A bicycle odometer (which measures distance xgvm/01 wynfan(p+ 8l traveled) is attached near the wheel hub and is desial(wf x+n/8gyv0 p1mn gned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on-gwho(lupqw * -wvxbk3 8u19y the rim have radial and/or tangential 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 l q luy*vk-1 39whp8wg(u bwx-oinear acceleration change?

Could a nonrigid body be de58v8zf nk gsds)o j+ij 3z47bo:tpjn/scribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forceamf2 v o/-x)+ ( ktdrzvk2qwa:? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind yonu(w mxfrwa2cr 7v9 70ur head than when your a ua(rw 9xn2 v0mcwf7r7rms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprocket( p*kd2d+q e y 1dwtukts6p;tpw5um2)s at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a lpdk5w2*smuedq) typ2p wt d1; 6k t+u(arge front sprocket? Why?

Mammals that depend on being able to run fast have slender lopr4tl;o vje:)s 8wnxkj+:g wjs.5o 2 pwer legs with flesh and muscle concentrated high, close 5kjg)+so:4xl wvt p;r8 e.jwo2ps nj:to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8– tux;a1*ctf3vi* 6yd9:glk by35) carry a long, narrow beam?

If the net force on a system is zero )xzgmxe -dzu/uq a2ez*c-.8j, is the net torque also zero? If the net torque on a system is zero, is the net force z j*2- ./mdxez)u8gazcuz -exq ero?

Two inclines have the same height but make differe*u)b;l f5wu ownt 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 greater? Explaib)luuw f;o5w*n.

Two solid spheres simultaneously start rollin. vf8(b//zoji wvq)en62dqhv g (from rest) down an incline. One sphere has twice the radius and twice the mass(6 ih jqzd.w/ve)8ofnb 2 vv/q 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 sammon7 t7f (gs05pwrrj7e mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at them7fj5 wn7s o(rtgp7r 0 bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular mzc d(3u62qyrhayig /j 7c6(i homentum are conserved. Yet most moving or rotating oh6 cic aj6uyd3 qgh7/zi r2((ybjects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, hsb/zdt c/49u how would this aff/4cz ub 9h/tsdect the length of the day?

Can the diver of Fig. 8–29 do a somersault without havj2i14w5ptdy eing any initial rotation when she leaveeijy1w24 5 pdts the board?

The moment of inertia of a rotating solid disk about a9a6 tdyf+d:ut;,x2lspb v 7r zn axis through its center of x2pbuzty,:s+av ;tl97 r 6ddfmass 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 j g(r4qn5 c36.cl-)munspotwxq t w,5 holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain(q 6 j-5.mlc4wq, s35g puwtnxnt)cor.

Two spheres look identical and have theygyl8 m)p33muice2t / same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is whic ym 8tie/pl3 g2u)ym3ch.

The angular velocity of a wheel rotating on a hori,,f d-bbk11z1fc6jaf(tqq zv zontal 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, descbfqvdz,z,16tjff- (qkc 11baribe 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 rotating turnta6 qs)er3utzr9xpf8p 1v,u-y s ble. What happens if you walk towp 9qeu) x63pstvf ,r8usryz1 -ard the center?

A shortstop may leap ip gqmq8j *q5pj2.b7yu 7w jv8xnto the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction j myq2 quwp8jvgj*7q .pb578x(Fig. 8–36). Explain.

On the basis of the law of conservation of anhe-: kkp5 q1yogular momentum, discuss why a helicopter must have more than one rotor (or propeller). Dpey-o k :qh5k1iscuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angn+dwffhp/ 8go ,lw3;b les in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coina*8u bnm00a9hu u f;fucidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen ; fnh0u0a fma*b uu8u9on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moonkh /v,(it(d *kd ), -rpebnjpu, 380,000 km from Earth. The beam diverges at an ang(d-uh n pd j)pbt(r/kekvi ,,*le $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the moto56jbve,i+,ww ipm-0bwz s d r(r is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleratspi+(rj5zv w-0w, , dbwm6bieion as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level4 k*bly6aj*-/9ad kt rsmquo- (sf2ng floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its dias ( anqumf6gj k4 dtkr/-*o29-*lsb yameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do/az +aggh8 4*zut rh 0mo2q* 6at8u,oc8g qqmb the wheelsgbrtuzao* +h qh/gqgc8 *20a6,z t ma848oqmu make?    $rev$

  (a) A grinding wheel 0.35 m i8nb21hmcxa fb 1n(e5 rn diameter rotates at 2500 rpm. Calculate its angular f2 becmbnhxa r(5 1n81velocity 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 compa5jjzop*ql,4(r wz p(lete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a childp,4(* wj zaq(rl5 pzoj 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 around theer0a0icqe e-n8yehhq r 64)yf*j*d-e vjy e(2 Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed on(u e*ph-,s ot.mp/pz f 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 axis 2gkjkal(jhz:o ;kb.f q z0v(18gaf h+of rotation is to experience an hk8ao qb(+jklh0 1(2zvk : .gfjfgaz ;acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center cz *(h(xi a ;nh .m d*idozu0w1hqi9:dfrom 130 rpm to 280 rpm in 4.0hc 0(:u dq;d(zh*nza .*9mdoiii hwx1 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 radiuskntc:-mgsdl g/+p1 -m $R_1$ 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+djdawm; qyv u-6r2-s aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution eve q-;2m+dwv6sdyaju- rry 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 1d isa8ztb: 9p95,000 rpm in 220 s. Through how many revolutions 9aptz:d9i 8bsdid it turn in this time?    $rev$

  An automobile engine slows down from .,* b okq73gw *stfk2gew5jxp4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying hiq1c.dlv 5c98 cpfj0dx ghspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn th98pld d5cx cfv cjq.01rough 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 diam*9vbol( sovhq n -m*:weter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on :*-(s*hm 9wvblqo v nothe edge of the wheel have traveled in this time?    m

  A cooling fan is turne; q zd7ex+ z6vvh+a4o)do*g pld off when it is running at 850rev/min It turns 1500 revolutions before it com ah;dezv)4oo d 7*+qg+zv6l xpes 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 dj :xa:m3 u .goa5r- 63fmrg2-hq exiyrevolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a dirm5 -mq-a gha3y3r.uj 6xe oix2g:df:ameter 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 spea/3*1wc1c e,v jyeb eded uniformly from 95km/h to 45km/h The tires have a / e wj v1c*,3aye1cbdediameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal whennck q),h+ sz(i climbing a hill. The pedals rotate in a circle of radius ,zn ikshq)c(+ 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 st bh/nqy n,5 f11h:aqj/xaw-74 cm wide. What is the magnitude of the torque ifhs1hx/f,5-b a 1jnwn/ qqay:t the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque abousm q8mpgkvra*+50 .h bt the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of r0 qbs5m. gp*8v mk+ha0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, ase// c;1i/7w uhfz9 ks c.xtidre attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net to.s zdkh/ew1 7;/ifc/ uxisc9trque on this system.

  The bolts on the cylinder head of an en * v+e6mqw5:.jvnfzqkgine require tightening to a torque of 38 6vn .jk:+ qvwe*fqz5m $m \cdot N$ 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 inerhxcv38irxjz9t2uxmc:t l* 7 * tia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is thro*9t x *:zx8mi3v c7rujcthl2x ugh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Th senfcr/1apv5i -lw6/ e rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored 6 /nfl/cv i1pers5aw-(why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizontaliy s i2hsztsd+(*w9 7e circle of radius 1.2 m. Calcul+zs(t* ihsd792syi we ate
(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 ang6iq/2 f*b9 m:jj gvetktjz2 q0ular speed (Fig. 8–42). The friction force between h:m2e2 bit 6kqq*zgf/tj0j v9jer 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 o d- c6a(wiyfsd(zy8(u bjects shown in Fig. 8–438u(zw ( 6s(cfdya-idy 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 lszbv7- o(a:ubc ):vue-7x rcatoms 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 sac)k+ 1qtb-d2se8m qeu tellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the qud2q-ke8 ebm+c1ts) 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 cyh l:l jb,z+2vuuq)n 5y:8klxrlinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculatkn5: lhyl +2q8 blju):v ,rxuze
(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, a(5wn bq mj.lv :o(nmb urm8*,fccelerating 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;l:ai* 9f96 2q uq2obl n dyb+xg+fkmh able to accelerate it from rest to a frequency of 15 rpm in y 6:+fi2lmbqgbxhnu 9*k+oq29fl;da 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 shutyx5c1/5z l dglsd3:x2rxmg r( off and is eventually brought uniflrz (xr xl55d2m1gx/yd3:c sg ormly to rest 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 acceco:ph 9 b g clsu:vgk7,x0b6/elerates 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 solell4z7((+-eukmli c t ap, kw ofa-2vyu.y by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The b. 7-v atekmc(fak2uzl4 lo(p,wui-+y all 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 consid7pt 8oug9t4n hf9a- xjered a long thin rod, as shown in Fig. 8–46. htp79g-8af4 xtnj9uo
(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 two masses, rwl.7 tsc.+gi $m_1$ 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 )leosc0wa5 k023gq zxthe hammer from rest within four full turns (revolutions) and releases it at akso xca2)l3g z0qw 05e 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 iner1pg) tsd*qho8 tia 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 2g ujf h:n)3 n:8ghn;e*bgu+hv80 $m \cdot N$ 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 wo bja8-4yvl 0yithout slipping down a lane 4y-abjvo08ly at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to t5m or:i;:;xw)nvy2o6 mbal d ahe Sun as the sum of tw :;mavo62d: brly nwo5m xa);io 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 rawkr *a)g08nsp hy8jt,dius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 8jw0tg ,kasr )p8*nhy1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm an74k5dfmg ojk) d mass 1.80 kg starts from rest and rolls without slipping djmf4 7o )kgdk5own a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and ityh/y6hw(at( e s lower end doeh/t( e(ahyw6 ys 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 fu,p -etqp,e) 9ew9k0 u4dzvf0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10,dup9vf- )t q,uzk 4fe0e p9we.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg u3htqocx 7v zw* y73c,npk/j5y niform cylindrical grinding wheel of radius 15c3k7hpzx c/yw, o73 jy*nqvt8 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, on a platform yl31(/w 2mj/ a5a+hq -umxrccy yeq/dthat is rotating at a rate of 1.3rev/s If hj3c w 21 (5x /a/l-yyuec+hq /mmdyarqe raises his arms to a horizontal 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 in Fig. 8–29-jyabfv d7t:pw.;/9ubvw9 p i) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makesb9a;p: twfi-/y vdbvwu.p97 j 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 ro b,7) mexma;9j*mvzin tation rate from an initial rate of 1.0 rev every 2.0 s to a final rat7 n;eivj9mxa m,z*b m)e 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 axis through its center at ww em7 b;hd;;na frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter7;;wh; nbme wd 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 y6(q2il9r +d2w fne armomentum 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 Ea.zdl(r*n(u *f pyucg+h6q 8rorth
(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 o4:zgobo9gya;0bv cd (nto an identical disk rd ob4g9v; ay: go0zbc(otating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2wpjf( : z;19l (ucw(ebnch;be.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 merr2upac;f 5n c+zy-go-round platform of radius 3.0 m and moment of inertia+cp5can2;fu z 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 veloci;/e5ca(4xqu xiiv eb,ty of 0.8 4 (;ebqcxxi/aev,5 iu $rad/s$ 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 eventualo3trq0zd3b *6q fo*on ly collapses into a white dwarf, losing about half its mass in the process, and winding up with a rad6f3o o3b*q 0dr no*tqzius 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 1qyoex u-)m0oj6 ,l1 fj20 $km/h$ 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 i 3z0nes/u.d ltq7c7i $ 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, 6h/kpe-oqw)/ 6rh9oja 6lrik that can rotate freely without fri) /hhlkr9a eri6qkj6/o- wp o6ction. The moment 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 pers 8gqb7ud m2qmi*lj y62on stands at the edge of a 6.5-m diameter merry-go-round turntable *mlbg27i2muy6 qq8j dthat 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 7*gw -- 5g.eauosf i;sh (jhxjl9x*sy on the ground 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 wi*glsofjh.x -asy iw;j7 s5g*xu -h(9ethout 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 p ia26ildmd -;side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentumi-62a i mlpdd;. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest atwsm(,y9( qr v z ueh-bqh4y2+m a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m a+3 q 7- wm8vroroqssv ushbgyb 1:d7i:30pei0cquires a rotational rate of from rest over a0qi udw-vb18sqh7p:3eo iv 7g30brssr +mo y: 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylind8ao* j -,ebhiw0( zyhrrical 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-awbi*( r08h -zjeo ,yhm-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 m1iozm2 9q ti-u+g5ljthe angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 uft9p79ws /xftimes as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational f/997xuf pt swcollapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for 3s70ic2v 2*rufehkqn it to use energy stored in a heavy rotatingf7c2v *q si02k3ernhu flywheel. Suppose such a car has a total mass 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 illustraypt)0uh 1w, 6 oi3mc5g gdy-xhtes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface ab)ti,rf70vy8oj5z8t oy)g c pt 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.5( qzdof y/sx7e., 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 thoq/ds(7x 5zfye 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 vertically8n *ekq9 e;i7 gtm(zk1o6 pj3zeyo.k v on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–5pm*eyegij .98e6q7kno tv(o;z13 kz k5). The step has height h, where h

  A bicyclist traveling withz qj6*vdm87v j*u ejn9zu7k c( speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the86c z*7qmjjzd 9ev7uk (*un jv 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 0.5ikbhz2.i /rx.0-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizox. kz bri2h.i/ntal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s zz. dht1,(pb.e7nfww body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular f pzn d1etz..(7whwb,speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly, h:6hjnw6f2 j/ ktvzj which consists of two cylindrical plates, of mjfzt :w k6n 6vjh,/2jhass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alontp6qgyt9rda xk3r/9,ty,ei6ps2fr . g the looped rough track of Fig. 8–58. What is the minimum value of the vertical hefs,/t.ap2rd xr3get 9 ytp, qki6yr6 9ight h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not hmn3 * 3w8dr8je-l xb(djq *vyassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 rel/g-bxaei fob,aonl+ 7 k /c+t7e/2lxvolutions 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?le+ /a alo+ / n,gbl/i2xfc-x7ok7tbe $\approx$    $rad/s^2$(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