A bicycle odometer (which measures diste7raawu1 p) 3lance traveled) is attached near the wheel hub and is designed for 27-inch wheelsal31 eu7ra )pw. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocig; jidc9w5eqr()w tl 9ty. Does a point on the rim have radial anew(9 qt 95cigdrlj) w;d/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 acceleration change?

Could a nonrigid body be described by )txe 4 zu.ok5njl 8e2ba single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain.1v13as, t)ns njepvz4

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 t().w h*uzdtw/k,kn hohan when your w,kzdukw*o. h/nt)( harms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal0uczz1lppiiid 77h8bi/ ci) ) 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 pedal, a small front sprocket or a large front sprocket? Whyz c8ui)iib1h7 dc/pi iz)0p7l?

Mammals that depend on being able to run fast have slender loep *en9t e:/l0urh1 gr0)u dwf8qqc3n wer 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 advantalq8e9ruud01 nqpfr ehwe/n0)gc *3 :tgeous.

Why do tightrope walkersh )lw6(kz 2mqwlsqn7di0 *9y s (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque alsogr , -qnd /t0cl.,rtzt.aclj 6 zero? If the net torque o ld-t tr nc6lt ./.qg,,jr0cazn a system is zero, is the net force zero?

Two inclines have the same height but make different angles with th o1(f.i btv2: 76lcqwjrf.ayee horizontal. Terfi.2b qwf7c6jat .o:(vy l1he same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously startd e;eoa w(c-p q -5/amwkx4ilhz81zy, 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 energypm-y(,;/h5xq 4oia d-z k1ea8 wczlew at the bottom?

A sphere and a cylinder have the same radius and them ak*etl8:n*ysdq +;f(b;mc f same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the gre;e 8myl:+t fsbdmnaf*ck;(q* ater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved/udo s+)ng)r y. Yet most moving or rotyud /r sg)o)n+ating objects eventually slow down and stop. Explain.

If there were a great migration of people toward t;53cdb-2exm b2cjz 8d j( dzwbhe Earth’s equator, how would this affect th(j5 e2wbc mbdbd;x82czdj-3 ze length of the day?

Can the diver of Fig. 8–29 do a somersault withoutv hncu .(plcw:wo58*neanv 5 : having any initial rotation wv.8* aon n5wecv:5hpnw :l(cuhen she leaves the board?

The moment of inertia of a rotating solid disk about an axis through itb qqk5jk /h1s:s center of mass isb:h 51k qqj/ks $\frac{1}{2}WR^2$ (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 rotating5l / (dsfgl8(vy/crdb)ue ds52c /adr stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the massfgry ()b52/du 5de/dsd8r/la(c v lc ses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is2 xwd)ixdm0:p hollow and the other is solid. Describe an experiment to determine which is whp wxm): 0xdi2dich.

The angular velocity of a wheel :v yzjuo7b5kl v7s4.zrotating 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 tangentjb.uv54z 7sz:yk ov7 lial 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 turntm .a*fk mqu+w1 le /yu1*(ztoz; hv:atable. What happens if you walk to1*l;az(/h mtz fuv +y1qoau *.w:mketward the center?

A shortstop may leap into the air to catch a ball and throx cw0p1w z0ko:w it quickly. As he throws the ball, t pw0cx k0w1oz:he 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 ofvds:f 2fy po90 conservation of angular momentum, discuss why a helicoptf fpdo29yv: s0er must 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 anglegl(hdwo (c0, es 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. Calculatey/rs0*vku wkuxcux 6+0d85znnd .m;v, using the information inside the Front Cover, the angular diy0/d685u .z n *+rxs xnvkcu;m0wudkvameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam yv 64zrgx2r ba(ithr,9l*hs /diverges at syg/ 2a6(hrr zhr tlx4vi9*b ,an 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 y -h 7;td5aucrw6r-a*k5 rye ca rate of 6500 rpm. When the motor is turned off during operation, the blades s 5-* hykc6arure;7drw a-t cy5low to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball o*pc nqks.t y d ;8-38d-o;g9(mhh sx:qjtbzz wn a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its ty( ;.sj:oqqcmnbsdd3 h z *p8h98kx ; t-z-wgdiameter?    m

  A bicycle with tires 6gvw4f,sx x6. qcvlc-,c3q ls 08 cm in diameter travels 8.0 km. How many revolutions do the wheels m. gcs6qc4,lfxv0s3 v wxlqc,- ake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotatesarqx:wnb a7x) * o/u,u at 2500 rpm. Calculate its angular wouxu,:a)7x qbar*/n 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-roundt 7l 9b3r os61sk:gmzr makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed o 93rz b7rl6s:k o1mtsgf a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Sundc 5weevdn3t2jq. p0bq/y( -e    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spw,v 5jx.u0dt xzh/i3seed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 97id yk6 v 62+j2quupj axej8kcm from the axis of rotation is to experience an accelerationk2ju +j7iyu28696pjkqaxvde of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleratf*h7pf7kcs u .33n l(h wdsry3es uniformly about its center from 130 rpm to 280 rlyhr 77*h(fss3cpu3kf. d n3 wpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius zd8q (nf4lg;3y zpe:m$R_1$ 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 Apollo spacecraft)tuib llq8o1vo7uo 6 ; put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start 6 t8 qlobl17u v;u)iooof their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Through ,9 ipiz0y3ka 4o)xw6 silc(yd ptt);x how many revolutions did it turn in thkz3,it4xx)py0 ia;i9w sol ( 6d y)cptis time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Cal/6wqi qw4se qzvj-8b9culate
(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 ( ttkr/ m1c d;g6htyf1jets in a whirling “human centrifuge,” which takes 1.0 min tr1hf tgt1;6c/ y(kdtm o 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 gxw v0o6y.o *iksb+.i 240 rpm to 360 rpm in 6.5 s. How far will a point oi.o+oy sb0*v 6gkx.w in the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850ruf5;q4 vficf,bje 36qh im5+h ev/min It turns 1500 revolutionie,i b f6fq53 j5+vhfmqh4u;cs 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 mrx .h/(x/h)k+akddnnwo,ol 9ake 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/hdndolrh oa,+k w/x k/n)9h(x . 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 revolutionq8m- j+u s;55hhff mays as the car reduces its speed uniformly from 95km/h to 45km/h Thu5;-+ah jhf8q5m ymsfe 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 p3ku8oy 1am.rc uts all her weight on each pedal when climbing a hill. The ped.y8rk13uc maoals 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. What is thh01pxha+bs tf+o 6 8kwe magnitude of the torque if th+ks+8o0th w b6hap1 fxe force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel b sc-*4 +tvyovshown in Fig. 8–39. Assume that a frictv*o bv4cs y+-tion 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 massles+rwvki4u* ei vyn5/r:s rod which pivots as shown in Fig. 8–40. Initially the rkn/v+u:ei4ry v5wri *od is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require38rvmc e7:y hq n)ic*l tightening to a torque of 38ervic 38n m:h7)c*yql $m \cdot N$ 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 +nmkv gshk8,ua)*pk4 qn/+wi gu t :sp9jjb55 sphere of radius 0.648 m when the axis of rotation is through its ,ku9gk+w a s4php gj8v)it5bj 5:s + knunq/m*center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in div-k0t387kiju1i 0nptepfnc uqve+ 10 ameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignjv0tcv+in1qi 8pe-3tu n k 01euk p7f0ored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, 10vc l-le3dcmt; gs8ylight rod is rotated in a horizontal circle of radiu8 d0s;ll gyvcet3-1 cms 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 5uef9vt 1j.-ljp parq1: 6ea son a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force1 :japa5le1 -st.qp6 evu9fjr between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of3 wgxb8kqgvgw(66tf 5 the array of point objects shown in Fig. 8–43 abowt fvwg k3qg b(g656x8ut
(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 who:wpsy29 9dn.nn ,9as8u/ ha au7woq kdse total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning a4owvf1/y2xj xh:g . :mlyc6cte cuk (0t the correct rate, engineers fire four tangential rockets as shown in Fig. m2l1g:cj:xv6 wkf xuoyt.hcy 0c(/e 48–44. 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 radius of 8.50 cxl g+q+2z, nr p6ermk1m and a mass of 0.xn,2+r1z+ epk6qrglm 580 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, a cwk ll.z70yf6isn; a;ccelerating 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 small hand-driven merro99xb 4+zpxm.egng 3ry-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, neglectingrp 3e9g o9nmx.b4+xgz frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off a5 ea2:weq- x3 uwmwh3qnd is eventually brought u23mex a-u3www5hqe :qniformly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–j1c7qy96xfut66k.;def mo h(ns d st6 45 accelerates 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 tr .fo2o2p p5 pwe8o3zehe forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45weopf 2 2 8.5pez3ropo. 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, a-p tr8:q,.vpo+ m zh mjeizn(/q6; wqas shown in Fig. 8–4pz + h;imqmr( ,-v8zo.qpaqtnew6:j /6.
(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 m4)3ez)6m;pv2 oj5s of;o qujk si+tfm asses, $m_1$ 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 28q n*-yeha b;j+igiewax3 2ufrom rest within four full turns (revolutions) and releases ij; w32iqy8x *bg-nuhe ia+e2at 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 ine itc,w3jw(6li roz9 )drtia 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 qz, 8ayihn0 1+brr1e k5y hl;pa 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 wit:vxfu 7 4lc3eip8x dr*hout slipping down a lane at 3lfvie c p *ru3dx7:8x4.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of th q.9vod9,vmlk9/iw /uc s:n2ykube 4ue Earth with respect to the Sun as the sum of two teruoy:d. 2vnq 4usi9e9kw v//lb mu, c9kms,
(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 rat/(nao ji9nx8 hbr1 h0.zmt8 yanty+gw4 bq6+ dius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it im1t xg/i4(n6tzqy b htj y.anho+r8a8b n+ 0w9s a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 k0q)fjd.qfzmv re 7.0n g starts from rest and rolls without slnr0qjfm q f.)0zde.7 vipping 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 bala*un:h ypplfq:ccce19ae;dpd 8o,0 t.nced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole justta:nco,hcply 1dpe9 0.;c:d*f ep8 qu before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on th9, jse ixnk w+) c/wqr)lth*7ira 0gv/e end of a thin string in a circle of q/xswnik )wha0,ier/l+ gjv79* rt)c radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylind,o5t0l:npb5 /m acbz bh5q g yl(p0lsp7c,2ejrical grinding wheel of radius 18 cm when rotatp5g0y7 :/(p5l tceqp05 az jhosmbl,l2 n,bcbing 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 rotat1 bfqeqes7 v ifd;a:/)ing at a rate of 1.3rev/s If he raises hb af7es 1iqf):d/qe v;is 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) )k9.fw qdo3lncan reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1k 3l dq.9nof)w.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 )fx)gtiin3yj j yt:.i5k,x1 ofrom an initial rate of 1.0 rev every 2.0 s to a final rate of 3 y ix,gtn31:koi5ij)fx) y. jt$rev/s$ 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 axir pw:.ij t.rgkv v+8cqp+(xdj 8d3 f;qs through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameterpxj+id wkvd38tqc ;prg: +. q rf.8(jv 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 b.q/ipgs z5m :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 the E n4q sv4a2sbl gcr7-1iarth
(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 cylindricw 32gn tbx: a5sn,gnob26lq)e al disk of moment of inertia I is dropped onto an identical disk rotating atn)63w2sb qo5tan leg2:g,n x b angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.tf8 28xasj 6uh4 $rev/s$ 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 pla 7wp9c(g0x3b0foiy61q ibh+cp/ gvcwtform of vc ( 0 6qc7w i0wypi+/3gcbbo ghx9p1fradius 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-round is rotating frej5 dwn9h +h:te6u,ba5qp uvz9ely with an angular velocity of 0.8h6q:9jzapht9nuwbe,u+ 55 vd $rad/s$ 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 dwagw+kj tkfbl.l;;(dh qq *2 sp;rf, losing about half its mass in the process, and winding up with a radius 1.0b ( h2.w*+;s;j; klfgltqkpqd% 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)$\approx$    $rad/s$ (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 pa,0yv7, as jtkzxo,1-i b*c/pec/pb2c h :goin excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass d/ +t.(yijer.dj ;xsl$ 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,f2 6pv9 + by0wc-wrrhieeu/o6 initially at rest, that can rotate freely without friction. The moment of inertia bi+p/62yrwu9 fr6hew o-cv 0eof 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 standxz*2d-mg5uk 5lev h*wu/5s j ws at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless beari u /uzsw5l*h-* evk2mwx5 5jdgngs 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 ons xgxokbn1 g h6h(*+q0sy+h,e the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, hole(6gs gh *bk,xy qn0os+hh1 +xding 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 Eaxr85kko5)jzd km9/ibz*x u baps 1 b*.rth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angu.a*5) /ksor5x*u bi j8dzkbpm9xbz 1k lar momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates yhjfc4u l14 ou/+v v:vuoj7z;from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquire 9y8uc jl ieiy(rg67 qmd27;cps a rotational rate of from rest over a 6.0-s interval at constant angular accelerag7i7( yijl2r6cyq 8m9u dec; ption. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, e:ud v /du+jgfgz k2e-5ach of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calcula2v j5- uuz ekdf/g:g+dte 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 ix79i7/pc ncxcs the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but wina )1(bu/nbpxsjt2u: ;qq5g vth mass 8.0 times as great, were 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 proc (aj5gxvqb ;/n2 1pnus)tb:quess, 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-pollutiokw(mod6 xdm9v;y0sp8n 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 unif0m wpsx9d6 m8o(v ;dkyorm cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an*ise7 ,:k9hf rw t8yv,jah)cg $H_2O$ 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 (h4 gihlmm)p v)9oop) 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 -wl.;kqcfa g )M and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8lgw-;qf. akc)–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, aazt uf+ sia3dd /n4if u7(qjb)s2rd5b e )eq54nd we want to exert a horizontal force F at its axle so that it will climb a step against which it restsd7ifddua) nr4jasit5sq( ee2bb/ f) 4 q5+3uz (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on +;x:ad 1i d zam2tbyu3a flat road is making a turn with a radius i dz tuay xmda1+2b3;:The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling ofhl2ugc- t3ib 0 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 req btghluic3-02uired to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin r,o w9gt va6d+8 rxcd3*p 5qupfods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly a xg ,d9 dpao3tuc*rw5pq86v+f gainst her body.   

  You are designing a clutw .q9t*s kiw;lwe2iz1 ch assembly which consists of two cylindrical plates, ofil*we1z.k2;q wt s9wi mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fd+io0m3 coey(ig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the hcye0(o3+di om ighest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ot:q/ffwy8fvc6; .: zwoqgdm-28p w wassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the ca;t. )l 3,s /f)xyspll/dyztdbr reduces its speed uniformly from 90km/h to 60)fyd).lps /,ttb z/lxs dl3 ;ykm/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s