A bicycle odometer (which measures distance travw,pv 5j0ll09f/tnn r seled) is attached near the wheel hub and is designed for 27-i r 5/lp,9tnfnl0vs j0wnch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocityns6o*e4m4c xkg0 1p nt. Does a point on the rim have radial and/or tangential acce4 ntn61eckp gs m4*ox0leration? 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 describux q) s1u-b xw0,ofg;tu) yci)ed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greatehhtbr jj- )+em -yp qd;. j4.bup3rkn(r torque than 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 hancnhg1pv 64vm/yh); ad zk(n1fmc,k+nds behind your head than when your arms are stretched out in front of hzm,v(;c61)d/4p cak h1 nk v f+nmyngyou? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at th2es( yu9iyr3 xe pedal cranks. In which gear is it harder to pedal, a small rear spr9(y xr23uisye ocket 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 being4qe rvw, 7z(bm able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution oqb vrw, 7e4(mzf mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beh9n70d17cwtcyl w l/j lwgy:0am?

If the net force on a system is zero, ix,tlfxq+6 pjtrjq960m*.m k ns the net torque also zero? If the net torque on a system is zero, is the net force z6jl 96 mf 0tmntkp+ qxxqr.*,jero?

Two inclines have the same height but make different angles with thg7 e2bnkp*lm1 zk)l8be horizontal. The same steel ball is)2*mp gkbl71 enzl bk8 rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rl q 7-ewaymk 0 *4gqa4dr*bh6eolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the g4mh* qq6ybe a kw4-7e*0rldagreater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same m7v 0 ,/c0pmo2mheak, dycmrp6ass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater sp cm /hkdvoeac, pr,7m60ym0 2peed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentu4c dhem 1nha0j3/(uhc m and angular momentum are conserved. Yet most moving or rotating objects eventuanj3 emc(hh/1 uhca40dlly slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, ho z;(whueet( q.w would this affect the length eq ((eh.;wtuz of the day?

Can the diver of Fig. 8–29 do a some kt9j2 xatb mq4j+s-0ursault without having any initial rotation when she leaves the board?- mb tqax0+t24ju9 jsk

The moment of inertia of a rotating solid disk about an axis through itt9:rsv ik 9ifgs7b/a 9s center or:fgi99ass7b t k9iv/f mass 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 hr tv)*4dm0-ru ybe+w2-kg mass in each outstretched hand. Itw 0m-) y*er+vbudr4h f you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and zqv )e597fdx9+bs 1wqkb n4lanri27vthe other is so4sw x+ean 91df2 kl7rqqn7bb5zvi9v) lid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. Indna u; btu (/wse63td( what direction is the linear velocity of a point;t(a(36dnw /t sdebuu 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 th3fu/ypw,8b e*jas qi*e edge of a large freely rotating turntable. What happen3wy,u/q fbpi8sae *j *s if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As hn1p0ufuvgh5) vydc ; -e. gt+se 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. 8–36dvnt) uvh.+puf cy0 g;1-eg5s). Explain.

On the basis of the law of conservation of angular mkrkmc9 oy /e+:omentum, 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 helicopter stable: oe/k+y mc9kr.

  Express the following angles in radianou+p kh.ysl64sv c2m- s: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, using thgwy3q:l8b: s oe information inside the Front Cover, the angular diameters (in radians) of the Sun anw ysglb:o:8 q3d the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. T 6c:8 hzvip5g ajv;cp(he beam diverges at an avcv jp(g6:ip8 z ch5a;ngle $\theta$ (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 hd: jz kn;n/3mof 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the :zhnj;d/n3km blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anotr aub1cx3/fi8); bfqpher child. If the ball makes 15.0 revolutions, qb1;up/xra)ffcb8i 3 what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions ws zqxko5p t40cpr 417z2cu 2hdo the whwu2rzo75 xt4cp2k4h c1pqsz0eels make?    $rev$

  (a) A grinding wheel 0.bd:s e)ht3f yhw5d2u435 m in diameter rotates at 2500 rpm. Calculate its anguldt y2f dewh:)sbh543uar 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 revolution in 4..+dh.9mtt 9i4- wrdpbgt,x c tcdp9m40 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m frt+ c99 gm.44td 9.dt -thxc w,drbpmpiom 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 the f3ev*upku 9rr:; pd )vSun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a14ez,w8 ymwg0 1bsnvo 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 partgfx/ws( zh1q4 icle 7.0 cm from the axis of rotation is to experience an acceleration of 100,000 1gsqx zhfw4/( $g’s$?    $rpm$

  A 70-cm-diameter wheel 0rai vru )j 3vpqm+*b7accelerates uniformly about its center from 130 rpm to 280 rp*aupi jm3qr)vr07b+ v m 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 3y; fzj egw)0)fn 99)xcdx laq$R_1$ 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, astl h81b2gj7/jb*z3axm l m 7a t,iyyi2eronauts 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 in m /y7 zm*3bjyla2hiiblgtj2 x17ea,8 terval. 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 8/fyt.r wl:w bc-*7czn pdz+srpm in 220 s. Through how many revolutions did it w8 b+fzt/cc.n:wpy-7 lrd* z sturn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcul cvj;;8uzxse,ate
(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 8 6p6/5esk1fxqcilrnstresses of flying highspeed jets in a whirling “human centrifuge,5i 166/ xkpfqcs n8rle” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelehr;au4j0)r0d a xm8gurates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the rud)0 ugxr80a 4ha j;medge of the wheel have traveled in this time?    m

  A cooling fan is turned off when nzj1whqfhvk5,bf 9:pjp 7p7 -it is running at 850rev/min It turns 1500 revolutio :7jfh59pn fwkp z7,pqv-1 bjhns 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 pguzd6+ .c tz)uu*0dsrevolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires havgdc0utzu .s+ u *zd)p6e 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 as the car reduces itnce urfc-xhb)0y0utg1i-- 1ts speed uniformly from 95km/n0yr feh c110bc)-x--guttiu h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bi* d c0zqcomv53*s cq+*u p7xwnke puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of rw73 0xumd+c*cps 5 qznq v*co*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 )mlyh,+ a5hvfend of a door 74 cm wide. What is the magnithl, +5 f)hvmyaude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fik8h8ufg5 w4ojkdak5pcoh5w0;y 9/9 hwv s /hdg. 8–39. Assume that a friction torqu8 ;okwh5yo94ujh9a5g/hdk/vs0 hckw w5d8pf e of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod which*pvt tyq /9 03 ;pxc6uuaor)tr pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the neauqct pvt/906ru*royp; )3t x t torque on this system.

  The bolts on the cylinder 4 bia6i-q9jzwhead of an engine require tightening to a torque of 38 ji wi- a9bz46q$m \cdot N$ 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 s)a72y xryt ph,phere of radius 0.648 m when the axis of ro),h 7apt2xyy rtation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. yjvi- 7uv q5h)The rim and tire have a combined mass of 1.25 kg. The m5j )v- iuhvy7qass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end ofi3,sbo kf q)k rw,4i tui;)3pl a thin, light rod is rotated in a horizontal circle osp3k3), 4 luii,fiqb)wr k;t 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 pk3 itz 6(8ruwaue7omr3w 4w-wotter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her haewwuwm w84(u3ri7k rz3t6-o ands 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 arx(kt*s wwm. 2bray of point objects shown in Fig. 8–43 a. (wtxbksw2* mbout
(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 ofqt5 ija-; -bsx two oxygen 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 correct rate, enginwocz8trd/wjf 0m;3 f 4eers fire four tangential rockets as shown in Fig. 8–44. If the sac8 wj0ztr; o4/ff3mw dtellite 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,+ 9)fknk-dqc:ylden/y9u yv cm and a mass of 0.580 kg. Cad,fd kcy yl9yvn:9 q/-kn eu+)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, accelerating it from rest to 3u/69ekhr r0y z5m v-st4 x1ke)k ggt1w 0e;iwi $rev/s$ 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-round and ir 0u j:gwq+(rq+3rlgzs 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 on t+0rlgwr(q:qz + ug3rjhe 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 eve5omx -r wi2e(1v . fvxbn00a(dw6 clmi)of 5ibntually brought )-0xod0lmmxawi15vi fbw fb6 5n(rovc ei (2.uniformly 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 accelerates a 3- x5smkr6cs8o.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 the fo4tqi 1vy, onmpb.yfnja4 ,k5*rearm, which rotates about the elbow joint under the action of the triceps musc*v .5q nbj,1kno,a m44y itfyple, 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 rod, as shown in Fig. 8–48nin jgk8,l d x(z41.b*tbldk 6t.ndn* g i bl48(d18kxlz,jbk.
(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 two masses, jsrmsi-4a5* c$m_1$ 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 bn7+ev mdxc. *from rest within four full turns (revo+ dcxeb7m*v .nlutions) 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 ofso lyc43tubqe4f+ 9+ o $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine i**me+ h+0kc +o2tyvg 6olq ktdevelops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg an 1bm86)nry,k bo9uhgu d radius 9.0 cm rolls without slipping down a lan9 oh bm8, gu1uy)n6brke at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic enex 0rmzttghu97-az42 k rgy of the Earth with respect to the Sun as the sum of two terms h2gm79 xk-4ttz azru0,
(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 kddvqh2i+igas72b ou5i d *f 9.t 3+ldcivq./ag and a radius of 7.50 m. How much net work is required to accelerate i5oli3a2qdd*+i./i. v uviscg2 t d d7a fhq9b+t 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 1.80 kg stm/ pdrs.mbzd. g )jr*.arts from rest and rolls without slipmrg.p.r *z /smjb). ddping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip.b: qmwf)) 5ss:w dyhl8 It starts 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 hw ::s b8d5wy))fqslmround? [Hint: Use conservation of energy.]    $m/s$

  What is the angular 7vhiu(x/e y1 b/w a+jbnxg61pmomentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.17v yx wx1bp/eg+ uhjai/(bn160 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular :u0q7be0j gpw;eb9ibmomentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when roteu0g b;w0bqiep9:b j7ating 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 a rate of g9jv+eknt d 73eq 5)ot1neqv5 3 j7t+oted )g9k.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig.( a h3sgru7g.-ku y)bh2 x6e oa7srhe zae5.d) 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular sph)7ao .r 25ges u(a6)eudbxgr3s hh7-zek . yaeed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rots6jt:d(oava 0a g2 j(ration rate from an initial rate of 1.0 rev every 2.0 s t2 (:j g6j0atv dasr(oao 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 x8,sym r yx;bms ,vx(3t48 f(ozgu1gtaxis 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 of843s ,;xgumzbx(yymg 81vofr s, ( txt 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 anguly,(lo wt(wk +n9:iq ozar 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 momengyr :tc;6bepff8db,0tum 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 ofhh:q;crgi8mv . )huw5 moment of inertia I is dropped onto an identical disk rotating at angular speed rq.mhuchw;8)g vih 5: $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.40d( h;lq8au0dqf0y e j $rev/s$ 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 rotatj y*.fq s-9ebkj1r3 h1t6lxgving merry-go-round platform of radius 3.0 m and moment of inerti13 9 k-f1.xl eyvsjbtrhq6*jga 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angulmhp/ i3(vvq 9iar velocity of 0.8m/i9 p (iqv3vh $rad/s$ 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 dwarfp8eh wt0j 3ad -vr*h4c, 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 angular momentum, what 0tpdv*4h8wj hr3ea -cwould 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 win)7ab0m0; kud ragw1tv7yc ,l ods 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 wkov u92x7:ya1iu7qb$ 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, initialfo- 3c8rk s 4syu*pnn+ly at rest, that can rotate freely without friction. The moment of inertia of the person plu 3fcnyo4*rn8k-s+ ups s 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 6hduu0b92ikg er6ef (8 .5-m diameter merry-go-round turntable that is mounted on frekg8fh9e uu2r di0 b(6ictionless 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 lying on the/fpt2-x 3wt6 nzk1m bck39ueu 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 bxkp 6uk/3e9z-1mut btfc3nw 2ehind 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 thatn 6k+jxbzd(35 oz+kz z the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. xj3k z6 k(z+o+ zbzd5n(In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist acceleratea6 :5f-l6h*dyrg komds from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape zbj8cb.xh1 q1of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by t1c1 8bz.jhxb qhe motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindricalcs -wcc5d8,we eg1(ydyk e/d) disks, each of mass 0.050 kg and diameter 0.075 m, joww18eg kyec,5)s-ycecdd(d / ined 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 rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speedistbjnbhtylv */+v283 7:j d j 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 wj zha1bj; n7x0wwt;35lo m ao+ith mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to oj3ojtb 5h0mnxza;l+ 7w1 ;aw undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a lowv 571- dcrp1n-w3os*qp . yhvnhxsso)-pollution automobile is for it to use energy stored in a heavy rotating flywheels nw5*scor3shox1-v n.d-v p p71) hyq. 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 “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 ap70rzu bnda.f04zix 8niw*a9 n $H_2O$ 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 horizonr )zdo1l jsioc4l5;q5tal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore friccwq;zp cn 03.qup j +/ps0iqv2xsk-5vtion) 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 (a) the angular acv3 +w q 50u kps2z np0;/pi.xcqcjqv-sceleration 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 rad4limaeq9wg og,- eakoah60-r1x2lbf3l2) v n ius R. It is standing vertically on the 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 ha, )a r 1ighllkafe-e n2v0l9 xo-owq 2amb4g63s height h, where h

  A bicyclist traveling with .qx-d we6cmlu7kfvn 4ap1j 6a((.dau speed v=4.2m/s on a flat road is making a turn with a radius The forces actn (a.fec( k6.wmqd 647av- uuljpa1dxing on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of l4dw7k44aij w(c vt;.zi g:v gy)-mcxt ength 1.5 m and begins whirling the rock in a nearly horizontal circle above; t44c:jm tdz) 7w(kvxwgc -a 4iyv.ig 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 cylind3; ., u (bey0xovcpe *ggblz j,f-6vsqer and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the u vsv0gjbf *;. 3p(z6,eql bx,- ycogeratio of 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 cylindric.usl3r. ogk, ial plates, of ma. kg.io3usrl ,ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 uy v 0*fyjqd3x;*h3 fa)j75q m;y pejlthe looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track7 ;famh juj03ly;fqjpyxe5*v*y 3 dq)? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assu4yah5tn.ae +c)d+ajr me $r\ll R$ h =    (R-r)

  The tires of a car make 85lw )kir7y1x k5 revolutions as the car reduces its speed uniformly from 90km/h t157)wy xrki lko 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