A bicycle odometer (which measures distance traveled) is at qhr::h-ze9 ehtached near the wheel hub and is desi9h: he-ze: rhqgned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a pointkn0 xu5;deb2l64sqs j on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangen d25kbn l0squje 6;sx4tial acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular veh - bbqpipr, e.2yu*vurz5l0 x/oj:1llocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forc 9l) kw /kjlq(7i,jrhae? 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 withx w)0-a;bnt 3dq.tgnol t7h17fmgat/ your hands behind your head than when your arms are stretched out in front of ym)n. /d 7qtn17bx0f awal;got-tg3thou? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rearfemac2/*4 rtv wheel and three at the pedal cranks. 2*eat/ rfc4vm In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender low7hbo(i3:xjk jq j6k0 nfah w9)er legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why thisb k(hqw)j kj09o3h: 76jfinxa distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) z99f ba ,7fa kt(3vhdp.gnvy: carry a long, narrow beam?

If the net force on a sf pf07/nl s8unystem is zero, is the net torque also zero? If the net torque on a system is zero, is the neff80n/ sun p7lt force zero?

Two inclines have the same height but make differentflb )wg; -v g*s/mqjz ):.wjms angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be gr .ss/zqv m*w:mwg )jfg-;j lb)eater? Explain.

Two solid spheres simultaneously s/ 8t;fquo- o9os5bs e8xahr n0tart 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 o8an;rxfo5-0s q9e ubho8/ts incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the 8) 7jbga r n:kj vryocan;4dte/-69mnsame radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total ;n no jv a89r/t6:m7andy -g)ejbrk4c kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momen x fm*l3x(/hgftum are conserved. Yet most moving or rotating objects ev3x *ml(hfg/ fxentually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how would a4g28td m e 6g)qzkzy) 94tpji +xh-tsthis affect the length oepq6) tt 8mjz9g-d 4g+t 2xi4 ask)yhzf the day?

Can the diver of Fig. 8–29 do a somersault without having any init78uulzfly7y0 ial rotation when she leaves the boardl 87y 0fluz7yu?

The moment of inertia of a rotating solid disk about an axis bx8tv (t+ee)dthrough its center of xbtt+8de(v) emass 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 2-kg mass in each ou)jlsdgd.v9rsb* 5c+ owp 3i1 r4o k1potstretched hand. If you suddenly drop the masses, will your angular velocd3b k4op *5 g1lcosd9 r)ro+ps1 v.jwiity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow a 9k5fbif be1m9hc( c)ond the other is solid. Describe an experiment b)bkceff 9 c1hmi(o59to determine which is which.

The angular velocity of a wheel rotating on a hmyd: 1 wv);0 p;sjw,wokb9oyhorizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this poin; 0ys )vo9 1b;dkwpj:h,oywwm t at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on/3elx flh 5wpb0w v6l; the edge of a large freely rotating turntable. What happhl 0vllw6f5 wexb;3p/ens if you walk toward the center?

A shortstop may leap into ln d6c11) x0vvma hoaen+w,0uthe air to catch a ball and throw it quickly. As he thrhn0 duxve)n1 am awvlc0o ,16+ows 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. 8–36). Explain.

On the basis of the law of conservation .abt 5.niklx56hf6 riazbv ,m1h *ho3of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second pro.ob6*ihl ,mv5k 5b ahx16.ztf3inharpeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30dxke9 o1.1ah/ g+k rnf kc+3ty $^{\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 coinci,d.l1aar5mmdjnu* ps4g yw) 3dence. Calculate, using the information inside the Front Cp3 mnmslr,d51.)*aja dy4guw over, the 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 Moon, 380,000 km from Earth. The beam dive(odiz6uvpxoq+t(; - y rges at an+idv y-zxp (o;6(u qot 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 ao53ol6ras *vmq 8 h ,juk*r,gdt a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the anguo,o *d8jsl*krvr 3a5gmq6,hu lar acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another c 23o ;umwt+rgdcqxyi*6c3 k+d hild. If the ball makes 15.0 revolutions, what is +qcrti oxc*6d3mu3g+2w d ;kyits diameter?    m

  A bicycle with tires 68 cm in diameter to56ua arcqo1-/9 ndkpzv5dg9ad+ od-ravels 8.0 km. How many revolutions do the wheels m9n-5rcqk a6 d1v od9/ ag5azuoo- dd+pake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates:mga o3yfkg0;h7zb o9oh0ky- at 2500 rpm. Calculate its angular velocm : 0;h-oo kzkgf3gb7y yoh09aity 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 on *(jks90)o*o 2tv hk9lqf nszte complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1* ostvljshqo nkk()ft02 9z*9.2 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 orb3lw ;b6irkjm*it around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speedhvk +gn.0m2 cm 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 pb:/ij1b-e wfoknvk86 c+ wy-particle 7.0 cm from the axis of rotation is to expew/vpc+ojw nfe 1-bkb-k8i6:yrience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its ce1y-+38c1xx y w6j1oiuyxir k inter from 130 rpm to 280 rpm in 4.0 s. Deji 3yiu wc8116kx-i+y1xxyro termine
(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 radiusys4gl ,5rsf:g/n ser- $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard theagu8rjr v3;k jb8u ,8k 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.8 ;r,vur8 g3kbj8 aujk 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 unif+pu3i(n)d7e sjt v 27vrht;sp ormly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in thie hntv(p +s)r jusp2 77tv3d;is time?    $rev$

  An automobile engine slows down frogz, b+4*z,jrj3mxjmikt lz67 5rur+f: i+h in m 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 highspeedi)b9,j1wq81 d ye pq c882uiyy5htw pu jets in a whirling “human centrifuge,” which takes 1.0 mq wujdi8q 8 2 9p8y)y1pue 5,ib1twchyin 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 36t,ei7 h*q43aqlo 9sezgwt*09.28fw pdsajlg 0 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in tgp*g3 je fqod. t9qtw7 9wea lh2,*sa4is8z0lhis time?    m

  A cooling fan is turned off when it icw5+**giio l us running at 850rev/min It turns 1500 revolutions before ic g *lou+i*5wit 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 caru5 md4 vss s 3jihwb)s)g7m.m7 de/tid68:ubb reduces its speed uniformly from 95km/h to 45km/h The tir :w7si m)7bveuih mtd48.3gssd)m db5 ub j6s/es 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 maky 2n: ;:pwsffwk;ue a3e 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tirep3sw;ffw:2ke ;y a:uns 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 ut2/v8r2owj * rtmf /5jh*nfa weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 c2trrna*/5fv/2mj ot8* uwhjfm.
(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 thei,xbrchewxtq bz,nzq7;/6/o6g 4nq 3 end of a door 74 cm wide. What is the magnitude of the torque itoxb/4bn 6/ 3 zqq,;7z ,6qxr hegwncif 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.wmaaguta.. r.dzn 4 e4haf.54 8–39. Assume tha5af4zaeu ghr.m .a.w a4n.4tdt 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 attach kyd1a+7yc ajw5sbjusr1b ,13ed to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal posiyjb5jc3 +rdauswkb y7s11,1ation and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine requir vyfmxxkb7vcv4 l 7 6utt785u6r3 0dgle tightening to a torque of 38 vl8gxdx7 c 3 vvtu4fuy7bm675l6 krt0 $m \cdot N$ 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 theeilh0: bgw) n4 axiw h le0gbi4:n)s of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. utqmd2.zc2g/kk3q4 s The rim and tire have a combined massm.dut23qk4 z g2qs /kc 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 of a thin, light rod is rotated ,t9s(j gv45o 0onpudm:c uab* in a horizontal circle of radius 1.2 m. doub*a vcp,(ju5mn:4 so9t0g 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 bowlf(zf*;kg:o6 u0 zyo cl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction f60fu ;fyl zgoo*(kcz:orce 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 array5 y3ar0wsvnc1lmfn 9-e am+6n of point objects shown in Fig. 8–43 about0sf-5lnc1mavr3y9ne a+nm 6w
(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 total c719 yn w:hz 5toru*g)ndv vnqz n0.1umass 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 satec/4-ocyp d6tedd xw neb/;*k ).)mmy*f ml x,xllite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–4xfyynmt-mk;d, ) . cbx dmp/ee)cl *4d/x wo6*4. 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 radius of 8.50 cm and a mass of g+h4 w/72kk / hl7eicq h*zh2vdqtvw8 0.580 kg. Ca+d*v7t 28hew/4 qlzi2qkhhkvcg/w7 h lculate
(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, acceleratinz . 6eohyl1d(i298o7rmipy uqg 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 .iuj61br8+oivzj dh 22 ch iw(hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other 8jr+c2u(i di 61 .ojz hwhb2ivon 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 1c:d y.-b fyp(u0,300 rpm is shut off and is eventually brought uniformly to rest by a frictional to -:fcb. pyu(dyrque 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 .iooi6+e u( uzi w1+pz7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, whichr3 me.) nsnk)g rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 en. gm3)rn) sk$m/s$ 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 f/b sg9.gyo3cshown in Fig. 8–463gfgso/ .9yc b.
(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 con3caox jok1l 8l z,(47xrh4sumsists 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 hammgop/ ai/ ug;3 rkcga+(er from rest within four full turns (revolutions) and releases it /ua ;gg/oip+ca k3r g(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 hasv*-h0g+uqw.*/me z 3gltz sbb a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque un+jclr7cmzuqn+x3 a ) pq9/+ 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 radius1 9:nkbx46 k)uegveo y 9.0 cm rolls without slipping down a lane at 3.3ex149kk o) bgn:6v yeu $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as thc/(5i 5(hi h6bpnqs yhe sum of two te/hyn6 bhi sci5q(h(5p rms,
(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 rm.x6di2e j+xw;lx5 nm. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution ;de2.+lxrx w5n i 6jxmper 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts ta)ts+se28z (rrv+)qn - ae wk.m +ovrfrom rest and rolls without slipping down aet + vrs+)+etaomn-8kas w. rzqr)(v2 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts , a peu *ynh/ig4+nbn-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 g-nnh4ua/ ie,*+gbp ynround? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating onw0-ek.lv ddqg osa,gk2/; p x+ the end of a thin string in a circle of rad lo k.g/s2+x0dvq, g- wdp;kaeius 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. h4kk5db )5ozl8-kg uniform cylindrical grinding wheel ofdbk 4hzl55)k o radius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rot+k.bxz;/ns ,tku i x9jating at a rate of 1.3rev/s If he raises his arms to a horkz ,.ut/x sb;+xnj9kiizontal 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 momg a6nv3uno6*bfrjl ) r93i.ac3p .nttent of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes ovr*p)3. icr33fnb njtaual96ng.6t2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her ez ex70/ri iy49t3oksspin rotation rate from an initial rate of 1.0 rev every 24e0 z9rysi ti/7koxe 3.0 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 vertical axis through its cdihz-19 srf12ziw j4 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. s91dfh-iric1w zj42 z What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentumi6v-1(2ej5yylipzl s 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 momentumdk58 dtzv.9lur(l jo : 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 3kw,yn.rjc4t3mk x5 udisk of moment of inertia I is dropped onto an identical disk rota jtrkn3xyu ck435w .,mting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at oiw;hbkvi 715 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 ste2fz*z gwv+o.4 1uy naands at the center of a rotating merry-go-round platfor4 fvga.*z wyz2+eo n1um 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 merwpfx8bdw2 6mj6 qs) uybb.1 sq w7:tw/ry-go-round is rotating freely with an angular velocity of 0/q12t6 q:pmfw6 wyb)8bsu7jwxb.d w s .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, lp :j01qc*g h/llz3ljyosing about half its mass in the process, lzh 0jj qy31/:gllc*p and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds ot,.+rtgw+x(*/zv96 u8 v usxubeibdin 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 ).q3 bz5m 1tm clx6,:fq oxtgp$ 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 freely witl2ogun*apz-d nvqhkp850 6h q;u i +s6hout friction. The momeq5 h * nzk;qlus8p6 d+o2p6 anv0hui-gnt 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 wm-a3wc- e; z6mnh4g j6.5-m diameter merry-go-round turntable that is mounted on frictionlesz 43g- wmjc;anhew -m6s 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 on the ground with the end of the rope ln13 g*8n ow;vc r3hfluying 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 spool? How far does the spool’s center of m;nhr*ng f 1vl33oucw8ass move?

  The Moon orbits the Earth sucp:e o)c7lztn2ln4ym xk i. w;)h that the same side always faces the Earth. Determine the rp .mt o7wn xk;)zlyn 4el2:)ciatio 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 from rest at a rate of 1 m qwza,rt 4 e9; lqqgtles(pwk707m,- m x)9lfe/$s^2$ 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 ow-kh2) s n-;1arqkuz of radius 0.20 m acquires a rotational rate of froz u-n2hsr- kqk1;w )aom 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, each o2ib3v8mal 5ofx2zn e7f 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 8 bxflvo5en2iam3z7 2 speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of thk,tvnau :swa oibb. u q;2r03mi *;6fze 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 times as great, were dc3y6-jd ,04 tuqzlocrotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of r3d tz, d0uocl4jq6 -cyadius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollutimu,rsn k( ge8ur;8 b(w/l 5icnon automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a unil m(5 cru8wgibn(r/skun8e ,;form 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(w x;rn,9wvjb+l7 zu p 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 (hoo4dw3ajtu.4 , gl1aa4hcnef9e p) is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L c6i d:e2ls,su)5 eyif san pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine ( 6il:sdee, su2i5y )fsa) 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 verticaly mvo37mij9a0fx* )ku mx bq,7ly on the floor, and we want to exert a horizontal force F at i3o7) mvqkui97 a0m *jx,bmyfxts 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 ;+uk12 mczwma)7 anht=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the m72 za;)k nhwuta 1cm+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 yojrl 8ycg0 ,,f8p d1oand begins whirling the rock in a nearly horizontalyforlp y0c8oj8d, 1,g 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 + ktzbs/(ntr0gav:1m3m- zz vzh0g z1a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then ca 01k z:snzhbzm3 zg0t a(m /+t1-grvvzlculate 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 assembly67zjs 9 chjd bzfbc;u)+wig 54 which consists of two cylindrical plates, of wz)i bdzfsg ; j+6u754c9 hcbjmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ro y/(.gljkh:( qf0ulq 7m,npx rolls along the looped rough track of Fig. 8–58. What is the minimumqp .y07l/lng( k x, (hu:jmoqf 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? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not a86fkfnv zx:q 2zp g4eaw : dx4r00-zeossume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car (v ef+uca-t86/z. 4ha txvcyb reduces its speed uniformly from 90km/h to 60kct(8e4/hvuva.f -6 t+cb xazym/h The 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