A bicycle odometer (which measunw21yvyqb v,qa0/ kdeq 7e0tp,,c- syres distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it qp -a2eckqyyvd ,q bn0w,ste107 v /,yon a bicycle with 24-inch wheels?

Suppose a disk rotates at constant an9 (3kl +0xuoaahvy2xg gular velocity. Does a point on the rim have radial and/or tangential acx hu y093ov x(aga+k2lceleration? 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 of the angular velocityxe h7q2kc (wte,ue9 b( $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Exp/*o ;diwah 8malain.

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 wmuaq4 oacou qk1s 3qb:577hm/ he6,lwith your hands behind your head than when your arms are1esaucqa q4 klhb7,:h6/w7m ou3oq5m stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the 0fhugiy ;g.p72vk bh2 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 to pedk;.pgg hvbh 2 07yi2ufal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to h : s1t-;.ektu lbb*znrun fast have slender lower legs with flesh and muscln; kt1b.z:* -bhetlsue concentrated 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. 8–35) carroks89up ci3 3by a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net 6,kvc.cr 0z9jo sdb2:ee ate3 torque on a system is zero, is rat9 d :ocvsbe,3zj6 0ke2ce.the net force zero?

Two inclines have the same height but make different angles with the horizotd mth8 ogyax 9oj.0hwo-p0*oc5 s*h ev;jl96ntal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greato5 o*o lp0o*vt e8 w ms0dxjc-hh.th699jya;ger? Explain.

Two solid spheres simultaneously startr h9f6o.hhtor oo4op 3**zpzj1d 23oq rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reo1 3.qj dtoz 6oho* r4rp9*z23f hhopoaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and th7r;4x. uqy dmv.q/ vwg:dx q5he same mass. They start from reddvg:5mu q xq.h7.ry w x4/q;vst at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum hkog99;ais2kup)d 0 dare conserved. Yet most moving or rotating ak h09 k9idgo s)u;2dpobjects eventually slow down and stop. Explain.

If there were a great migration of p(ks- l7asv0td eople toward the Earth’s equator, how would thika (sdls0- tv7s affect the length of the day?

Can the diver of Fig. 8–29 do0md6 89xb nxwy a somersault without having any initial rotation when sndwb0x8m9 yx 6he leaves the board?

The moment of inertia of a rotatii+i toz9hj uu t)e5e,1ng solid disk about an axis through its center of mu 9j u)ez+et,t5hi1oiass 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 inix8ya o) i2i5 u1h1jqq each outstretched hand. If you suddenly drop the masses, will your angular velocity increaqiuj 5ix1 qoi)82yh 1ase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is ho0 cwct /hw0b/n) le9zqllow and the other is solz nl/)w 0cqhc/t0bw 9eid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points pxdw5*fs lw96west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acflx 6 w59pd*wsceleration 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 xl+b;8iub 4v m4 h((brw6g5cjam :il v2ea0ahthe edge of a large freely rotating turntable. What happens vr; 6bh b w45+ li 8a(lx:(b40iav jgmhe2muacif you walk toward the center?

A shortstop may leap into the air to catch auja/1/u*j3 fws/tb vwvo l03 h ball and throw it quickly. As he throwj j1u*alh / uv/os0wt3vb3wf/s 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 o-qdh f.)su2bp47dbl conservation of angular momentum, discuss why a helicopter .uq-l o)bhdspfb47d 2must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following anglr -c9s 8ysip; ltk9x;ges in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Eaxj1 pjsnos2f-3 lr1i7rth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Ear 1s3ln1 isjfrp27o-xjth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km frp2fq.*s2 0blt)i xybm rszj4-,e gb(hom Earth. The beam diverges at axe(t- yfi.jss 0b *4blpbrq2g2 h)z,mn angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When /+8nw ay7gkz c6p2dgothe motor is turned off during operation, the blades slow toy2 dgak n+78 cgpzow/6 rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball mys d7a*0rrga.:ac 0 vnakes 15.0 revolutions, 07r dnya0gscaa*.v: rwhat is its diameter?    m

  A bicycle with tires 68 cm in diameter travelx/qs -*;rr.dtekv g: ks 8.0 km. How many revolutions do the wheels mak:rx.-/deg; kv*r qtkse?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate igjia3sbg . nw1elszt-:2 6w:r ts angulae-bls gni:z61tw .a2jr gw :3sr 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 revolec am fl*t/m:m -i1q+eution in 4.0 s (Fig. 8–38). (a) /meq mc1em+ft:-a* il What is the linear speed 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 aroundsw a2oa )o.r5t the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of(vyw;w7l 45 si1nx7qt ;lfjlw 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 1nq;a)wg rv0p if a particle 7.0 cm from the axis of rotation is to experqgn wv ra1p);0ience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelzho( b3y3 979b +a kkkce7rthlerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determin 7by b9z9k3o7 krel3takhh+c(e
(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 radius6q txu(gf o z2.hc*nr+jz4* zq $R_1$ 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, astro,*mztn :jswf*nq a6sx +f0ka4,1 yiignauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecrafzf,j ky s1n4+,wfa *nasii t*mx0gq:6 t 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 3p0jk w+ kt2i/-osp mytxy.(r15,000 rpm in 220 s. Through how many revolutions did it turn in tkw+ x2o-t (ky0s. pr3ypij m/this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm 9pb*ia+fx f* gin 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 stressesx1g44we t 9pfa of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachi 4ef4xpta19wgng 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 acceler,7re +zd4n uk(yf,k c2oyniqt 1,mk(tates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this tki4q ty,n2+71( zdoumt(f,n, kcry keime?    m

  A cooling fan is turned off when it is running at 850rev/min It turnbc59sh3 my, moglgx u1(3qb q8s 1500 revolutions before h9,lqq bm x8(m53uy1osg bg3 cit comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its spi.gw whdo z6x;:h-0(qw*8lx b)firwueed uniformly fromxw hgd( :w rf-6 hb;.u)wqlo* ixiw80z 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed u mou..vdk(el0;* dzxjniformly from 95km/h to 45km/hdv* lxed (ujkzm;0. o. 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 pedal when olgt1c45:uva climbing a hill. The pvo1lcu5:ag t 4edals 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 +pepd,xa g6*g. What is the magnitude of the torque if the forcpdp xgg,a* 6e+e 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 q8b7:8hu e daxf62tz8kt: grk 6 d;dfhof the wheel shown in Fig. 8–39. Assume 7 ug6hdd eft8 f;xtqz:abh k:68rkd28that a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a mass89u98ycudrc nless rod which 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 net torque on this s8u9rcu 9y d8cnystem.

  The bolts on the cylinder head of an engine reqthw1l6 un(, fruire tightening to a torque of 38 (tn61,hu rwf l$m \cdot N$ 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 o ef35hw/cersv mm;x.9y9 l5rqopc5+ -kg sphere of radius 0.648 m when the a h5vrcpwc m5fel;9m93or.xeq5yo/s +xis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bimx)/ ry;f1flm cycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub yfx;)mr1 f/ lmcan be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball onb((sc0 e,w cqd+xdlrj(+ juc 7 the end of a thin, light rod is rotated in a horizontal circle of7,b s(+ c(cjl( +qwex0ujcdrd 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 vr2: rp-/ kwi1qg fsl*yp.uk- a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the ck-:qyp/rs vlipg 2r .*1uwkf-lay 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 array of point objects shown in Fig. 8–43kixd:( a70pdn abna 7(:ddkipx0out
(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 tot4k8 za c-pdjn;al 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 sp02cjcf dqq./2y8ay9i gmwo93u )mjv oinning at the correct 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 uc89. q2oo/ji 29f3dywmvgc)yaqjm0 .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 cy-3b9g4u ksg j abm:oe*linder with a radius of 8.50 cm and a mass of 0.58094 3eb mbgo*kuj-g:sa kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, l 2n9 bi/rrj ybus/fn8(t583sjw(7h we cix t2accelerating 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 4xq fn1 7dfa1u)vi iv.small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 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 frictionalfq1d 7)4.i1nf a uvixv 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 7i k/a4ixhqy-gth+ l5 2mum+veventually brought uniformly to rest by a frictio4h2 a g/iivy ++5m7lthqkxmu- nal 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 acciyd +iu;8ltv(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 the forearm, which /4 hapd gtw8)g+,xcq +okrbj) rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated u ,) 8wr/kh+pqgxcbd otjga4+)niformly 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 shoaf+ 3yt-tm t 9n6hb/b(.8jf tvg/ocsk1 melx/wn in Fig. 8–46 6yl f8- 3xbe./mb k jntsh (t+ga1oc/9ttfv/m.
(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 consistswslxl1et5*zk 6 mi /e- of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer 7jps+sd ;ypg g*u/rf3 from rest within four full turns (revolutions) and releases 3*g pjg ;rfu+sy7dp/ sit 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 momeno3, if gn kkn-57zvod2t 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 of//h+ x c.itta/l4 ht+yxrw;4 rqmm*ig 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 slippiv7vtfe;)b 3iing down a lae3fb 7vi;vt)ine at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of t np(zaw4-yja8u*u9i)gdb9 tfhe Earth with respect to the Sun as the sum of two t-u (4y nu zd9tawb9fgpa)j8 i*erms,
(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 oz:jpj3aohe 5+f 1640 kg and a radius of 7.50 m. How much net work is required to accel5pjj 3:e+z haoerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starhukx i )jv*.y1ts from rest and rolls without slipping d jk. *i1vhx)yuown 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. It st flzdfn0 7*e)farts to fall and its le7ff)z*0dfnl ower end does not slip. What will be the 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 momentu,zvdtxcu56*v5 of 6 zfm of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10d t*6f5 5,vx6vfcozuz m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of + 0h63n ppiq-owbx i/7fn.hxla 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotatfp7.ob 6-3h qixliw/+xn 0hpning 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 aj,uwh; f2c)yf(8ml f8bjd j5.biat, t rate of 1.3rev/s If he raises his arms to a horizontj,( ;cb8lf5,tim f.adw hy8jtf) u2bjal 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 moment of ie3a ufa 2mlrl,:uk7.xfsd-d8nertia by a factor of about 3.5 when changing from the straight position to t2x fl.rau:de 8lf-3kmu d,7sahe tuck position. 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 rotation rate from an initialp1z0 bz (jkl fmin9:;ih5j,z v rate of 1.0 rev ev ;i(vz1z ,j l:pkjfi h5m0z9nbery 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 through its c 4r2jp8kv250u-i 2dijxjj qvf enter at a frequency of 1.5rev/s The wheel can bruj2ij2ivjv8 fjkd p02- 45qxe 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. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skksmt)1df .goy4uc4c 2 ater 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 momee*x .l0kh( h m59vvzqg 0tr-nentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an identi8xpdhh,grp m:qs/r1zv ,-* an calr,qra: 8x- g/pmdn,p1zsv h h* disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 mktkr ,m x;bf;bsfd**bd,*/t$rev/s$ 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 standp/o zfo69e1)ndym usqu 1*bq :s at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertiab9pdy6ume1o)n q1 qouz:fs/ * 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-goia,r 2e3aw0b.64fibygwox3 m-round is rotating freely with an angular velocity of 00. 3i2af w3bxgiw,ybome4a6 r .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 as3e r uuwyv:+vczt3zzz.h-; m*:p/qa white 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 angular momentum, what would the Sun’s new rotation rate be?(round to the nearest intp;+3:w u /:3.- hevyqzratvc*szzmz ueger)$\approx$    $rad/s$ (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 ex dv ai,aw /-i;acu krpj1-*c/f1i2og vcess 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 arj igvn f( b87-z )aww+z,vq)$ 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, l-+ .*ghhywr7 aapjx2 that can rotate freely without friction. The moment of inertia of the person plus twj+al p h.gr2 -*a7xyhhe platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6. pqjrg*; 7.v ;scqf:xwd)qqk 55-m diameter merry-go-round turntable that is mounted on frictir;d5xpvqgk7 j *:sq;fw.cq)q onless 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 ls36oa,4uc5w/ p mo r/miwp-rv ying on the top edge of the spool. A person grabs the end of the ro5/rc-o/pam4s w3wrp o, v6 iumpe 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 mass move?

  The Moon orbits the Earth such that the same side alwaysft5+ *(fh z)z*tr cpbb3zskd; faces the Earth. Determine the ratio of the Moon’bbt;f5(zf+*rzhs 3 d c z)ptk*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 ofaleiulyq4h(1 m5zmvk -k3u3/ 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylindl2b:b9-vw fxwz 4tsc. ix jc63er of radius 0.20 m acquires a rotational rate of from rest over a 6.069s wi 4f3blxcjbz2. c -vt:xw-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solidu7egss:x4wyh . dqu,( cylindrical disks, each of mass 0.050 kg and diaw y4q.u (s7xu,edgs :hmeter 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 rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of c+j t2.z. 6hixwnmaa8the 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 tim 1nmz*6evw l 07vv+kmk (r2mades as great, were rotating at a spee0vd(r6* k+am2 v vk emzwnml71d of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? 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 g7fc6zl sar/36 3mhw :y4wuswn a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.5wz7crww 4:l h agm6 /33fsy6us0 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 an z)iwyjc cd0sf;. rk5 2$H_2O$ 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 ty,gx6tu5x: dar66auty zz9: 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., wqss7bzwts.f ;+ u09d ;po 0luoe ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the 70 ss+;ool0up. btuzsq9fwd ;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/0totsdtcrg ) j ,h*z/. 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 steh)osc d/t,gt 0r*t/jzp 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 f l/qh675 x,3v waybbqhlat road is making a turn with a radius The forces acting on the cyclist ah6hy /awq 7bx,lq5v b3nd 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 iu8e ey(randgd xte.8bt 8* :c*nto a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does a*.ntgdyurd :8bx(*eete8 c8the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cyi-ywr,p2 nz8 fjzu0*o bj y *.szi2,nclinder and her arms as thin rods, making reasonable estimates for the dimensions.oc,fz,8 ibuy.jp y 2 -w 0ns*j2inr*zz Then 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 clutch assembly which consists of two cylindri (tkl s51mhf*mcal plates, of mass 5lm1kf(hs tm* $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 of kmlk4upw+ zkn10x:v3u4g,t o Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it iskv4ux3uzmn+o1:k0w,l t pk g4 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 assur 2i1/s5tbz9z.6yt mea.pfr 2ctzcj x8 *vui:me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car redux+ r4nten/nqq c07) vaces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angulc rqn) ven/+7qx na04tar 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