A bicycle odometer (77cif*1x p9 vs+v qy,vb t6wdmwhich measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you wfy vq7*ixb cp9s, vmd1+7t6v use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doesbt 4phh*u ru/-cda((d a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, doesb /4 dt dh(cpauruh*(- 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 denff y9oxv-xtmfh )i l)o1(w,.z bf0;rscribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger cp h;z +-t8ottforce? 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 j0;6k0 rzcjj 5d-yb;wgc2zzg a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram ;- j5w0gc2r0jdc z yk;jzgbz6 may help you to answer this.

A 21-speed bicycle has s7ir .xj5jti-* 7j ynrpeven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a smal jxrn75jipti *-.r7jyl rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have ex dbe3r8e*vug)2 cg4slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advav *eerud geb)283gc 4xntageous.

Why do tightrope walkers (Fig. 8–352mo01 p ulsmz-tc5tfj.bv48 u) carry a long, narrow beam?

If the net force on a system is zero, is the net torque alsolpkd, gix. sz:-mdor/ :8z )ni zero? If the net torque on a system is zero, is the net x.i- /s: )kird8z,d olng:pmzforce zero?

Two inclines have the same height but make different angles with qskq* 6 hds.*vx9(umxthe horizontal. The same steel ball is rolled down each incline. On which incline will th9xsvs*u 6* hm(. xdkqqe speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneousz6f x7ykd2t c)s nfq3s4fn)7fly start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic entnfss x)27f7n4f6 d zq3fyck)ergy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start fbie0j*j): hg hpf73sfrom rest at the top of an inclin h3sgjbpe)h :0f7ijf *e. 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 msvb4q7fz y(a3zm 50mlomentum are conserved. Yet most moving or roz0svq4y7z5b3(m lf m atating objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, hoy)r 41pg jmk6aw would tpjr) y1km6 4gahis affect the length of the day?

Can the diver of Fig. 8–29 do a somersau6ryme if e(a*d+jgy+he49 aqx,fk(w7lt without having any initial rotation w 6hgerak+waym((fxi+7 ,9jq4f eyd e* hen she leaves the board?

The moment of inertia of a rotating solid disk about an axis th uwnahd9c4e x:dqfl*1h0vs 6)d 3h)ukrough its center of wdh6f9*1k: vqd0l) ues4hduxn)c3ah 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 fs3dq0nkbx3367ugpq5 hn3l 4 espnd4holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or 4pdhn433nfb60d q3xgn 5qs7 epklu s3 stay the same? Explain.

Two spheres look identical and have the same ma/;az jme75 qwdx,n ih)ss. However, one is hollow and the other is solid. Describe an experiment to determine which is whij)ae7mz5;d/wih , nq xch.

The angular velocity of a wheel rot( zj co2q5;7 t hw0gp5w+ek; qwyniq3 fw-jc2rating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasiw5+cp2 nk wjjhr;0cq y 7fw qgq5(te2o-z3;wing?

Suppose you are standing on the edge of a large freely rf9*fzw cc5o8wotating turntable. What h5fw*9wz ccfo8 appens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickhm)qi ,*ni-ck ucx(y(ly. As he throws the ball, tkiq cn)*chi xy( m(,-uhe upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angued*ai7n( xmvde (z 0d fma8nz9o(y2a0lar momentum, discuss why a helicopter must have more than one rotor (or z n(d908 xoyn*02e7(aafeizmmad dv( propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians:9pmpxp ou)s2.nb az /) (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. C 5 (qph.o-hdf js)f,qy5wd )aialculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on5fhqp a(qyi d-ws)hj.5) fdo, Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from4bt r9dtgi gzg7glg:,2 k9g(x Earth. The beam diverges at an ankggd x(tlg,24t7zir9gg: g b9 gle $\theta$ (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 8hni g oo7x*yx w5v2m bg+5aj:y .d,l1bptuy7rpm. When the motor is turned off during operation, thxl 255bpv d:.x* j y +ta7g,8onomyu 7yhgiwb1e blades slow to 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 l iog;yg-adjlx*mgse9n 643 tg7 *n3z cevel floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is -a3gj7eo6yng xd g9mgctl; iz3*sn*4its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km1pox68; fg) xl( mv1/iqyvq pw. How many revolutions do the wheel6/go qvfm;p yiwp qv1 1(lx8x)s make?    $rev$

  (a) A grinding wheel 0.35 m in +v.;rk kd9;sndk2p vdls l7 l.olv5m/ diameter rotates at 2500 rpm. Calculate its angul sk.. ol v/pvk9d+25svn ;lldm7d;lrkar 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 revolzz5xz 3a pzccm7w)at3m w:.x-ution in 4.0 s (Fig. 8–38). (a) What is the -czmpwc73z5xwz 3 mtzx :)a a.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.hz+ zw t3jy 8vptdk*qd*-k.x its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a po-j xe lsqu;4*kr/h ep7c++ rplqu r.o0int
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0q7l1xeypggp;9 zp8 x1h w8(hl cm from the axis of rotation is to experiencyxg8p7 p (8wexlpl zq;hg 119he an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acce+p/fl8 n xmv2a;sp b2rlerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Dete2sp 2lpa/;frmv 8+nbxrmine
(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 radiub*x to. 2y,1+j tgaukn:jcs y5s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboardt8)rd3eoh afv82g)c r 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 revolu 3f etv)8hard8og r)2ction 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. Throug- ,k7 s,9wm fvesox., w0gkhrrh how many revolution- 9e,,xhgssrmor f.0kw7v ,w ks did it turn in this time?    $rev$

  An automobile engine slows down frv5:asoddob8 0 veyu;/6(i nsjom 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying ezir)vd9 4)rpg m38 /fy k/d3m mu,lixhighspeed jets in a whirling “human centrifuge,” which takes 1.0 min /i3 d38up ymmfil,e) 9mrdg/rvx)4z kto turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpmglar :pu cz.). in 6.5 s. ): l.urcagz.p How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min Ite-ih2xj rfify1xx n1.b.t/n( turns 1500 revolution yfx.i2/1ni ehnj1xfb- xr (t.s before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the carp4,hwk /9alkn reduces its speed uniformly from 95km/h to 45km/h The tire/ah 9lpw k,4nks 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 cargctbs4mt3c/h03 lculya2x; x 64t: bg reduces its speed uniformly from 95km/h to 45km/h The tires have ac6;msc uxt hl4 b/taxg y32tbgl04c3: 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 u he h f,l37yc k2jj28z1vr4apclimbing a hill. The pedals rotakea f1y 3hvc2l,jzr4h2up8 7j te 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 l og .wva)e,g jx4v*kw-zb;/jcm wide. What is the magnitude of the torquez-b4 ;egg,kv/o*wa .xjvjl )w 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 torquetk,2cgdy+ omwq pqt18kyb8 93 about the axle of the wheel shown in Fig. 8–39. Assume that a friction toq83tkq y 1dbg,2+89op ckytwmrque 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 w;j(w98 *yboun j/mdw nhich pivots as shown nw mjj*d y9w;/bnou (8in 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 system.

  The bolts on the cylinder head of a8 bnw88w z,m(elg*yj*n 0ts ptn engine require tightening to a torque of 38(w8tp,ty ln newjzmg8 0 8**sb $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m yicu-)f lsz03qf,7.oyta x.z when the axis of rotation is through its ceni0 7,al.q .)oyu-yzfs fc3zxtter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 6k3pw:,m1vn jo6.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The k:j,mn pow 31vmass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on7:o5 7xx d n3j/scuya .vmyik nbr23i4 the end of a thin, light rod is rotated in a horizontal circle of radiusiv: 5 4 uo.diyb72x/n7 3 srnkajcy3xm 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s whejw, 7-cqd7,t* zph,l-ppq8 scudq ud1 el rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N total. djp,c*zduqcp87, 7 qd, h lptq1usw--
(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 oc esi8)zg kf..f point objects shown in Fig. 8–43 age 8.k) zcs.fibout
(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 co+ 8ih*x5n ihzernlx 9i-v+c -unsists of 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 cylindriv9;.sfuy8n f-oi wh7 kcal satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellit-h89f.ou;ny i svwfk7e 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 of7+ bq l-gnsiu: 8.50 cm and a mass of 0.580 -7gs+ni:qblu 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, accelerating it from rest9oxb7scnak0aq qf7xn /j 4r )5 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 ava dt7i2g(myhuy-3tz9p4u:r 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 et ugz9y7yt(:ap 4-2 hi3udvmrach 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 rotatin*owoj6gcg g pv 0p9n:-g at 10,300 rpm is shut off and is eventually brought uniformly to rest by agn-9pwpo jg: 0v o*gc6 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 a;g-unn(vy 3ae q/z ik7ccelerates 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 forg/xbslq;nq4 eqghg:g8( .o: d:e3xq tearm, which rotates about the elbow joint under the action of the triceps muscle, Figsge nq:dxeq/o3g.: (8 t 4q;gxl:bhqg. 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 shom,a rhkbp+8c ;accl (3wn in Fig. 8–46lccamhr3(8 kc,a +b; p.
(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, ;:qf/yuow+tmt;kf qh l );9ye$m_1$ 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 (rev)v(ft0kcy2n;l: pd mbolutions) and rtdc;bp2kl 0yf)(n m:veleases 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 inertz:y 9) 1ixgh3hpd9 jkoia 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 1 t*ukg4b2fu69: ikdv vahv 6kdevelops 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 and radius 9.0 cm rolls ;zgo7vqmg+ cco a9c+nxah3 8hd86ek2without slipping down a lan+e 36hgahxo2+na c8k7 v o8gd;cm czq9e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic eac6 y3pz+guqo3, c72r rey(zenergy of the Earth with respect to the Sun as the sum of two terz+7 ap3qyg(u orz e23 reyc,6cms,
(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 nete3 9hv6j k55+emg - wmra exa+v2-)n2tt qwein work is required to accelerate it from rest to ait+vqe2k eg-xnw -heeja5at29 m5n)+3r v6wm 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 starts from rest dnj,.l+k - hpwand rolls without slipkl-np+ h j,.dwping 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 *d1 xwgc2 dwm5u1nv2 eits tip. 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 gx2ge 1ndwd25*m cu1w vround? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the86pg gv:cxm ,p end of a thin string in a circle of radius 1.10 m at an a,pg8:6vx gcmp ngular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform c2srdv,u, ai 9so0pi 9ejs3 kt8ylindrical grinding wheel of radius 18 cm when skd2 s3 ii,ustpjv90,8a9roe 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 plak5hs vhq9 f2r5tform that is rotating at a rate of 1.3rev/r 9 2vhfh5kq5ss 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. 8–29) can reducex ai- :tv*j09bku6jye her moment of inertia by a factor of *tk0jy6- :x9aub evij 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 speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an inic 0rx)f-kw50 1ux afw3tw-v yqtial rate of 1.0 rev every 2.0 s t)015xf y wx-k 3u0fcrav-q twwo 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 thror,fpjb4 cf, fv,pfc( gw 71u+pugh its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequecp 7,bw f,c4 fvf pr1fp+guj(,ncy of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular mrxqa0 il/u9 u8omentum 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 theql-heq+:dw oym8vik+- rm8r 3 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 dropped0n/z.h:e*jffi 2fe z i onto an identical disk rotating a2 :.in ffe*fj0z ze/hit angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2 *4l juhjuuv62.4 $rev/s$ 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 mer v l eti;7r.o(;t6flhvry-go-round platform of ra o;vliehtrf7v6t l. (;dius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-evxajh* jydq7a 6lwj0a84qhj 4l;8;sround is rotating freely with an angular velocity of 0.8qj;a ljha6vh08sj*q4 dwlya7 exj ;48 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about zh1qzuw2 y1/ qv1vrl: 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 integer)z1qu2 yl1 vwv 1h:zq/r$\approx$    $rad/s$ (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 excest,kf;a sw43yzm3eaefl 6w 3nnc oe2 *,s 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 zw-ls :tje,9pwwpg bbd3 9ue7 89xw1t $ 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 platb pt:ln)+p/ rmform, initially at rest, that can rotate freely without frictionrtm b) l:pn+p/. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-ri .x; xqzm.rwxya e.)ejx4i4cy326 uc ound turntable that is mounted on frictionless bearings and has a moment of inertia of yj6;x. eaxi i.er.wcy uq2z4c4x3)xm 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 ropegdwp*is.p x4(31 fki *fi oiz9 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 person withofx(i .fk*3p izs1di4pwo*ig9 ut slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces y0;f3li l8ci4 jh:gnd the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case,yl l0ihj 8igd:c4f;n3 treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratp /*7ybkmsh/ l ,qn3vte 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 z zh waq3+cd )4a8mfv4shape of a uniform cylinder of radius 0.20 m acquires a ro f 4ad ah384mqzcz)v+wtational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.05wyst1cgq8n;k *ax/2p 0 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 ofy qx n1tkg8*;p/a c2ws 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 angry81i0hm: b n jtftoz*0 5m:xgular 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, we6 rlv klrls r:4wvq0ip;6: (wkre rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and t:kr6 (6r:l;qp4lwkwr0v ivslhat the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for akr 7/ii+f q7m*brmk klm;ki:7 low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, iikb/ k7l7*m rfr:; 7m+iq kkmand 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 y51lwfjji0p /e (eox y;vefi2ip;xy:04vx . pan $H_2O$ 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 pmao4a cz ;4:vsn+/plfjzf.v s ,r99p) is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length gxaey5x 4- 6gzL can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (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 ma6-axgg5ey z4 xss of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, andbwueo1 i5m a;9(hk vg-1wb uf1 we want to exert a horizontal force F at its axle so that it will climb a step against which it resto5w fu;gv9(ae11k-uih b 1wbms (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v.bm v4h/ jy/ b7;cubfm=4.2m/s on a flat road is making a turn with a radius The forces mb/hvb7fy/4u. ;bm cjacting 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 .w) 9 fb,ziu2r1mt,inqnun(nb40a wbsling 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 rpm4bmn2n9w0 ,qz nw b batfi).i,(nruu1 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 thiushlugn3w qkcmdsi6).2-e. *n rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds s.k ui2s). hmdn-gew u6ql*c3for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates, 4s+qmvl2lp hc4 u 6gkh8+s-c eof massuvqcmc64 l +s shg4ehk-pl+82 $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fig5tbc i ds4)+sz. 8–58. What is the minimum + ts) idsbz4c5value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do n8r/fjscsy r xwp51 0y+ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces itslx*3zyu4n8 0j( ft hng speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tiyfzh4(ln 830 nuxg*tjre? $\approx$    $rad/s^2$(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