A bicycle odometer (which measures i*lzigd 4ij3y x16 h;mdistance traveled) is attached near the wheel hub and i43;ii*m 1 zh xg6ljidys designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular . ::jthwd6lszcby6 ,9:v xyt pvelocity. Does a point on the rim have radial and/or tangential acjtlz y6y6wx cd9v : hs:b,.:ptceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value o(jhgobv0 -nv77q fa. uf the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a7jjc snn7/8z d *ig,da 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 behind your heyf ;foqn ,9.tclh r+*kad than when your arms are stre,c9fh l +fy k;qo*.ntrtched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprr.6e2ug825 sict0.u elvmcve ockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to2vcg0it.es2cl8v5ue 6r uem. pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being abxgfq*7hb(14 cn// t5 lxncnfmle to run fast have slender lower legs with flesh and muscle concf1nxc nh g /*xb5 tlf/4mq7(cnentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8– ke7zlk/ 6pq.;yw:avk 35) carry a long, narrow beam?

If the net force on 20 qlwop t4,wq(er 1bfa system is zero, is the net torque also zero? If the net torque on a sqqr2e4pwl to 0,wb1f (ystem is zero, is the net force zero?

Two inclines have the same height bsmcktrx+l(mesj1 8n +g-cv . 1ut make different angles with the horizontal. The same steel ball is romtce+.c +1g(- 8n rm 1xkjvlsslled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) do9zo9 x*jp5 )ams:m o.k5kmdbdk q:d 3ewn an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which .:e 5oba:5q dd9*9)dmkkxpm3 skmjz o has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the sal3e gs,2k7 ro4hwpux3me mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bowg, kxr2ps3l3h7 4ouettom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving or ro hjo p747 t5f vn,yk8u(grf+n+1f rdaotating oo f71 tufv 7,gnyoakjn8+dr4r(+5pfh bjects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equa7 pq46 wpfc lm2vq8h.vtor, how would thh8 24pfqlmpq.cv7 6wv is affect the length of the day?

Can the diver of Fig. 8–tny g(6 fck *cxbla96729 do a somersault without having any initial rotation when sh7*b(96lgac ty6nfx kc e leaves the board?

The moment of inertia of a rotating solid disk yg 2a.x5g ;yot4y7asm about an axis through its center of mass is7 y2y4. symtx;ggoa5a $\frac{1}{2}WR^2$ (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 rotatingce+i(*s)7y w.h;vrq k rkjt.g stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, krhj;7rtq v(+g i *cyes)kw.. will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and6)+ dw7wpg;w tik h1rz have the same mass. However, one is hollow and the other is solid. Describe an expek1wwr ;gzwdp67+ ht i)riment to determine which is which.

The angular velocity of a wqugkb7vpm )6;heel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheelg6k mb p;qu)v7? 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 i2.tf7 o0nhxdhap2ohn f7di.0tx pens if you walk toward the center?

A shortstop may leap into kv*7 w+io oup)the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 7iwo*)+u kop v8–36). Explain.

On the basis of the law of conservationsxkx7 ww agn d8j43q9 w16d1qh:e/xvn of angular momentum, discuss why a helicopter must have more tha9gw8e w6vwdh3jnds7xk x1q qa: 4n/1xn one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 3:js4 mm-hy4 io*kmx j(0 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazin 9kt c)3lpx l:;i0l8ddlzdw*b g coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as sexl80idd9 l3zl );:btdcw p *lken on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges ab0a*gm7sj* g; d, ma5yjoquv5t an angle;b7j uv5g* mmqdaja5o*,g 0sy $\theta$ (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 1y /6 bzciwg,0vskbw9at 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 angular acceleration asi w,0/wbvyc19 6bgskz the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anothentdh:hs5w 3 (c8jbrf;5hn+ blr child. If the ball makes 15.0 revolutions, what is itsbdb 5; c8nhw(l+h sfnht 35r:j diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revorii/.jxs p2 8flutions do the wheels makj8.2rx/if ips e?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calt- g w44)i 1ryn) ;douirjwd3kculate its angular v;dn))i ou 4 jidrkr wy1-3twg4elocity 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-rounxbqr:bs76 hgp5bqyq-pw* 4x 1 d makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear qbxhpr 1pqbx4-:w 5ysg*b6q 7speed 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 velocity of the Earth (a) in its orbit around tzqbei+6 ;de n(he Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed 5l9 i uk/u nba-ioa:l,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 a2law ycj9agk57pr6y 0 particle 7.0 cm from the axis of rotationlypwa7y05g k2 c jra96 is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly--wb* tbomwv 4k6jmn) about its center from 130 rpm to 280 r b mtnw b-6j)wv*4k-mopm in 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 radius zgs. ,5ryh , 7qsn8yve$R_1$ 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, astrona+jmf wgx*,nr;uts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, mnwr,gf+ x*;j 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. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Through h0myg)j. kn.w3u)d l ghow many revolutions did it turn in this t)yldmnhg3.uw kj.) 0 gime?    $rev$

  An automobile engine slows down from 4l t /;m;9onhfl*8fk 7bute n,i500 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 highspeed jets in a whir;)o i5z :kjaurling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolution;ozk5auj: r )is 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 zubt 2,nb;8k axfx ng3:v 5db8rpm in 6.5 s. How far will a point on th:b3ufbtx8a ,b dz nxk8nv25;ge edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is runniqvdotm+n1 /6*bgdc5 s ng at 850rev/min It turns 1500 revolutions before it comesd n6b5*gmqdo/tcv+ 1s 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 r4ddma,fp 4; p1ri xiqhtl73 bde 14f*pevolutions as the car reduces its speed uniformly from 95km/h to 45km/h ;e*1xp4lq4pp 7d fiiab,ft rdh 1m4d 3The 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/z)+td,px ci:2y puun revolutions as the car reduces its speed uniformly from 95km/h to 45kmy/,xpdzin+uctpu )2: /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 her weight on each pedadlu1pw py1,k.o +1l wbl when climbing a hill. Th, 11 dlww .1+kplupyboe pedals rotate in a circle 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 door 74 cm wide.cvufjkk2:8- 7+ ch demr gp.7w What is the magnitude of the torque if the fo rgd72v u.78cf- j w+hkme:pckrce 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 va3)eboj kn ((the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 0.4boa3vjk)(( en $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attachy+u.tmsq ,4v/6 khgff ed to the ends of a massless rod which pivots as shown in Fig. 8–40. Init vtf4 /q su,6ykgf.mh+ially the rod 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 fiz l:ta 5ej nz0smz*(ys 53c3torque of 38 3l*0jaif yc5z:s( tzsmn z3e 5$m \cdot N$ 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 mru/ z7gu-wpzkcr:b2g8km vk1,++ ca d when the axis of rotation is throb1 u7 pdr+r2ugkk-g8cmc,: a /vzw+zk ugh its center.    $kg \cdot m^2$

  Calculate the moment of iner9 c upe17c3foitia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignore79cecuo 3i p1fd (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotateu0kcng; h3jp 2d in a horizontal circle of radius 1.2 m. Calcu n;pju0hg3k2clate
(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 bowuwc.p s71um f6l on a potter’s wheel rotating at constant angular speed (Fig. 8–42sf.uuwm1c7p 6 ). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the arrak fdmj2zfs0 :5uoc8t9bii :p3 y of point objects shown in Fig. 8–43 abcoitm fdkzf 0 :iu8:p29b53sjout
(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 od769j,nj7 ougx-:d.smvj x k:ejzo5xxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the corro ;w(iiq:37 xfzvn e9wect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket fw(v 379iwxqez:;oi n 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 oxaw5s+g( hsc3icd2zy8bd+9 pf 0.580 k(b+h +g cswdy p9dz2 ixcs835ag. 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 f;jphv5 q)ce- from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-rounlul fo:w /x/q, xsuke2l0a0x9 d and is able to accelerate it from rest to awu /k ,ol0/ xlxaeus2q0x:l9f 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 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 evepr g1f(7gip.dfw850mlu/s b 6.y ag zyntually brought upgfp.z 1/rf5(.i s gg0wmyluyda6 8b7niformly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accgn4 asp1it6:gv.mu x ,nfp,l/elerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of 4/r0)ka tni8emuqnv/ the forearm, which rotates about the elbow joint q /tirkn /v0en8u )4maunder the action of the triceps 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 considered a long thin rwt49 2 nxynl8qod, as shown in Fig. 8–4w2tx9q4 ly8 nn6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of twornf -n+p /hnr8p k07bmly .spl 0e//im 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 (z .orq,e ebnm(nb7t )tfa;7c3 revolutions) and releases it at a speedmor(b7. ,;t )7 ca feb3nnqzet 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 ine, y qxv1z )4cp4hlh3ecrtia 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 k(,nm-owft xm ml0u34of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without y) 5fumlbhe hbe9(td9; /x,z w+h6 eulslipping down a lane awzh+l9d/lh)e6(5e yu 9;e,m ufbxhbtt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of(ja zmg ya.1kw *m6*vm two tak6gz .m( y*jv a1mw*merms,
(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 ne.r 3/nhuq/8hj +3 vh(u i-;zyzesriz5jo8z l zt work is required to accelerate it from rest to a rotation rate of 1.00 revolu ; (.hv38 ioz35zhnz-lzqrs ije8+zhuy u/j/r tion per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg start ekx8))z -hxutl.jxe4s from rest and rolls without slipping down a 30.xex . k)x8lh- u4e)zjt0 $^{\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 verticqj// al gy,b,xally on its tip. It starts to fall and its lower end does not slip. What will be the qbx,ag, jly// speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the eb /1ih.5lanp4ph t:lynd of a thin string in a circle of radius 1.10 m at an angula yhlpipbnth4 .:/la5 1r speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cyl 3 ddad4f69do3:yfbtctea9 le;jk/. windrical grinding wheel of raddda;3. te/:ad dw9fo3t9b l 6jc e4kyfius 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 rotating at al ys: s.(m /xz nfyue4ig++vb8 rate of 4 mu xby :n.s(+ f/i8eylgz+vs1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the on vhcyxrh28ynf+ z,0nr.ab 2w* e shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck po*8y c+h20xzy, 2.fb rn hnvrwasition. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rota *nc0 ahtal n3zu;;sdi,2(cwttion rate from an initial rate of 1.0 rev t0i,azsd w3 c;c(h;tlu *nn2aevery 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 vertical axis throl3s fb22h-p ;k7hgj tyjzj;pm6 2u 4duugh 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 pmkl 7yj gpbjp3-2j t;24hdsh2u6;f uzotter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular l.f y0ox2h ik.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 of thef 7dj+-u8lxmtr yv/t f4wie-- 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 inertiao;u4:, ln gclx8eyhr / I is dropped onto an identical disk rotating at angular scx8l,egn:lhu ; r4/ yopeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns cxtcj9 xfzno6ukpp bd.;i 0k7 uc 4,64at 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 the center of a rotating merry-go-round platxf4*jm-ekao/ -pz58a5u ppqjform of radius 3.0 m and mo -p/aqapjuk585ozpx ej f*m4- ment 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 4kect6tj5y +/wpmgut 0ey+,l freely with an angular velocity of 0.m,u6ew+/ p4 0tygtj +lky5c et8 $rad/s$ 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 ugdetn1/flxv2 b q7; ;dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angu1xq/gevfl2t bn7u ; ;dlar momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess ,komw. sd )oi h9w1qz/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 /u/1m 5en0ubyyo;:db ;qa frr$ 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 platfored2y3 uy4ph,c p )f/ktb9d /qkx ffu81m, initially at rest, that can rotate freely without friction. The moment o9ch/f t uykpfe8d ykx3fq),b2/ up1d4f 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 nsoav. /yy5qw2bwqt4ev s5h s)+bh18at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertiq/bo42.)qvv1 5yyw5ss8h aw ben s+hta 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 rope84 ar/lh5l d(,ad luwa rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the d4wlarla5/(a,h 8dulperson 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 samlhp;zcg ,cz uh 6ie7m,.0y o3op)qpl* e side always faces the Earth. Determine the ratih ozloely ,pc;,7 0 zucpmih6)q3.p *go 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 )b(3tohq 2tgs2xv1 wxat 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 inkt)0sf7* h3 y qafsy1l the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular accaq 17fs *)h3lsyy f0tkeleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made onkjsh+ph7l2+ f two solid 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 string, if it is released from restn+lk+h hp7j2 s.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, h+u 0ud aac6c*gow is 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 our Sun, but with mass 8.0 times as great, were rouaroklfuu1+qb4 4) 7k tating 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-quartersu )ukf4bl+kqro au741 of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored i. nk/ p x(ivv1ayd-u-il:7gbsn a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel ynvlkix1. iab-: g dp/7u-(v s“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 illustratovqj93a f si20es an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3.3 g.jwx m(7; fdb$m/s$ 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 ignore frictionetd(*)t nb4e9pc2 h rs) about a hinge attache2e *)tdcrs(e9hb4tp nd 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 vertically on the flood 9xvd7.tsb33y9gb qbr, and we want to exert a horizontal v3dxbqsg9b7t.9 ybd3force F 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 is making a turn wn)/;jizry78f uq /e0fgcizm f l+y n97ith a radius The forces acting on tzj;e r7 0iucf/8 )f +n z9ynimlq/y7gfhe cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begie5jh (cha1ez5l8 1uy: va ve(ins whirling the rock in a nearly horizontal circle above his head, accevi8ac1le: zaehy(1 ( hj5v5 uelerating 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 her arms as thin rods, m,2 4xqwf-z6xi3ywr z vf46,cxjp)iyq-b5k ah aking reasonable estimates for the dimensions. Then zchiaxw3 kwqr,yx-bv ,iy6f6 4-zp45qjfx2) calculate 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 clut7 zk5yeck5()s 5zoxsq6n ,nscch assembly which consists of two cylindrical plates, of mass7osnecz5k()56qc zs5 ,s yxk n $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 track ofm3 1e bwtz3;an Fig. 8–58. What is the minimum value of the vertical height h th;ena t 1wbm3z3at 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 assume 0,ln hhc6t ov5 fpxr43$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifol29- cx.omxeermly from 9.x92cm e-ox le0km/h to 60km/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