A bicycle odometer (which measures distance traveled) is attached near the whe2gsl p2ft3d o 1d27sbyel hub and is designed forfsp7y d232gs2lb do1t 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity.y7gua(*3gy zo-c rlg e ;iu8dmtyn9:8 Does a point on the rim have radial and/or tangential acceleration? 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 accelerati :r ;y yuonmlg38987 ugzt-y*(dag ceion change?

Could a nonrigid body be described by a single valuevo7c;p *8alrc s,b6 mw of the angular velocity $\omega$ Explain.

Can a small force ever exert a greak fd0gg1clpxo( d6l)2xf t0d0ter 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 hands behind your head trf28 kyzuf6ka14 .xhvhan when your arms are stretched out in front of vf . 1fz2hkkxury864ayou? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at tho:uz7zh -nln(o ajq 98e pedal cranks. In which gear is it harder tola( 8-nzz9qh7 unj: oo pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able tbon 7;xdf+4kvba- z(u o run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis a 4kz n fv7xu;+(dob-bof rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkerjeloyfqyg 9xq9 3k;q4 n+up0.o;gt,v s (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque al0+ q9 e*meyryxso zero? If the net torqyy+*0qme9e xrue on a system is zero, is the net force zero?

Two inclines have the same height but make q n3e :qhd db,+k+(rh;rlmd: zx v0jv-different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottdr jkvd++h0qz3hm ,d(ve:;:qbrn l-x om be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down:s(b l mk;ezf: an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bs; k m:f:elb(zottom 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 the same mass. They start fjg hwx+jgnu7,hlo sv3.-8b, hrom rest at the top of an incline. Which reaches the bottom first? Which has tlh xbn7,jg + .,ohus3wghv8- jhe 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 momq/+*i lry( vqsentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Erl/sv*y+q iq(xplain.

If there were a great mig 7nxv*c )ho /+tvhdfz x7 kpjidk0,i3(2v iv(fration of people toward the Earth’s equator, how would this affect the pc, 0k(d ixki(ivf7n7* zjh+fv vd2 tvx) o3h/length of the day?

Can the diver of Fig. 8–29 do a somersault without having any init :01q 8g1h9zronq3-; ojkezjtp dmgy1ial rotation when she leaves the :gqn9 jgp1d1- h1;mztj ry8z eoq3ko0board?

The moment of inertia of a rotatingrh6/.gela tgcy+,xufl * qv,)bmw, )v solid disk about an axis through its center of yrcf )+g,aq),*lx vgut6./m wlhb,vemass 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 2fbhr jqqf04 -r4ds3rj b /4ex,-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay t0qr b4-jxe4 , fhqsf/ rbd34rjhe same? Explain.

Two spheres look identical and 0f4-9.ayvzk i 7mpnie /tz qz7have the same mass. However, one is hollow and the other is solid. Describe an experiment to d4 0z-v9pif7inz t.k yq7azem/etermine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In wh2/ y5k4krg- xq p:9zuyta,jajat direction is the linear velocity of a point on the top of the wheelkpyy zq5jtg r/:kx,24 -a aj9u? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Wm/ f;37jhsk bfhat happens if yo7;mf kj/h bsf3u walk toward the center?

A shortstop may leap into the air to catch a ball and throw it30b) 21j/1y5yyhsgrqh bbn b d quickly. As he thrj1bs3b0y52gqby h y rn/1hd)bows 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 ofslbbdg3m:zhd;c l3(o ,:-acp conservation of angular momentum, discuss why a helicopter must have more than one rotor (od - :(c3mglblbazcs,;d :ho 3pr propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angle/mq) 3 o06)j.,2uwv6dql:g rclhgkeb9 dmb aus 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 Earth because of an amazing coincidence. Calculate, usi jrfj56ffazr vh3.r8 h js5y74ng the information inside the Front Cover, t 4 z vrh7aj5s6f3f8yjf.rj5 hrhe angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directedl uq9ui cu g:m(r2/pb( at the Moon, 380,000 km from Earth. The beam diverges at an angiup:q9b/ ucl((2ug m rle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blendy1q3a- (c9 xz9f qsjgzer rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slowyf9z zq jcs13-qgx(a 9 down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to ano uo o6:j/h7mf3 ap/vikther child. If the ball makes 15.0 revolutiu37 vh pif k6/mo/oj:aons, what is its diameter?    m

  A bicycle with tires 68 cm in dp-,wh .u xqeur5 o-umkct,5n;iameter travels 8.0 km. How many revolutions do the wheels mrmx pton-,e;hqc w -u5u,k.u5ake?    $rev$

  (a) A grinding wheel 0.35 m in diameter :ywoi 7fgqx8f + oy;8dj*dhv/ rotates at 2500 rpm. Calculate its angular velocitjwy;8h7 f8i+doxv/y fg*d: o qy 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 p ibx1 civv;;4complete revolution in 4.0 s (Fig. 8–38). (a) What is the lv 1bv; 4ipxc;iinear 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 o)z ns 4(3bgq/wce.:xvquehrh) w;* gvf the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pos +4e 1ifpfn(bint
(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 7.0 cm from the axisq)e pcmj6eerv(8o* c ( of rotation is to experiep(*8cv (ee)c6oejm q rnce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly 4au7ll 7kl3mn about its center from 130 rpm to 280 rpm in klmua4ln7 7l3 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 radiushxd+le3 az g .0,r2xyf $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apoll6 x3pzgnj4r4dhlh)g, 2*ni 8d (bbssuo 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-dns2 nbh( r6s j*x44ggp,l)hz83 ibudmin 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 unqiw-u2 v 2g;tgiformly from rest to 15,000 rpm in 220 s. Through how many revolutiq- w;u2tggv2 ions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate4 +9uzkqbreo5m1x l, q
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirlinwm q;slq6u0 *5 b6 k(*wqjn/mbr;( itwap /ssmg “human 6q ksbw*/w m5;lbj0pq(/imm wa6nt;s*(uq rscentrifuge,” 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 accelerates uniformly from 240 rpm to 360 rpm in pi*f8jy-t.h0xw 7 hvu- ;rt7uodu7os 6.5 s. How far will a point on the edge of the wheel have traveled in this tii 0vto* j.- dyxwu7h-upfo77;t8s hurme?    m

  A cooling fan is turned off when it is runn*ylfz+)na c /gm2x-dning at 850rev/min It turns 1500 revolutions before it comes t* ny-2mx )/fdzn +glcao 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 speed uniuo(r z.+ib-c fformly from 95km/h to 45km/h The tires- z .ur(bcof+i 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 uniformzv rk 7c;7 gqtm0(0uifly from 95km/h to 45km/h The tires have a diag0qt7 ;f 7krmu0 vciz(meter 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 eacs-wms9bgvayy4 g 7+6ql;m5v uh pedal when climbing a hill. The pedals rotate in6qvbssa9;ywu- m + 7v5ygg4m l 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 74s t9u(5;qi gs5 n6v964xn:;viahh1nkpz goi t cm wide. What is the magnitude of the tsi6x1o;t9z5; qkihiag 5nnn:6gh u4s(9 ptv v orque 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 qut1-i lrs(4fd8,u qmof the wheel shown in Fig. 8–39. Assume that a friction iu 18m d,-srl f(4tqqutorque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass 77hyz3v 32+ wvejtzxt m, are attached to the ends of a massless 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 torq2vyvez7w7+xj t 3z 3htue on this system.

  The bolts on the cylinder head of a 0i6 depn(:ejgn engine require tightening to a torque of 38 djen:g0e6i p ($m \cdot N$ 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 it.b7(zs383bu bi 3vcwsi.rya2 xf/ d snertia of a 10.8-kg sphere of radius 0.648 m when the axis of ru8rxbv af .bz3 d ts3/7(2sw.si3byciotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inx+ih- nfi9 c;sertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.2hc9 nf;i+-x is5 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, ligegem;ll ktt8sl:c7*+ht rod is rotated in a horizontal circle of radiultl* mct;sl g+8ek:e7 s 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 wheel rotating 8ww 7o6y ob 2drdub;k)at constant angular speed (Fig. 8–42). The friction force between herbdw6o;wk7ro b)2du 8y hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of o2 :b d h 9x53ma.hcook,7st gouuc1;c-v 8dfginertia of the array of point objects shown in Fig. 8–43 aboc, 75ag 8f k;2.ucvo3gohtm:oxbuds o1 -cd9hut
(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 tota:pc7+ p nk 3xqk) /k)aymked)dl 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 ate7zk ld,,rl8.kld 1 u9h x :upnn.a8vq the correct rate, engineers fire foul.p9 hkld7re,klu.anvu8x8n1 :zqd, r tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket 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 vsg)0( ge;qn7 * kr(yq eb 62kttky/jgradius of 8.50 cm and a mass of 0.580 kg. C( q2st7nb e kg/(yvtyk ggqre0j)k6* ;alculate
(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-yiwh ,8cg nokg,x4p9 rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven ghk rz4ari;37: zdn c,merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merrczn ka,r3 d;4g7: hizry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut o- l(y1hlfcac6ff and is eventually brought uniformly to rest by a frict -1cchlay6lf( ional 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 acceltfasfmfe,,, i* q/w e)erates 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 balb/9o 8aqen6ah l is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–4abe89 qh/6 nao5. 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 cixs*h(8ov4m.3lf5n pukhe-s an be considered a long thin rod, as shown in Fig. 8–4 s fou(lve.-5xh 8ps4h m3k*ni6.
(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 consist/ l:)s13 mvk7uwhgol ss 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 from rest within four full turns (rtbh+ *-ij,i(ofpp ph(a:)nnqevolutions) and releases it at a speed o () (pp*hnb,qfati pjonhi:+- f 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 mozal pnqc /scx/ q1k1: -pi .d/med,pu.ment 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 280 1slyu mpjfp *.+w xbolk)j;)6$m \cdot N$ 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 rols0:p7 0 wy5ds8fco)ecykf5qhi m 7op5 ls without slipping down a lane at pcdiy5fmsc:oph8qo)0w e755k sy 0 7f3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the suaob t/3ha kn+fa/-h d1m of twoh /h3ko/nft1- a aad+b terms,
(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 mj 8r)iye+v5 )mnflou+ass of 1640 kg and a radius of 7.50 m. How much net work is required to anm)+)e y lv+oir8fuj5ccelerate 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 k5-/w7x). ad1losghqag.p sj eg starts from rest and rolls without slipping down a 30.7.opdq/wxg as)s a5lhg1e .-j0 $^{\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 startspss6+e1ma8nza1t4mcvy y6xf d. l; e( 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 ground? [Hint: Use conservation ofpa1c41vezay6ldn6 ty;e m8(sxfsm +. energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating omlq5 tjb lq /3e p:-w8mhhy0k9-bmy9cn the end of a thin string in a circle of radius 1.10 m at an am0c 5 9-eq 3wtym/j8mq :hhbkllp-yb9ngular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding t6x .tkj 5;v+jgk2xhl1a dksc0ue-g 6wheel of radius 18 cm w xhckxkj6152;l0djt-t ua+vk ge6g. shen rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rat1xqw2*dakuxs/bz ;- bl2h q3fe of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–1;/x3sbu2*k-fhb qlqaz w xd 248, 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 rhur7)goy m(cm+s 1 8kleduce her moment of inertia by a factor of about 3.5 when changing from the straight positmu1 m7hls )o8rg( ck+yion 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 xy8( zrgyr0/hd.5 aosfrom an initial rate of 1.0 rev every 2.0 s to a final/yx(h. 5ga ys0r oz8rd 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 center ati2l(c+* oaftl 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.i+cl*ltfa(2o What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3.5 kcct2 2 c8vf1c25dij a$rev/s$ 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 mome5 kijob:g-2gbntum 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 vqd .o 2d12;t1rieah 6 fi*up+zxv8 auof moment of inertia I is dropped onto an identical disk rotating at 6u fov2+uev1a d2aq ;ixhpi*1 r.8dzt angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.x3 n(o**jth :ovy7zwq4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round platfo6mt;:yc (*w ,hxpo oyorm of radius 3.0 m and moment of inertia 920 ;6,wxmo(yt*: pcyo ho$kg \cdot m^2$ 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 rota(ibh -q gb4rh: de.t2ating freely with an angular velocity of 0.-g ahti4 2d(eq h:b.rb8 $rad/s$ 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, losing8+uo+q 5xh cdcht++s m 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 c8thhcd q +u+ os+xm5+no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excessf)(q7gb-mzx i 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 6brwyfs eeso0:z;*5 k 9 lrz*v$ 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 platfolzv+i-f k veh ovl z-ht59md)84r*3uq rm, initially at rest, that can rotate freely without f 8h9if-uqvrv dv*k 45l)zz -he3lt+omriction. 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 thedwh2 yf4y(a2g edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of iny yhag2d(24wfertia 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 th-z+hpek+b*u i e end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope ezup i+-+k b*hunwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same siofapgm8i u**( de always faces 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, treat the Moigpma f8*uo*( on as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 19 w1no .h -bfm1yiprq1 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 cylinder of radius 0.24dd5 : yiof4t+dyxi; e0 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calcu 4ide4fd+ yto5:ixdy ;late the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass fs9i+365l)2w g;i 4hihmuue ) ohzjbq0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conseh29;iobjzw+ g)seim6f h 5i)q34lu u hrvation 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 the rear w) qx xa9).g hm7leyur(heel ($\omega_R$) 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, bu10tu5xkeo;qhau , dj,t with mass 8.0 times as great, were rotating at 5,u;xude,oj1htq0k aa 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 that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pb;q7b e cm7c *4q5wcv, glao5dollution 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, and should be able to tr7*q, owba5 c vqe5dc47mbcg;lavel 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 illustra x3nrqafq6wug+9 kfhb22 cfh08inym sa3r8 :8tes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a ho+ci m)d8/dm /i61q. kuzsw.m l pk8vanrizontal 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 57a r4edgu stl wp (hgb+yl30-i-r/pp friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At thp+u / ber(04 psr-l3 g7-dhla5 tiygpwe moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertxrxvpm x(3t -7fp58 ptj/r (vqically on the floor, and we want to exert a horizontal forj v8(pp (p7 /vq3rx r-mft5xxtce F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling wit:zjt3 kppj ( nb(,orc/h speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cyc :tkjbjpo/, p(c n3r(zle 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 lev qfrrw0.c3sn, +6fg cngth 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest toc.rf cq3w6g nfrv,+0s 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 -h(1-o 3 e7xronhxe zt -,rwoja solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular sp1 tjn3o-, h 7xw-r-(eeoozh rxeeds 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 cylin l32rsu; ksdh+dhy2w: drical plates, of mu2+:yr hd3hdwssk;2l ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Figdkhsusw:y( oj73kwh37 ,:kyc. 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 t7 ys : uk,sjkc3w73dywhh(o:krack? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bgwt 0/u*8eepkut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reo y2 yvi(yw--,w.aa waduces its 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 owi(ay2waa,- -y w.yv 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