A bicycle odometer (which measures distance traveled) is attached near the wz ltej)y/w1z7n zu / yc6)x6hcheel hub and is designed for 27-i7 / 6z1nxzuw) yht ejz/6)lcycnch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at cons,yg luc9 ( y44qcux;iotant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radialyu4, c;u 9cio g4qxyl( and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body b-m.ac/f)x *72w l 9.cxdxrdkuw(al w ne described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forcgk/- vwavf3 /me? 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 handsltw6) kfh0utjj(-.tj behind your head than when your arms are stretched out in front of you? A diagram lu(j0f- h w.j ktjt)6tmay help you to answer this.

A 21-speed bicycle has seven sprockets at the rear whe +vgpow8g.s j4el 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 p jv.g4g+w so8front sprocket or a large front sprocket? Why?

Mammals that depend on be2qnnpw7j)2b oltk /-hing able to run fast have slender 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 advanqn-b/ k2o2w7h)n tljp tageous.

Why do tightrope walkers (Fig. 8–35) carry a b;fx3+kt o7s:vi*; rxr xgwc8 long, narrow beam?

If the net force on a systhbq 0uwpt++g0 wsj3swq49 2yzem is zero, is the net torque also zero? If the net torque on a system iswjhu0+g0sb42+z pq9w3qw sty zero, is the net force zero?

Two inclines have the same height but make different angles with the horizontj 2k) 4g9dds8 vvcby udzs0l2r9f86c:dbn )kxal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be grrc84fv90k)gn8y)926d vdx bj:dls kz ubcds2eater? Explain.

Two solid spheres simultaneously start zexmy.v nm xiw/r2 .9ddb/0z1a s9xin 88rx 7qrolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has th.wx 0nazv2sym 7q8xr/xbirn8 9.em1/x d9zide greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the sm4 s0;pi *0boz8a9rz qzhvk /aame mass. They start from rest at the top of an inclinei ba0m/9voazzh r k0; sqz8*p4. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular moment7,7 ezf u:aphxum are conserved. Yet most moving or rotating objects eventuafz:a ,ep77xuhlly slow down and stop. Explain.

If there were a great migration of people toward the Earth’s eqkpg1czm ;eesid42 // nuator, how would this affec dpe2sinegzmc; 14k// t the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initialz;o8+f u1 bsagr cpv l6pgk*n/(haq22 rotation when she (pg 6s; k *g/ phclnu2+va2r8 bazqof1leaves the board?

The moment of inertia of a rotating solid disk about an ax3qt 4ydo 6,us17 aufk:,d+yswjno w(gis through its center of ma d, ua,fgjou ty+d ks473(6 qos:ww1ynss 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 rotati6/jg 1*epg 5ca( ff tycmk;h0dp 2vu*ing 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? Expla1p6y/gvd kuf(*emc cf* p0ahi2 5tg;jin.

Two spheres look identical and have the same mass. However, one is holh: 3ick6*1 wtxf 1htldlow and the other is solid. Describe an experiment to detert:1*k6 tlfx3hhdw 1icmine which is which.

The angular velocity of a wheel rotating on a horizontal axle points we-o(att3 an)wkq,osfsy 3n h ;+st. In what direction is the linear velocity of a point on the to)(nnoafas y;o3 t wkqs h3t-+,p of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntabpcxr6*df6no ;le. What hxpc 6fr*;nd6o appens if you walk toward the center?

A shortstop may leap int7 (5xrk l7 juofq;b8aho 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 j lfuhr78x7; qo( bk5adirection (Fig. 8–36). Explain.

On the basis of the law of consexs 7bwzi bf cgp9:qsnzdo6q4o-8 o+) /rvation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller).o6 s:oi74q+-bwfb c ngq8s)dzz xo/p 9 Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles i0o4la;h +f:ca0o bcuc c k1(zin 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 amazin:+y 9of n:m5gx9 qfugzg coincidence. Calculate, using the information inside the Front Cover, : omg5 fznq:+y fgu9x9the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the veo(w.7vbn3p q 7 1qmjMoon, 380,000 km from Earth. The beam diverges at a (7 .vqqb71owpnve3mjn angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a m58f d(9gp8xs 2 uyhgjrate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.sfg pxyj( 29 h8mdg5u80 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. 1 n*k6to zz-xbb(cu5bIf the ball makes 15.0 revolutions, what is its diametkn6b 5(x*1b- czb touzer?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Hownb ilb9)uw 10dr 2xpqe.zl:b( many revolutions do the wheels maq. bd(lwb21x9z brnl:)pu0eike?    $rev$

  (a) A grinding wheel 0.35y3e5hrvu , az5 m in diameter rotates at 2500 rpm. Calculate its angular veloyh5e3zur ,a5vcity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig.5pfrmzv*/ kbe.1n m :c 8–38). (a) What is the linear speed 5b kfz1e v.p:*mcmr/nof 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 9timvd, p*(vearound the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poivwxp,) c9cck*i,k(lo t+hu 7jnt
(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 rotakzfkj af-c4cd-e3 sn e :m4r)9te if a particle 7.0 cm from the axis of rotation is to experience an acceleration ek3edk9mj4csn -c :r-f)afz4of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates ubw lg9 .orw*mr +/k ,qbw84rdyniformly about its center from 130 rpm to 280 rpm in 4b9w8l brwqgd/r .rm*4oy+ wk,.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 g ebpcnmtq)6 c+t 1.e5$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts ab+8juz+6g mj(;ev il+k z) jgwtoard 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 thought of ault;+vj gjg+z+)ek (8m iwz j6s 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 8;5wv-yqe.zkd-.sv: f 4bq ;odz qsqh15,000 rpm in 220 s. Through how many revolutions did it turn in tho8;.qsb qk w 5-yzhv-qfv.;sqzedd:4is time?    $rev$

  An automobile engine slows down from 4500 rpm toqyl+*9g 5o+7dvm8 wdqndqy8q 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 whirlingau8u aho0; wmx,4d y*v “human centrifuge,” which tauy4, h u*;mv a80dawoxkes 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 rr:wrz.ro td ys42st4wup)s7h j.p*d6pm in 6.5 s. How far will a point on the edge of the wheel have traveled w4yr7pwts.dro6 z *j42d ut ss.r:)phin this time?    m

  A cooling fan is turned off when it is running at 85bjr sgs w5cu23 ,(z1vt0rev/min It turns 1500 revolzstg bs cj2u,v(1w53rutions before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car makeq9 u4 ms:,-pd/vmyoktyli vyv gu-)6) 65 revolutions as the car reduces its speed uniformly frsi-q o/ yym 6:vdvvtkl9upuy,m) 4)g-om 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolmh9rov0c1n 2o 6- shkcutions as the car reduces its speed uniformly from 95km/h to 45km/1vms -c0h koh6o ncr29h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when cqd.l-1 bszp o9+ hpou9limbing a hill. The pedals rotate in a circle u.o p-qs9l9z+do p1 hbof 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. Whakfo3lm t6urg50 bt/n .t is the magnitude of the torque i m 3/fbnl0otkt.56ru gf 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 about the axle of the wheel shown in Fig. 8–39. Assu)9yuwht w trey31cv; t+e(u *vme that a 9* )(tuu ewy vcr;wh+ttey3v1friction 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 lwxk1ill40q:,vkd 6gjqr: 8 imassless 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 torql4:r,6 d x:i0k1q8qkw vlglijue on this system.

  The bolts on the cylinder head of an engine6v- xpirm 75ka rr2l0uio5 2mp require tightening to a torque of 38p257xk 2i la 6r5-urmm 0rvpoi $m \cdot N$ 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 inerj/6rcx i* bbhaf w/.b+swb4c 6tia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its cenw*ib./h6wc b+bxjarb cf4 /6ster.    $kg \cdot m^2$

  Calculate the moment of x7 pyhks:b-8i inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.s:yhkb x8ip -725 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod iao1t iavj sl( e4x;.;ws rotated in a horizontal circle of radius 1.2(e.l tosw ij x41;aav; 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 hkp 5 h, uvxr/zko8t-learx.7,p my (/potter’s wheel rotating at constant angular speed (5,yt/x/lvhrk ,rmzk(e 8x uh 7p. -apoFig. 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shoma -. mt h/r:qg*gs mqbt::;o6fw6rbvwn in Fig. 8–43 abou/t.mb m6 :mrbv;g s q:qa*thow-:6r gft
(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 oxm0 q,mb/dzylv73 u0yzy/; g3 ohtemj 0ygen 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 cug o2z8md2 md vtlyf), w(3 m.;odnx3satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Figgx 3 wm)zdm2.lc, odm3 dn;y(o2utvf8. 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 a uniform cylinder with a 2ulk )xd1 :fhs)ho xa5 w7n7pgradius of 8.50 cm and a mass of 0.580 kg. Caux7xhp ld7o5k 1)s nw2:fga )hlculate
(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 bat58+2 wgj,i okks4ydkd, 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-go- *x6cdqo hc7z co;j3 6k8:hrgeround 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 chzcdgh*co6 8j6r oxk3:q; che 7ildren (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 q0, e1advd-wsidjyph1ry*g)p/ lg )nv1- j6q rpm is shut off and is eventually brought uniformly to rest by a fri ,s-vjdyyd)hn 1gwpi-jp q1lg1) rq6 *0/avedctional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball at 7 ovsvwvkxsn3+ m7:a,6 1k z2ow$m/s^2$ 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 tpv3*)p 2yo gjghe action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8py*vj )o gg3p2–45. 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 Figdief3xf oxw*q ( /e1.m. 8–46.f1* q3x(omwx ef. /edi
(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 o n3dyipzi/z 3+)h0q zxf two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within fourso baqa7i88w : full turns (revolutions) and releases it at a a7ibqw:os8a8 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 inertia of ms*zkpb *n+9x $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine 6 )om;(opjap1 8flfuz develops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slippingo*ts7 w)ddjr.qe( l di3m:k/l down a lanejl ir)ksowmt/( 3 d7 e.qld:*d at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the suy; 7nsz;q g6zvm of two tv 6ng7z q;yz;serms,
(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 m4pli9. 3jpjgk. How much net work is required to accelerj kpjg3p. 9i4late it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and masskc qr0m42n)zy 1.80 kg starts from rest and rolls without slipping0zm2 qr4ny c)k down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It stew9b ungu7h 6 ms(1wdr97 fbq;;*hhwjarts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energybu;bswrm9 *efn;h hw 7qug7 hw9j61 d(.]    $m/s$

  What is the angular momentum of a 0.210-khlgxz57 p(ao sf:,upve *9ru3g ball rotating on the end of a thin string in a circle of radius 1.10 m at anz5lg ,7osp:ux(rp*9 ve fuha3 angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical g )(cvjwjoq9 t8c( t/a(liy*u u3 5 ay,zti/uvgrinding wheel of radius 18 cm wh/3(cz5/jt( 98,igjvqcy)tiu oa(wu yvu t *alen 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 that is rotatinx4 jzwuy-hs+ r(xen- 9g at a rate of e+-9xzj4 -nru y(xwhs 1.3rev/s If he 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) can reduce her 36 tiq-x5i 1,l8l wbcik fmxb:moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuc ibqk l:-f,5t bm36xiwil18c xk position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotaticm,96kub z gz j54(qcmq-aw:oon rate from an initial rate of 1.0 rev every 2.0 s tou5 cc,zk:9g4 6 z-o(mq mwjbaq a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis th l5a;-l7chtgid l7o ;0pzg n/jrough 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 wheel. What is the frequencygpnl;/ c ztilj7l o-75a0;d hg of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentumdf yc 59-ujam; 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 angularg ut+c .w3gk8i momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto anc:d(m ;6)t aldxiss:s x 4fy4v identical disk rotating at ang:s4yx4(cfls6v m :a si;d dtx)ular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns p7pdyaei4):p *za8guwl ov 7/ .kvim4at 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 1rw3 hskb/u, qat the center of a rotating merry-go-round platfor bqurhs 3/,k1wm of radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity c8xzjzd; j r 9r.;n azeo8/n4bof 0./dr bo8n; zej ca4r8n.9;zzxj8 $rad/s$ 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 dwsccfdx)z)sv 5 x;i*0k l tye9:arf, 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 e5 v9idl))c:yks s0xt fz;cx* 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): gtkbhh5e ncq2b-k5 winds 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 u7j1m5l/zm1 0vfwc o8 qkr6pq$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that c9 drrqia)j*i( 44fozfan rotate freely without friction. The moment ofz9ji4 foqaf rr4i* (d) inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diamete9l uwu +4b6zas;8dryt k(icp2r merry-go-round turntable that is mounted on friction dy wp zl2cbk(t9 s4;ua+6uri8less 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 o. ,-b ,w0o,ietg pr4py t*idsjn 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 without slipping. What length of rope unwinds from the spoo.4 ,ior*,spy- e,pt0 jdigtbwl? How far does the spool’s center of mass move?

  The Moon orbits the Earth such t0: p3hj m1t8;e+x,df4oezyn e5 l0iktd kxld 6hat the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about0kjnzfdy5+o4i dplek, 6h;lm 1e t:x8e0x t3d its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate offz l;g;tv/ )(u o0xi 4glgofeu1q1 4hg 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 radik82*z yp4nf zmi1 u1l;c0mtsqus 0.20 m acquires a rotational rate of fro kz1iuznqcs*p0 ;lfmmty2 4 18m rest over a 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 cylindrical disks, each4(nym6aid 4 f +*jqqwa 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 whma*fwnq(64 jyd+a qi4 en 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 of the rear wheel 7/tjf *4j 0xdo6lhh (h,jzmjm($\omega_R$) 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 oftnpyp/r ywlk6: )u9 2z our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revp pwk 9 r2ny6y:lu/tz)olution 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 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 0 msa:y4zhlx :a low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, us y:s0x:zalhm 4es 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 d721zcyy xi4h 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 hoahl(9w so yb+bvr6a+*x. tj;urizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignor,.: armx 6m:sqzr + teiy;yc61cyhh; qe friction) about a hinge attached to a wall, as in Fig. 8–6ryesc.h +z1hmr:;c ;y ,:ati q6 qxmy54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is simhu3mo* jifcwprf9x6;4 26itanding vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a stepum6 pi3wmh;ij i rf46xo*cf92 against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2ci1h0-irr8gww 2nl+s m/s on a flat road is making a turn with a radius The forces acting o sl1r0rigw 2 8n-cwhi+n the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock i-y ,ktezdjb llvfr u:42zs-(,nto 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 th,jl kuz2 :4y-t,rdlz -(f bseve torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thio622 ylv qqml+n rods, making reasonable estimates for the dimensions. Then calculate2qm6l+ly2 qov the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch ass5slbdry lm(.0embly which consists of two cylindrical plates, of ma0lr5(.b lmdysss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 t56d a 8em* lu)hjcxj+orack of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the traexh)5*+ujo 86ad jc mlck? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bunnwcsq2n l48zd)y nw9--do ;dxq ri.4t do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduchx 0yk :tgqz:/es its speed uniformly from 90km/h to 60km/h Thx/ gz0hq :y:tke 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