A bicycle odometer (which measures distance traveled) is attached neara6g ,- p iq7uow5x8;dmamgg0m the wheel hub and is designed f ix85 g am,6p7ad;wo0gmmqg-uor 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doe u 5pmbxzgg),6s a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/)5xg6 pg,zmbu 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 va,/ h8 bs,bhw)ivfe ;veiv(l9 klue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larg3e3l( wr)ut9s4fus 0:vgn i xler 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 sit-up with your ) f.hyn0m.ik; c1;y5xg pxwp khands behind your head than when your arms 5mk;;ciy.)x1 p pxhg kw. ynf0are stretched out in 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 mrzbw wo5 )7tc1bn4*athe 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 7 b*zb)c1tn4rwa5wompedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs w,h +y8 c gy4h02sjrqgpith flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this dis8r+4 pshcgh ,y2 jqy0gtribution of mass is advantageous.

Why do tightrope walke (mzgi sty3+2xo5u )awrs (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torqueb 4jg0f07y1ezv b- pxm also zero? If the net torque on a system is zerobgpmz4y17 -j0b v0 efx, is the net force zero?

Two inclines have th83gde2m uy1ld04bk pke same height but make different angles with the horizontal. The same steel bakm2l pd4d081g eyubk3ll is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simul9 (3)sew,d ebi 1to3 thp/rvoptaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the it, psio91pt r/3we dvheb)o(3ncline 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 azyi+x w46wrna+:y7/f klaj 6k nd the same mass. They start from rest at the4yz7k +/lanyx6:wr a6wk f+ij top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular mo*wtmxz)/v5t 8ss m uae/bkt+2ip+)oe mentum are conserved. Yet most moving or rotating objects/*+)+w)a82e i/5otzpsku tbvxe mmst eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’sl5( yid* z2jgd equator, how would this affe(d d5y*ijz2 lgct the length of the day?

Can the diver of Fig. 8–2: tkw ;o n4j( 3wwhzy(;ukxs:e9 do a somersault without having any initial rotation when she leaves t:( wstoxy;nw; ue: 3k4z(wjhkhe board?

The moment of inertia of a roteb/vrqk, o p;0ating solid disk about an axis through its center of mass isvro; / qe0k,pb $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in eac x1bjr dt1*sg;h outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.b;gxd st *1jr1

Two spheres look identicaud3k 4qn rb.ta xkb:7)l and have the same mass. However, one is hollow and the other is solid. Describe an experiment )3.kxdrua: bkt qb47 nto determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. y; c*me:x7an q,f suc;In what direction is the linear velocity ofc;:e,nas uy7*x mcq f; a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the enp z50 ,/ab +qx8pgzprdge of a large freely rotating turntable. What happens if a /prn gpzx5q+z0b,8pyou walk toward the center?

A shortstop may leap into the air to catch a ball and qd ik1(6rvax,9m zw o,-tffw.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 (Fig.fri1 xd.wf9 v o,(qatz k,-w6m 8–36). Explain.

On the basis of the law of conservation of angular momentum, discusy)1kodh*s cb 4iz s(/p f9v,6fcmgz-gs why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the secon46 gh mp)fgbs1v, *dfz/ iz-y 9cc(sokd propeller can operate to keep the helicopter stable.

  Express the following angle lz( v iv80smqjr,jmb2/7soy5 s 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 coincidence. Calcula n:jr. v, gpd1vmqr9g6zara 2: 1)p9hjg3qz ente, using the information inside the Front Cover, the angular diameters (in d9.n,a1:m13z vepv 2hg9 g p):jagj6rn qzrr qradians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 k dn/t3 ye2t2hhm from Earth. The beam diverges at ant 22dn/t hye3h 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 ratbytn swej7,b(6p +k**c: y6fhxfii:pe of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades s:j*t x,f67f kpb6h (y*wi eyc+pb :inslow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m tc- o8hqxfc;f. o another child. If the ball makes 15.0 revolutions, what is c. qx;-8 ofchfits diameter?    m

  A bicycle with tires 68 cm in diameteke vq mw92:sy,r travels 8.0 km. How many revolutions do thvsy :m9w2qk,ee wheels make?    $rev$

  (a) A grinding wheel 0.35 0r(6nv4 mekid -d/nnr i+tl7a m in diameter rotates at 2500 rpm. Calculate its angular v0v6/47ekn m t -rdr( ni+aldinelocity 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.)5 wht1 qlu gk9lf*65gexb 3dk 8–38). (a) What is the linear spe gh9be*3ukfq5t6 5w xlg)d k1led 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 vele3tg4x4 bk+dl ocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spfapa1 rbr604+ iv z2ueeed of 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 je,m(rg ;sz2 vparticle 7.0 cm from the axis of rotation is to experiezv(sg, 2mejr;nce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from -fip v10pwrg.130 rpm to 280 rpm in 4.0 sv-.girfp 1 0pw. 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 radiuyq h11hqxa n* :6)i6p/a rajrcs $R_1$ 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 Mov-pn6mbi6 t3qg tj(c(;k8ybqo39m )n3 gs l beon, astronauts 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 every minute during a 12-min time interval. The s)6bmye3 on;vtbq 6-9c mktj(inbg8pqsgl3 (3pacecraft 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 t ,pjf37vic* owo 15,000 rpm in 220 s. Through howifv powc7*j 3, many revolutions did it turn in this time?    $rev$

  An automobile engine slows dozjr8-.l :nbtxwn 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 of flying highspeej4 ) r:m1*va(pxktvz zd jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20zpt : x z1)jm(v*4akrv 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 6.5:-l a*ec ,yoxp s. How far will a point on the edge of the wheel have traveled in this time? ep,* y:cxo-a l    m

  A cooling fan is turned off when it is running r zr/4pie0i*q at 850rev/min It turns 1500 revolutions before it irz/r0ie q 4p*comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniform qkszp-hb*: x8ly from 95km/h to 45kmxhspkq-bz8 :*/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 revolutions 4d *sd1mmaf+kas the car reduces its speed uniformly from*k dmf41 sdam+ 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

  A 55-kg person riding a bike puts all heriz4;) q,vq efy weight on each pedal when climbing a hill. The pedals rotate in a circ4q), ivzqef y;le 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 doolyu1p20ttn r j9g;sn .g1apn(r 74 cm wide. What is the magni2syn(lja1n9tp n1gt g r0pu;.tude of the torque if 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. 8–39r /kam d0 l)qqht/fgw0 :hj,k.. Assume tha d0arjh/,.q:g)/kwf qml0k t ht 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 massless rod wh8uvyw5x a s.o0ich pivots as shown in Fig. 8–40. Initially the ro5uavsw0 o.y 8xd is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torqr8vu t aflbp0 ay(,-9hue of 38 ,y a-l8 r0pbvfu9at(h$m \cdot N$ 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)u u6zmyf93lwf): u;k hkd2 fy sphere of radius 0.648 m when the axis of rotation is throu2fdkf6 wyy z : u;)hum3lf)k9ugh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66 dld0g gb99ct-j/qh4el f:n 6t.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignor ltcql 9jb-:0ndh9gdtef /g6 4ed (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light ww;pxix*7fi1 rod is rotated in a horizontal circle of radius 1. wwix17f; ip*x2 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 wsvopj2, ;.mbh heel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N tota,;h2 bvmospj.l.
(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 shown in: (vly)/:c7hkcoppkg2u/r zeyv1ek1 Fig. 8–43 abylryp(2v1e oh:// kk )ue vkc c:gzp17out
(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 oxyge ,ozyu( lwuzlz;5*h9 jpoc( 9vn 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 cylim-p6 eiiy5s ,indrical satellite spinning at the correct rate, engine 5imps6e-i,i yers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder withxh8j5f8bp6r b5pq xqzdo.k+; a radius of 8.50 cm and a mass of 0.58rh. 85f5ob q6dxjk bxq;8+pzp0 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from z7 )a) xzgm2zmrest 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 merht g59ek6braqejb 531i z+tl6jg f 61rry-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 3j j1h+rt 9a5bgz66ltei5b6efgr 1 kqa uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brought+lgu):; dlfih uniformly to rest by a frictional torque of 1.ui) l:f ldhg;+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 avmkjg h/5w3; l 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 n yyax* -m;g8eby the action of the forearm, which rotates about the elbow joint under the action of the tr-x yn;ymag *8eiceps 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 cn- eqgxa ki 6h/8++lrd ip,*noonsidered a long thin rod, as shown in Fig. 8–46*-,kg8i6 iqo+dah +p/lxnn er.
(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 consistf/vw6qpf rx5- s 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 accelehd ralws;w.gx. q- /h8rates the hammer from rest within four full turns (revolutions) andawlqrxwh; ..8-shd /g 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 inertia 3 3lricw7c i 9cou ijp7gt7 p;c9s3v6mof $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280+3m1w h+0bcep.uwqlalwzx*, $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radiusi2m2zurdm omnk4y1-j( x*o g- 9.0 cm rolls without slipping down a lane at 3(nmz*2 gm x4jom o2-u1rd yik-.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic en9sx ob pm+(g*aergy of the Earth with respect to the Sun as the sum of two ter9 xg*sbao+pm( ms,
(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 aupucdma2 w(u*;u jd+ 5 mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate idc;u u5 u(+m2aw*jpudt 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 mass 1.8p7ckcim k0(q lp bi6mw7,q ,c50 kg starts from rest and rolls without slipping down a 30.0 76cm iqkwp(,0i,pc7kqblmc 5 $^{\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 ando-/gwir 6 inwy:va6 ;d its lower end does not slip. What will be thayd -i6iwwn/ ; govr6:e 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-tqymgg rl k.qop0s(4,ep:v8s/+h: g( wksho a 0.210-kg ball rotating on the end of a thin string in a circle of radius s/ps (klyq.t, s :khe 0 4-:qgp8vhog(wgrom +1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylifn5gv3j3 n nyl2 to3ck859g plndrical grinding wheel of radius 18 cm when rotating at 150n3 c23jo95y gnt 3 l8pv5flgnk0 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 rotating at a rwd+/ -qeo*a q;9dwfcjate of 1.3rev/s If he raises his arms to a horizonjaf woc;ddq- * 9+weq/tal 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 hervsm/l/t g2 uf) 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 /)g sv/ uftlm2tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her sw)ygi.k-nqywr62 5e y bzs:wyoy3 n:.pin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate onei:y s53 gn w6 b)yzy ..wyrk-y2w:oqf 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 f0c:v8 gg g ir+n:nms:a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rosgvg0 :+ig:f8:rcnmn tating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angula;efrgstuvd9. 7 o5g r)r momentum 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 oy5 -g .rwuho:wk,i4u bf 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 inerti xcs 5gk;5xu9ba I is dropped onto an identical disk rg cu5xx9sb;k5 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 2r 9e0bqvb e1s(.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 y6-pi +qgd fs(merry-go-round platform of radiugdqi (psf 6+y-s 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 iwa7j3hyjq3 7with an angular velocity of 0whi7y3jj q a37.8 $rad/s$ 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, losin )x3zce mt.);a gruy1t5ymu4t g about half its mass in the process, and winding up with a radius 1.0% of its existing radi1tu;.c u3ma5 rtxt y4mz)e)gyus. 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 h:72 e.edjaxz u2ou k.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 tq :z 2d/,,r jvji3z4xr bn:xd$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can r fnw9w211hkewr,cdfx3dr5e4c edior / 3//nmotate freely without friction. The/rwxedd 2wowrc m/13 /ir,5h ceen d193fk4nf 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 person stands aty7 fq + efbjr:ay,ue,c+:)*euhle m- q the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and u7e:flh*q u,je,e m)- yce :ry+ba+ fqhas a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the;vg5hxbv l ;d6ivmj :5 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 rop mgjv 5l6d:x;vibvh5; e unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwapfx blrf+vb+ 1me3i ,)ys faces the Earth. Determine the rat)x+f1mp 3elf+ ir,vbb io 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 orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates frgntqo8j6 -3 n a57ocdlom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of r/c*.rj bucd95 +vgmoo,f bsd/adius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleratio b /jo5.fu/,vd m9bcds gc*r+on. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solir8+tix 0q+ yuvd cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long striqu xv+ry it8+0ng, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angula cyu:hh hs03:atp hllcb -7o19r 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 times9t9)jjd+om m aa,/q+l:pmy ez as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, whattdmlm/9zp + :om j q,aya9e)+j 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 it to use energy sto xv px)1x(yq6zred in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, qxx y 16()pzvxuses 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 ap)igq , :i7kizn $H_2O$ 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 l c:4o1dnt(2kneb /wfsurface 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 ignorexwjuaip )t6b+ 63tb-t friction) abo iu6p+ja6x3tttbb- )wut a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing v2m6lwg uh:9avertically on the floor, and we want to exert a horizontal force 9: m2wlgha6 uvF at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road ishj5*qim(al u vozc/4mf**cs+ making a turn with a radius The forces acting on the cyclist and cycle ajm(h qm *+ilc*sa5u/ 4vzof* cre the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m rg5+n;vafi3 mzx1 9ne and begins whirling the rock in a nearly horizontal circle above his head, accelerx+3nv fa;e1 i9zgm5rnating 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 body as a solid cylinder and +xnouug h- v/c w9kr,hg+2k w:her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the 2xhg:r u ,9g uww +ocv/kkh-n+angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical platw9sb o4le j+4pes, of ma9pwb +sl 44jeoss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ther1: +8wdjvdpg)z .;9xmmk tpo looped rough track 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 track? Assu; w+o:k9mpz1dr .dv ) pxg8tmjme $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do noj)71uub9p kk oqb wn*o-y 6nh+5vp +-pi5zbnqt assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the ce4a2xovn5pw + ar reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of wxona+4 e5 p2v0.90 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