_1$ 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 g)s +/akq;vluaboard 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 time interval. The spacecraft can be th/uas+g)vq;lk ought 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 fe*7s rdn,9 qhin 220 s. Through howsf*q,9d h7re n many revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm t 0i9f9 w-n gslte4iwx-4t za*lo 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 in a whirling “hja933t-o i ;(brsc jvb,ba/ wouman centrifuge,” which takes 1.0 min to turn through 20 complete revolutwor,a/9j-sib;t 3vo3ab( c bjions 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 acceledfs n(fi79t: j6 kt7dbrates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edt : di7s kdj7fbf9n(t6ge of the wheel have traveled in this time?    m

  A cooling fan is turq4:b)xqxrgi,4tc1dyww6rk )ned off when it is running at 850rev/min It turns 1500 revolutions before it co xb:q6gix 1ctrr)w4wy)qk d 4,mes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed un/g0y sqms5 7stiformly from 95km/h to 45km/h The tires have a t/0 yg7qs5smsdiameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed un11qu c -lf4.m bu+pdxiiformly from 95km/h to 45km/h The tires have a diamete4xpl.m1u 1budc-fi q +r 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 g7b 1 aa1ui4yjon each pedal when climbing a hill. The pedals rotate in a circle of rad7 gijaby1ua41ius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What i0l8i.njh ;jve ea;8vc+ sg-m is the magnitude of the torque if the force is h;c8egv0+;e jijvsl am.n8i- exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39srzei 9fa7ap(ik/ rd 9ln;c+:. Assume that a friction torque r; d/n9slc+ 7p(fa k9:eziair of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are at*n.c 17*cq6fr lvni- g fk,klztached 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 magnit*ckv7 ,6rnzli- cf1nk *glfq .ude and direction of the net torque on this system.

  The bolts on the cylinder head of an tv8ahx8/ b0zrengine require tightening to a torque of 38 0tr8vzxh /8ba$m \cdot N$ 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 a1jh v..s,tf5a0mgtl)zr(s onc1 p v 2y 10.8-kg sphere of radius 0.648 m when the axis of)c2t.pav5g h( jr0,symt n1 1osvfl.z rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in dia3*j eys06) n6ju+ 7oyokx vbsbmeter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can bn6xs37* yjoy jbv6bo 0) ek+use ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, ligh lds 9:9dxsl/4no6rclroofu/+te6d7 t rod is rotated in a horizontal circlet/x nl6d9cuel6doo4s9 /+7ds :orr fl of 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 bowl on a potter’s wheel rotating at constant angular spe+ vnwg+ ;dx nzw-9c2fs--zijz s;+qn ped (Fig. 8–42). -+dnw9;zjvfz x +-ic +n-ss2gpnqz; wThe 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 obje0 b0/eee;akxgowke (7cts shown in Fig. 8–43 e0 ee;gkxke0w(/ob7a 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 oxygequ.-nnq 1/yz ln 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 satelc-3x(qiuu05 lo ihwi8lite spinning at the correct rate, engineers fire four tangent0wc85(qoi u -ix3 luhiial rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is abeu/3k( /8asmd7 ynjq uniform cylinder with a radius of 8.50 cm and a mass of 0.5dm(j3ukb/7na8 sq e /y80 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,1y1qeier yal ,5xw2558sscjz 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 small hand-driven merry-ggb/ 4d8pb 2a8; 9jvf7 qsjskulo-round and is able to accelerate it fr8d7fp9s2jb sla;/jgb8qvu 4 k om rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut ov: m;2wyb oaj0x1hw-06splckff and is eventually brought uniformly to res-a:hc 0ob wvwk 6m1y0;jlps2x t 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 n bubgm. h8v7+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 theh sj za8g6 sh0f6-hpsif;5a)i forearm, which rotates about the elbow joint under the action of t5 8p6hsffgsai6sa0zh;h)ji-he triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8wn,a.yu+v k844 ddl ;kkg zrc;bk29nv–46. a ,vwrldud9gc8ky4k vn;b k2.z+ nk 4;
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of tw nn8gsxxl)38p o 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 acceleratehzw8xv,3wyzj, wz 1ye3ktj(x /u, kk5s the hammer from rest within four full turns (revolutions) and release zjv,5k( ,ejyhwywk331 tzu8wx zx,/ks 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 arbl6mgni c9g() :vme2 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 de: gyv25brtkwlhy h ;ehp6a7./i 6q;ovvelops 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 raddn8ejr p9xx tj;507liius 9.0 cm rolls without slipping down a lane at 3.8 l xjnit5jepr;7x0d9 3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun crarz fejh r .x1ym8)23vg3. d5jvye*as the sum of two termsejje3arv1 c)vz .8ry h2d yx.*r5fmg3 ,
(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. Hr4 0ap p1) fqaaqi*t :v1k;vwuow much net work is required to accelerate it from rest to a rotation pav1twfq)* 4:a vp1a;q kr0uirate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm andzu 9fzbp3r .c:.w3b-h1xx ytc mass 1.80 kg starts from rest and rolls without slipping down a 30.0 9ybu.brfp- xz3hcwt:.1z3cx $^{\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 starts to *isit ( 3u6gzmfall 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 conservation iumis(63z* tgof energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thinv,lga6(q c) kb string b kcv( )aqlg,6in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angula-vak vvm 0z9 du 6z8kfs8py08wr momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm? z8 0v8w0ym zspdu8f k- a9vvk6   $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a k8tn- 3onccal x*qcgc) d5w2((xkj6hplatform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontkl dnn(* -cjc8wg x3c65 (thxq)cok 2aal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown lk o19 f oohnm3cqdqn:k mn.1hp6j0.; in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the s. 13qn h:o9dknj o.o6nph0m 1cqkfml; traight position 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 4gc/ eeftr:g)nzj * :xher spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a fi:4*rxngf/g: etc)jz enal 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 aroun8 9rxg2hogrumk8n) 2uuyra.ek p7 - e1d a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating w9h.eg1 oruun8rer k ukm 7 ypg28)x2-aheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a*sea-ely :fey-s(.n qhsohn (3 3 vdxc9la(2 q 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 momentum j6s0 1wa: hjmgof 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 cylindrical5f9ndf5 l 1 m/q6uvunj disk of moment of inertia I is dropped onto an identical disk rotating aum/1 jf5 dlvu nf95nq6t angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a 0yowss z*4z5i4jnq4lt 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating meru )7u 9t6vtvj6:z1b wxvcv3c cry-go-round platform of radius 36bw ):vvvtu jcu7z1c9xvc3t6.0 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 isfg j zk-i084 rmdyj(i4uz rc82 rotating freely with an angular velocity of 0. jr8fu8cgzjr 2zkyid (4i0 m-48 $rad/s$ 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 iu aql- 6i:u1vtnto a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what wouldtq 1ulaiu:-6 v 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 winfgajczn447e t p3q6is)h3.x v ds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass :bz71 tc q1m*tbg54hp ia.0ccgfnu4 q$ 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 platfvtbz*8qo)q -,ay x9il*rddr; orm, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the aqvz;i9q*l,-*oy8rx) tbdd r 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 el3*j 2bz mzwhivt-k.qh:w.l- dge of a 6.5-m diameter merry-go-round turntable that is mounted on fr2b w:zvqhll3--.w j. tk z*imhictionless 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 lying on th1my, n(eq;f30xi w 4e30oy7cjaa bmxze 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 thexief;x,my n1yz4 qabeo373cm0 w0 ja( 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 faces h p)23t9sxmf6k u.ub o.uxe3yxff/v8 the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbitaffxes.mh 3 .xvu8f) ukbx2o/uy3p 69t l angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates f.lhs/x ipqxyju 2 d+;.rom 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 ic+p wv ,/sy,isn the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delic /s,sp,+iyw vvered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, wan57 :e,k/jyr la*jfeach 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,/kn7: a*5raj leyjw f, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wj h*bk5a(xz-6z,xoh d heel ($\omega_R$) 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 o wg5l gpeao-rzy8w1y 67 .xiqh5+dctpp ei(85f our Sun, but with 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-rc6ez+ixt e1yo5p.8qhdyl8p i5gga7w ( pw5 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 automo-or 5p;rd ab.4 mfac;d1wd+wobile 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 cylindrirddw-a4+ f5; ; cmd1owopbar. cal flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an (q1yeh j5jyr/b21y sa3 :ewhh$H_2O$ 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 surf5yh)tb5ez 0mhace 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 fricti 9dz9.drqq +xyon) 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 accelerationq ddyrqz .x+99 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. sk snre3)+k3akr om6 ks :e:r(It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so tha eksk33o(ark:srrn k:m)6+ s et 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 roa 2ayvm bcjmu,3hs(/ w,d is making a turn with a radius The forces acting on the cyclist and cycle are the navu m/s,y(,23h jmcwbormal 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 inwdm5+ s,cy9rg to 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 torque required to achieve this feat, and where does the torqmdy r,wg5c+9 sue come from?    $m \cdot N$

  Model a figure skater’s body asox mu-9u5; ro eoz6d9o a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular spomderu65uox9o 9 ;-z oeeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly whid .iwj kod;/s0wy ,geu:b)h 1mch consists of two cylindrical plates, ofg ;/i0uy1dje,s.:)whokwmd b mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 the looped rough track of Fig. 8–b43cfvd rl* 3d58. What is the minimum value of the vertical height h that the marble must drop if it is to re 3rd4*3cflbvd ach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bzn)q5p9 f/8yhi)kl u put do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 nmu62:gnqfe x4x*i.c revolutions as the car reduces its speed uniformly fr.u6 * im4xx:cnf2neg qom 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s