A bicycle odometer (which measures z wr:bujq(l; 9distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if yozqbjlu wr:9( ;u use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constantr d 6d+ l5hkbh4alc/5uk6pga)9mh uc( angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases unifor+5 m6d9 hal)gcclp5/h 6(ua rbhkdk4 umly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be descri sgo(+o;gdy h5bed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque thanz ainj4,1 t zngdb.(4 9s-wlic a larger 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 hands behv tf5w.n4c5b iind your head than when your a4 5cf5nwbivt. rms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the reh 3*9oqxt 4bpzar wheel and three at the pedal cranks. In which gear is itho 43q* zp9btx 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 large front sprocket? Why?

Mammals that depend on being able to run fcas)ij13zk31 7;q1c ;hasvl yc nwl-oast 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 advantageousc13ai7 lwl3ha;vnj)cks1-s ;qy1oz c.

Why do tightrope walkers (Figkg mi hw1tot.k;th*(: . 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the nev6t,zk+gbhz)wy) k c:t torque also zero? If the net torque on a system is zero, is the n)z6vz,:bktcghkyw+ ) et force zero?

Two inclines have the same height but1t85k b /iradx make different angles with the horizontal. The same steel ball is rolled down each incline. On which inclr /ai85t1kbd xine will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start roll9w .8et 7dj myau,-urring (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom wd e7ua.8r yj,-m9utr 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 )s/1i4bvqb p jhave the same radius and the same mass. They start from jb/vp s)41 biqrest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and ad-)g6 fkghv* qv93vjq s 8uzj2ngular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop vv q3)d*hq2suvj9 fj8 -kg6zg. Explain.

If there were a great migration of people toward the Earth’s equator, how wouwkx/6 bf;3 .bjod lexxhbj;50 ld this afxd b;jo /5;ewhl x.6xbfjk0b3fect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initiasv,cp2c .5 wap 9deg8tl rotation when she leavestd g925 pweac.,8 vpcs the board?

The moment of inertia of a rotating solid disk about)hnbs(s8hm; q) l4dv(+ ge tgzbq4*ut an axis through its center of mas*gb( h+)4g 8 qvzeq ; (4mu)btnshdltss is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a +;d1rc(fy mjpp /(ebx2-kg mass in each outstretched hand. If you suddenly drop the masses, wi/ ryc; dp(xj1 f(p+ebmll your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one8mrnkdp, p3+h is hollow and the omnp,r 8hkp+ 3dther is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on 4kapg)k7 z g/y a,;sira horizontal axle points west. In what directionkr apzgi74y) /;s,kg a is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. What 1; l unxj;14tsm0zjl1 uhc+d+e9hqawhappens if you waljlzsj;+1hen h0;u1mq x+twdc1al4 9u k toward the center?

A shortstop may leap into the air to catch a ball afj(di 4ahk c73nd throw it quickly. As he throws the ball, the upper part of hisadi(jhk3c74f body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conseri4qggqg 29n ce:++)htohonx8vation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicogge+o8: o+hni2xqnqgtc) 4 h 9pter stable.

  Express the following angles in radians: 5+ze)e)m-x l:y 5p oxt zog4gy(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 bec /f(zjemq.u t x21ynd9d j*5rrause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen o2dj./ unt *dx5j9qf(1ymer rzn Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon,uv6 z jwz;m 61lz.rn. ibxl0y, 380,000 km from Earth. The beam diverges at an anlmzz.nv1;6xj z6 i0b. luw y,rgle $\theta$ (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 motor is turned p8ra1)kl-v;i gjwr 2e--b wan9qp. vroff during operation, the blades slow to res b9pa2r-k- v -l)pjqg8nrve.aw1wr ;it in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If 8p2jie + lwa;n 7t(rjbthe ball makes 15.0 rtl; +7(j iwrb2japne8 evolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels py de2yl*.0xsky:5 cc8.0 km. How many revolutions do th5c2py k y:.edyx*cls0e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angulhce ;sudo*zah:ore; 4 ct0p 6)ar6u z *hccteoh:a)0p ; ;s4rode velocity 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 revolutil4 pwkqm +bj3my)2c 2eon in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child me3qykm4b2lp + 2wj)c 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 y du2zld3e+xfi g g- l3-i sr5td(j6l4the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speeu8i 3)hac:t w4+sormt. yzma r)vu/a: d 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 particle 7.0 cm from the axisnft 5uo: a++nn*yal+y of rotation is to experience an acceleration of 100,0t5 y+nn :+yn+ au*lfao00 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpn6y3cg90arfkm3 g6qos wzw9a3u 7 up( ob4si;m to 280 rpm iangsaipfk 9w 3u3uo 0o q (g4r6 ws69zy;b73cmn 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiu*;0 ;ij gmbahma+nm m/s $R_1$ 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, e,nf:xal :oo- 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 spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Dea:n:- lofox,e termine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Through l1 my 5vi/h3o36rriuyhow mao/5yr3vrih 3m6u i1ylny revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5/6)tfaha1a rl *w0) a- 4chwlw2afco x 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 stressesnda 95 /9v)ca f/lvj ptg08ept of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions befor9adgtv)pn8p9vj f t0c/la /5 ee 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 j(ra dzsw15zn )dd 64,exim,gs. How far will a point on the edge of the wheel have traveledewsmgdx n) 15rz,(4d,d6a zij in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 z9mixgs 1e 2,5 inn)fwrevolutionsg)xi m1sz 529 ,nnifew 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 make 65 revolutions as the car reduces its speed uk x gs-ig2hnyf4o.hl.od4 6 a2niformly fromyi ho -al.h.d2n24kso 4f6x gg 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 revolutions a/+n.mmyqf5blo+tkmxfc 3 *o7 s the car reduces its speed uniformly from 95km/h to 45km/h T.mom*f3 y 5 k7nqcxbo+tf+l m/he 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 eachsvppm /ii4vmpv.6 0;e)i:r pd pedal when climbing a hill. The pedals rotate in a circle of r/smippiri ;vv 4vpe .dp) m60:adius 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 c1-meduw79hrdu l)9 l ym wide. What is the magnitude of the torque if the rl em)99 -ywd1d hu7luforce 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/jrw5byeg 62 w. Assume that a friction torquy/2 rbw5ew jg6e of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of t -kiktvekh *o /;73;e cqn6hdmass m, are 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 thenqkd;ck3t -khtei7o*nv 6 /;he released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engw3lg k6+s *w8odl,jg rxll 4x1ine require tightening to a torque of 38 x*8 s1rg+ol d wwll43lj 6kx,g$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the m5 *q3i;:euv4xlv-dqz uelp oh h:a:8axis of rotation is through xvp4:v 3 l8qou5 hzlhe:qa -ie;mu :*dits center.    $kg \cdot m^2$

  Calculate the moment of is)6 s.pu3m6 )7 (xocroljax ienertia of a bicycle wheel 66.7 cm in diameter. The rim and tire havx.s)) o6rjo6cpsa i 7e mu(3lxe a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of9gsduj6fs vpa88(5kosf lt j+d(y j(- a thin, light rod is rotated in a horizontal circl (s9uf+gfl 8kpj6 sj58a-vso(dj y(tde of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at const gy fveg1wrrm4 wa2- 5i9,wwdo gs0i*/ant angular speed (Fig. 8–42). The friction force between her hands and wdggr*o/,sm4 295vyw-1 arf w0we igi 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 ofs eaga( ob75 qa-cg(9l inertia of the array of point objects shown in Fig. 8–43 abou-lbg7a(o5qga cs9ae(t
(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 atoms whose iw t9i0s0ema f)jkrt3id;*.63kgcubft: : my 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 cafa 1n wtr)h/z4b4.vu ylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fitf h4nvr4b ua/wz1a ).g. 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 wi29govjpgp 6h/c4m,l ma p(5ons (,pnb th a radius of 8.50 cm and a mass of 0.580 kg. Calcucpvgop nlg ((/p2snb, a 9 m6o4h5m,jplate
(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 swin-ud, s.o514ds brusjngs a bat, 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 tangentiall u rd.. vm +1c,kwdbh,q6h(wogy on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 b.md 6 wk1,od,v.hqg h(u+rwcm 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 8+syg rkxjh0v/ ne0( tat 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional tv+(gytn/e 0khx0s jr 8orque 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 b8(q3dvh5fgt* nb4ry oph /0 n9(mr nea 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 threo,fatltw 9*,own solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, F tw*otl,9f,e aig. 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 b qr(ni1 *p8k- 5sj 4ipsp8pqwge considered a long thin rod, as shown in Fig. 8–4j-pskrp (iw1q8psq8*5g i np46.
(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 consistkeeq.t +fv3 rx edj+nqx+4 )(ms of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turns (,v z(+sh-1d ec a1cpq/km5gf, zg)hjlfc- i;j revoluti efk/1h +dzjjc; qaslfi-cm ggh) c,15-pz(v,ons) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia ofrk/maf*); 1w7v c z/uofyh9k h $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops m47zp:didgb,gahe 2pg*,s:v8v+ kzda 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 radaj.oy x-2a*za ciga 7;ius 9.0 cm rolls without slipping down a lan-7 jaca g;a.a z2*iyoxe at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the 0f2:b 5qdpb8i-w m( /knqr pfsEarth with respect to the Sun as the sum of two ten(iq:d sfkw5-8rm bqp2/p b0 frms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net ydhg9 i5u:k. swork is required to accelerig.k5dhus y:9ate 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 mass 1(7hi+jy1reinh d. lm)t6)si /c bg6d k.80 kg starts from rest and rolls without slipping down a 30)e(cgb.rhj 1 ksil)+iym6n/t 7did h 6.0 $^{\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 a0m3vpg6e 1q6(jqs lrg vrk.q9d b37d to fall and its lower end does not slip. What will be the speed of thv k vq 3q p61(dlrj0gr .me9sgb36q7ade upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular moments ncop87 7f:a7 qsrnb1um of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10.san qp 7: 871cno7bfrs4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel z4 ozn7i j+gh7ub+u x0tzym8:of radius 18 cm when rotating at 1500 rp48z+b:0+huu mnx7 ozztgi yj7 m?    $kg \cdot m^2$
(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 plai)(e-5bh qog btform that is rotating at a rate of 1.3rev/s If he raises his arms to a hob)(5bhigq -eorizontal 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 moment1jd5 eja(smw6 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 tuck position, what is her angjwd e56ja1 sm(ular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from ae72lcy,f- v r 1rkkj+lwb;l(mn initial rate of 1.0 rev ev (-rkyv l 12llk+ jfer;wc7bm,ery 2.0 s to 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 verticn-c9o-ih fjd l+;j c) u04wbbgal 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 rotao-w4ic cd+b;nghj 9j ulb-)0fting wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinn8xlw r , sr63 qiufj iq,b;bc;g+z/o7bm+xu4fing 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 mon4nwvbbb/b/jy 7 q-,h3ceizi 511z ej mentum 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 mome ezsciv(uy 9 is33*t-7xte me;m+yl)vnt of inertia I is dropped onto an identical disk rotating at anct+my 3e*-l;e szu mixste7i )9(v3y vgular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at-l( + 2hug6 byatn9r) p:mgmux 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at thuk*; rz zl8 djwzs/,e.hi6t4he center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920 hs,zi4zj 6 z.wrd;*/th8 eulk $kg \cdot m^2$ 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 merr8in e*:)ktpow:bmf) 3*h fj cgy-go-round is rotating freely with an angular velocity of 0.8 wchig3fetb :km 8) o n*p:)fj*$rad/s$ 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, losing about half its m2g688gun7eyhe d fj7 xass in the prou8d g6g j78h2neefy7x cess, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve wbl5 noxzg -3ci9s9 d7vinds 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 uca, dt0x2s+f$ 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 rotate freehqm xoylsy1yo6l7r 2,yz+-;ply without friction. The mom 7m+hyoo6ys2- ;zxpry, ql1 ylent 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 at the edge of a 6.5-m diameter meru8d lf 2jooha:foo7pd1d; (l 2ry-go-round turntable that is mounted on frictionless bearings and has a moment of (hdo j:alo2 8olf1p fdo27 ;udinertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end 7+qhof 4pw - ib3wpj8wof the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distpqpbjwh+ -7w4if83woance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earyw w3kimbw 3/7l/g .mcth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a parti3 mw7.wb y3/lgwkci/ mcle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of2pk nsscejbbc 8o )a():jw-,v 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 i0ec dw 50kskn:fjwp 3q)k we)2n the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from kdsk 5e:e3)pc0fk) 0qwj2 wwn 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 soli)s)n 6xx*rv;ru+ o )aa lqu+wld cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric;xsr *xlu +nl)w+ar)6 uv)oqa) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how aj ;z(,mmr8h ris the 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 ou2cf*tjckl4, fr Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be*t, kf4jf2 lcc? 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 fb vcl;;qykxh8owtyc8 wb:9y 8, 62vnoor it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass o cvc8 :o2 y,8;tx6 vynwqhb9kylob; 8wf 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates achwx9qe- o f 4mhng5-3n $H_2O$ 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 at speed s*(cu2,pivf z*1t oy7 cvt ot6 mk7;yt/hx b6nv=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 pivotu kbd4rynobo685 k3 fp20hob2 freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–5 42obbf3ondu08kbo56y2pkhr 4. 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 vertically on 5m hv;3.l lsnbfx)a9z-m )ua cthe floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has.fl vz-;c a uams)n3hx95mlb) height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat ao/-d;kjef-thlnbd u6y+u4+ d 6gi/q road is making a turn with a radius The forces acting on the cyclist and ub-a/ i;-6/j+ +kfny dt64dqhlgdu oecycle 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 int0 z 4,wk.dv u7mtg)sx eb:kdng 6p6op)o a sling of length 1.5 m and begins whirling tk6,p beu o.sp:)0d6 z4)xgk gmtv7wdnhe rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and hery0mshr esbq+qo 0+d-3v6t gq2 arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of 20 d oym+0 q-qqhtb sersv63g+the 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 cy i4og5 sdfbgq;18lpny g76pz3lindrical plates, of 8nb61fsq dolpzy igg5g3p ;74mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 tracp y5z;mgg2 w/hk 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 pohwzpymg5 / g;2int of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but doo1zk359i cwv)n yh 9 gbov9dr6 not assume $r\ll R$ h =    (R-r)

  The tires of a car make 8/w+zjoior-a to.h 0db9 o,h,q5 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.9o/o0 9, .d ha oj,wrqiz-boh+t0 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