A bicycle odometer (which measures distance traveled) is attached near 7m(kdet66l p, fp4jdmthe wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?j4pkmemp t76, ld (6fd

Suppose a disk rotates at constant angular velocity. Does a poin v,:x-zajdx k*t on the rim have radial and/or tange*,kxv d:-zxjantial acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value 0lf 2yy95jmu t6mi:k kof the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a la))ctbgnlr*u vk *;j9 urger 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 site gst/sx:s qy6;*(li6bxv/g u-up with your hands behind your head than when your arms are stretched out /ely s(:6; vsti6ggxsu q x/*bin front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the p1mp41-c: /ik8xoy cma apnhm,edal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sp1no 8p-,apmc:imkh1m c4y/a xrocket? 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 slender lower legs with fiu 1tr2)u d9 f/nosyp:9ycn:fh*ptf, lesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mati:9sf/d9, f:t o1 h cur 2)fnny*ypupss is advantageous.

Why do tightrope walkers (Fig. 8–35) carryi jis+x. u5uahwi40 ,z a long, narrow beam?

If the net force on a system is zero, is th p-wf 9d)qbt+xny;y,j e net torque also zero? If the net torque on a system is zero, is the net force zero?; wx,y-p)y9q+btnf dj

Two inclines have the same height nfeee8*so48iuk cx19but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greatf1su9k c8 *n i84oeexeer? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. One 1r:/ hqm6f9 possn+iy *e p-xtsphere has twice the radius and twice the mass of the oth-*f: h9sr/m x61iy +notppq eser. 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 rxcqdg: y;pqfepw70 ljk*sa 69. t*jj+adius and the same mass. They start from rest at the top of an incline. Which reaches the bottom faxj ;p7:e .q*ypwlj6k q0gfdjtsc9*+irst? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most x296p z)k4b bv8smv j/x*bgtmmoving or rotating objects eventually slow down a8b b tgzv2b4ps6m9)x xj/v mk*nd stop. Explain.

If there were a great migration of people /vbmub/4(v mg ye7*sctoward the Earth’s equator, how would this agbub v(m c/vs4 e/y*m7ffect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initn;lplg) pyns9 )q- b)k lnm--tial rotation when she leaves the board?); -- gs-nlp9)mlpl )bynqnt k

The moment of inertia of 4/i b:l1y fewma rotating solid disk about an axis through its center of m/w 4e 1mblfiy:ass 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 sittf+ shpyt:3 5-t 6siwkwing on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? E +-p3:ysfk tws65wth ixplain.

Two spheres look identical and + 84h ,;vjzpr cvj/r 7pr s65jd)gzqwkhave the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is w vk +dchv,8/;wgjpjjzr)p5 7r rqz46s hich.

The angular velocity of a wheel rotating on e;hz) osw/l7-o cz6at 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, -e zosh lc7taw ;o)6/zdescribe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotat9fkk;k6r ntq y(b: r5zing turntable. What hapr 5;ry6 bkzk t:f(nq9kpens if you walk toward the center?

A shortstop may leap into th+l b-(drtyl,i i5:sfte air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direc:lr(-,b i y +il5tdfsttion (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discu/+ dn ad5ut*y ph5i 3-m*twislss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicoiadn 3dm5 *iuwy 5lp/h- +ts*tpter stable.

  Express the following angles in 9icrqle3 )t -sradians: (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 amjh 2;qogk/ i1k 3h:mnoazing coincidence. Calculate, using the information inside the Front Cover, the angular dik1hjn/m3 kqo2i:go; hameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directje aosi(/;lzjqy9yxg3 .m dt7q3 *uk.ed at the Moon, 380,000 km from Earth. The beam diverges at an angle ; uz3q(yyge9 ks mtql3x.7ojd*j ia./$\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blend (agkypqx3+egqn4f 8)er rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow tn4g(qq payg8k3f + )xeo 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 3.5 m to (bo9sq;l. y t4f*7/6uk4kv tiadxi ;l mgjl)yanother child. If the ball makes 15.0 revolutions, what is ittgit7mu)x 4;( y 6;liyj.vq/b9as d4fl *kkols diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km.9ds.e;dry/ v rt os jw,(.yyg( How many revolutions do the wheels ma teo .;gvy,yysjr( d./srw(d9ke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calc v,2n+ w1a(c3z.c c:beuidetjulate its angular velo ,ct(.1+dbaw :zcu 3ce vij2necity 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 mak)tbg2)96m.q, jxztvzhote v kqo;:b7es one complete revolution in 4.0 s (Fig. 8–38). (a) What is the v7m,hxez9 vo.btj 2;bt6)k qzo)gt :qlinear speed 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 orbit aroundh3odr57zx;nj 0 .iqs*mb kxt ( the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poiv ap)j9 fc-6c)n zea3 ur-xes+nt
(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 centrifugd6no *6 fg2i tx/qki ysh7 .sz4)vkmk-e rotate if a particle 7.0 cm from the axis of rotation is to experih7zd kxtm f* 4onq26.ik)kvsy /i-gs6ence an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 ndcm jp)e-r2 vh-/ac) rpm to 280 rpmav2d) h-mjcc)/r np -e in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius nqo (+zbx3:al $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themseuo411 jta29kshoqk8 dlves into a slow rotation toa hd oo2s1t8j4kk9 u1q 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 uniy 1 p+uc(tr0y(uijq2aformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time(u py +(1iq0ytrcj2ua?    $rev$

  An automobile engine slo 1))ozloppld spe4(t:6gcq u8ws down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses o s ,iygmm5 qf+i*ms*;:dfq :zyf flying highspeed jets in a whirling “human c; yq:5 i sf*iqdg:+m,myzsmf *entrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.59 ) 0cb1ze )r1dlqkzwq s. How far will a point on the edge of the w1)q0wr 9lke cqz d)b1zheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It /y*r6lqu*r/9cb ju-j r+49wde utcyrturns 1500 revolutions before it come*y 69wqyc rej d *rlru /u/c+t9jurb-4s 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 )faoh 3;f 04bvl+rxh buniformly from 95km/h to 45km/h The tires have a diameter of 0.8 hf)a;f3vbhox r4b+0l0 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 mh2f9 rjg0aee1ake 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diareafjhg1e 920meter 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 e,otujl wpr x/( o3oflr)86r y.ach pedal when climbing a hill. Th.),rwu(roy8j t lolxo/r6f p 3e pedals rotate 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 of a door 74 cm wide. n wc5ek )2d-9 adusnoi84zfy: What is the magnitude of the torque if e uy)d:owin ac59k n4-fs28dzthe 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 az 1wuyzb) y0+vi3 cu9edfet50 bout the axle of the wheel shown in Fig. 8–39. Assume y0)zeziv5c0u9 wdu1f yt+3 b ethat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massle 4sut-t a;;ifrss rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then rel-4fatitsu; ;r eased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine requir 4dmqld tvzfr/+ fll609ta9)8ixwf4te tightening to a torque of 38 qld/x a+w dmz)l60i9tf 8 9fvft4t4rl $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radiugjbssnwv ;jsxmqhbm):2 g ()lw7(e4oq w17n ,s 0.648 m when the axis of rotation is thr q n(bswvbx)g ;hsjsqglomn :,)7jwe m(741w2ough its center.    $kg \cdot m^2$

  Calculate the moment of inertia ota.1)ggar+ksqw l/f(f a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. gw f/rtgq(kaas .)l +1The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotatt4p;/he dder8fbx7a 3ed in a horizontal circle of radius ;tdbdhex78r3/ 4pf a e1.2 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 constantd/ )mqpw*z ymeer56j2 angular speed (Fig. 8–42). The friction for)z 52mw/ydeqpr 6m*ejce 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 po 9afsz cq bf7* 93n/)+.ynsgtpj*fstd int objects shown in Fig. 8–43 qnt+c f)s j*p *stfgafd9n.s7 3 z/yb9about
(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 o 4ize(o y:w357o9aqz wf.bwrtwo oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct ratiq+unmqwk 3 7,s8x6 ate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 360nix3,7kt6q+ qusaw8 m0 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 a uniform cylinder with a radius of 8.5xq9w(,ylztt,9 .w hcr0 cm and a mass of 0.580 kg. Calcwrw.hx9ytlcz9, q( ,t ulate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest to 3 d l;5 )tu9m m)qlthb/zss 6nc/$rev/s$ 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 and74opss kb e2)7j+5rh bf. mj4a s:uc small 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) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting 7sb p:j+2n.fs5) 7eam cju rb4ohdks4frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor ro5k : xj)cdh+jvtating at 10,300 rpm is shut off and is eventually brought uniformly to re5 j+:d)khjxv cst 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. x0ipna+k3q*ud)1l ;o ct*kbp8–45 accelerates 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 a9.s wh7xl-(bbtmky+n ction of the forearm, which rotates about t 9 x(.y+blhkmwt7nbs- he elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod,d9satov1v9u0a+ w (: e5 ;ygxvdb8war as shown in Fig. 8–46.0;15 wddb9 ovr+ 8aaxev:st(awy9vgu
(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 con x-zl)b./:9 cwibfgamsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hm h)*-0h shb1/jzxd7 sca+ngaammer from rest within four full turns (revolu+acn7hb) h- j*sxs d/0hgza1mtions) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of ine/lbm ct0t ljxk7(a0a: rtia 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 of 280 wgj2j .qwb15h bkj v.noqnod;)4k ( 3j$m \cdot N$ 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 rol1ynt/u f8wcsc vko)3lpek)b p+l/(; rls without slipping down a lane at 3.rbtw/)n;ue )ofkcsc 13l + 8/p(v plyk3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic ener heyxd m6; (x7m/ xmrnn c9u9iuew0s91gy of the Earth with respect to the Sun as the sum of two terms9eyx / 9uix1rxsd;c6m(une7 nmh90 mw,
(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. dftvz a .y83t9sjw, q:How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolutdjy3s9 :,t. tv q8fawzion per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg startpv ;tqujz ( sh32x0w8qw6:jqfs from rest and rolls without slipping down a 30.p6v j( 3hus;z82xw0tqjfqq w: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 starts / zm(q d:rwk*bto fall and its lw:d (*brmq/kzower 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 of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating o kru9hl92rg 3vwbo16r6uh b/in the end of a thin string in a circle of radius 1.10 m/6rw1ihv3 9bk br2uhuo 9 6lrg at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unif l6f., whma0avorm cylindrical grinding wheel of radius 18 cm lva0.wf6m ,ah 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 .n qu br aji:uy/.wq7/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. b7q/a wui.jq ./ un:ry8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her1gudb 9z)x 4q3jcuevh+5j*ar cl k+.r moment of inertia by a factor of about 3.5 when changing from the straight posk .+ zc)dr41j u9jgab+l *5c3ueqhv xrition to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation js cu63+s.8rptgz8w g rate from an initial rate of 1.0 rev every 2.0 s to a finaptsu38s 6.jzg wc 8+grl 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 rot*y4y .fs( v 6btvwpf/pating around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat (*w.vft6bppys4/ vfydisk 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 ig iu eb)rmh,, 1/uzy7momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the mq8og jci7f. p77u0 taEarth
(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 droppedxo6 (myyn5yfc )zgp5-s tfa:9 onto an identical disk rotating at angus z- 5x5p (ynmc6ff9 )ytayg:olar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4me)x g x:6ej9p $rev/s$ 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 merry-go-round pt jztzn+,vnd)cziw/8,tzz;3jz /e5 l latform of radius 3.0 m and moment of 8je idn, )zlcvwzztt+ ;zj/z5t/n ,z3inertia 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 angula m9jmhmx /4n-mr velocity of 0.84j/h9mm -mxmn $rad/s$ 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, lh mufu7+ks+e;yb) p( 1*swwo +p)av gcosing about half its mass in the procesgssaymp) +)v *wcu owp u7e++k;(bh 1fs, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 120 678(hckz h qdn$km/h$ 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 ,- sr1 n(tdy,kfmbe7p$ 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, initialll2wbt, xe79k fo e;vi8y at rest, that can rotate freely without friction. The moment of inerb9v2loef7;e 8k it,xw tia 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-go-round tu)(m 7g/ast-eefg+ d vorntable that is mounted on frictionless bearings and has a mome/fgv+oa t eg-ed sm(7)nt 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 hfuau6. si2dlr)j7e(+mh4jk of the rope lying on the top edge of the .2s+)m4ajhr(kj6du ufhl e7i spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same sizn3 elzz:;jq+ de always 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, treat the Moon as a particle orbitj ; qz+n:3lezzing the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratez sw)t+9-z ukpp- w qv0shx-h7 of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinde qnsu3k7w 0a4mr of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Cnq34 akw0us m7alculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, emp)18bpuf m8 /ertpb; n;gwx6ach 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 bt) pnp68 8umewbgf x1p;/rm;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 angular speed 0phg9gv nn0x5ph8 s q e0wd*:vof the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, jbd;fr4,y1ynbut 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 threeb 4f ry;y,jn1d-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for i ,op xhy-9mp9ocpb9pfaenu 9 3yz;g*6t to use energy stored in a heavy rotating flywheel. Suppose such a car has a total ma cxyo936en bpmpf9uopy *;g h 9a9,p-zss of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without 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 admlr;-kn ;q-0e eujs)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) is rolliv/-1r gmk-gagj7 ; mbvng 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 h yvl;fb8ds9 *dwhu;y,4ie5g L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is hev hy;wb4*yus ghlf 89i,dde ;5ld horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically zqp;od rb5*cv9t3p2e- d uja 3on 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 hzdeu;2 o 3jap v*5c-d9bqtr3p eight h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn wxm4+ xkhc6iccao acrrq0h c:0.+ s0k) ith a radius The forces acting on the cyr) q.0:oai r0kc6hamhc0xs4+ c+kcx c clist 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 7nxer:edq ys0 p;+x;sv8sj ,f.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 sxn :fr7 ps ;yv,dxs+;ee0q 8jto 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 thinf 4mzj:l0clws k; :xgy*05gvz rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning smgsy: 0wkg z*cvfl j4x0;l5z:kater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of(hm76pu kp 5zslm: 3eo two cylindrical plates, of mass 6(pu5ms37m l:ekzohp$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 raok9 ulwt35mtx wm )*s:dius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble *3uowl ) 9mk:tsmwtx5 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 not hibm yul -w/y4r 0 x;eyb871-uux cb3eassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly f6z k+vzs/eep (rom 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accelerzz(e+epsv6 k /ation 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