A bicycle odometer (which measures distance traveled) is attached 4zjaoa qm3-r3near the wheel hub anamjz -ao3 4r3qd is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on -hsb9ygcvw z)i4a0l9q k*jy6 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 woulj ywgl6- i ) *caqbs0zhky94v9d the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velop4tcnik/lc4aa4e 8 0 ocity $\omega$ Explain.

Can a small force ever exert a greater torqpp *w :uv0m6ifd74k+dtxsb/uue 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 yoq* 8zdc.9w2eh omju)y ur head than when your arms are9 ecm q2*z8odyujw).h stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the zp( jb4);wivjbiys 6f2.z/ix rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to .jxi z wi)2y zps6/( 4fibv;bjpedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with dc el*;+,f jrwflesh and muscl+ ;c,ldwe*jfre concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a longep27j6so- iy.w a9 urm, narrow beam?

If the net force on a system is zero, is the net torque also zero? I0u)c +q /9eabupb1ogu f the net torque on a system is zero, is the net force zua p+qeuoc ) 90bgb/1uero?

Two inclines have the same height but make different angles withtdz7xbu /i3jr 1/y3xxn(3 qnn the horizontal. The same steel ball is rolled down each incline. On which incline will the speed ofj (7xu/1i3/ndn3txryzxb 3 qn the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inclutkk w(fa;e; 9ine. One sphere has twice the radius and twice the mass of the other. Whi(wkfk 9a;e ;utch reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radymq 9 t3;rjk 6zx0vu/dius and the same mass. They start from rest at the top of an incline. Which reaches0;m tz udykj93qv/xr6 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 bwp a+/7akp0cj htd 88momentum are conserved. Yet most moving or rotating objects eventually slow down and sto tacdp7ka+8b0p jw/8 hp. Explain.

If there were a great migration of people toward )w 2gv7i/ww6 uchs. jaa:ow1z-s0n orthe Earth’s equator, how would this affect tw0swa ).wa v/go nurczsh:21 j -67owihe length of the day?

Can the diver of Fig. 8–29 dx3xl)s eekv0) tzq bq ep7 n0 svzw*4l;oo,+z8o a somersault without having any initial rotation when she leaves the board?0z0ze+v3o )q;n kl 4we8sxpvq et o*x,7bzs)l

The moment of inertia of a rotating solid disk about an a9tazr( yb3w qsqq 74u2xis through its center of masz4yqt 9w sua7b3q2rq (s 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 o ke0pzpg -2mvezf )/mq,9-euholding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explaiz92g,/m-voeeq )mkf pez-u0 pn.

Two spheres look identical and have the same mass. However, one is hollow and tt)* vkmxu (qe-t1i(y7 2udkvahe other is solid. Descutv(t 1kqiku*(ema7y-d)v2 xribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle poikna+ 1*3a52y vqawq cvnts west. In what direction is the linear velocity of 2cav 1w5y*+a3qn vkqa 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 decreasing?

Suppose you are standing on the edge of a large freeydjll2 7yqsk qth a26g4:hi8:ly rotating turntable. What happens if you hl:ah 8q 4 :gsqk7itlyd26 2yjwalk toward the center?

A shortstop may leap intomc *q.ofoz4r *(;m xs h c8n6rsa.+ikqk1mr7m the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you lookrqncm 4rz* f(. mc68xho oksr*+ki7 sm. ;a1qm 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 angular mo-; 3wfnnu3o:+ poz soomentum, discuss why a helicopter must have more than one rop -now3suofn o3:;+ oztor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radiat1*ttkr .w l/xns: (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 642gf hljoq(/e1mp lb62eq iecoincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon,ibe22 e jq 64f6hqgpoelml(1/ as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from E7 yipmqc nys3d; 9n8c 9wmu.b,arth. The beam diverges at an angle p9u9mc 7,m8swyn bcyi;q nd.3$\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When .vu; kzyz)y(sog wnf :e 61lx1the motor is turned off during operation, the bly 1:z1ynsfu.(xz) vogl;we6kades 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 level floor 3.5 m to another child. If the ball7xwr 9(i)w e-shp+99kr zvj rq makes 15.0 revolutions, what is its diameter? epx-hww)r+7q9z rv( rji9k9s    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Hci aqs/ i :gqix;f4/4fow many revolutions do the wheels 4i ;igifsqx/f /q4a :cmake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotateso(x1bqbbq)p34ixq v 6 at 2500 rpm. Calculate its angular velocity in 1bq b4bq(ip o x3xv)q6$rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution jg7f25.lzq 2ddw:xya in 4.0 s (Fig. 8–38). (a) Wha72:x 25zyql fad d.gjwt is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around theqmxsn6d2 xi:v;ux 8, o Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poinyeh d/ 27s 4qfuog 3pshts-6p*t
(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 axis cw/l0x)hy ,br of rotation is to experience ,w b/0xrhcl) yan acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130g )sxlu5o1f0df ,4h hwtj,/vc rpm to 280 rpm in 4.0 s.o1x,ls 4t vf/h)ud50,fc hgwj 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 radiute/rt lx3xcq 8r: 0a(ys $R_1$ 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, astrona) ;e czge+z :7vnucin:ivyq41uts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenlycz;ivneiq7cz1evyn +u ) g4: :. At the start of 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 resj (czyjh2+x 4h cu7e/ :qmey)z-odk9 ot to 15,000 rpm in 220 s. Through how many revolutions did 7(c4zed-yuczj :qoy) k+ ox2ehh/9mjit turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm t1kqys (si:9e 4adei *do 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 highspeed jets in a5ls:xa* wzk5b whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching lk5:5zxabws * 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 )yqx ixa1si7v28oye sr+- u1o 240 rpm to 360 rpm in 6.5 s. Ha2sso18x-iy ory7+q)xiveu 1 ow 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 850kjb 3xu:dot6fib2 w1:rev/min It turns 1500 revolutions bj:k:b6wu3 tfd12oxbi efore 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 make3,ast64 :q/: 0 mub( mw+gycrfsvzri1mwbw/u 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tus0v 3cyw b,wmmu rm/:fsbr4iz/( gq1ta+ 6:wires 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 shyyle 0n/znc+t2un(e.0-ssj peed uniformly from 95km/h to 45km/h The tires haz yun2-s/ecjy0sn nt 0el.(h+ve a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbingy(u vbvegjv6av3 tv 3.82qpz8 a hill. The pedals rotate in a circle of rae2(8gtvuz vva863vj3 vpb.y qdius 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 wideyx g vl05ydgw7h1zq)rq; am.0 txkv 68. What is the magnitude of the torque if the force is h706) vy5zx8w.qgxak1m;q gytrld0v exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Afp :gc ;.ej() nitwcs5ssume thani. e) 5fgc:;cpjs( twt a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached t8m j.; 0zv/mvu8 aiordxsp.v8o the ends of a massless rod which pivot r mj.vzim 8xpdu.0 v8/;osa8vs as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of anu d3slc3,u;pn engine require tightening to a torque of 38und l3;s, u3pc $m \cdot N$ 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 0+.gnjki c-7q thw( nyof inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is throu(i+ygjwnk.hn t c-q07gh its center.    $kg \cdot m^2$

  Calculate the moment of inertia. .jmhow: n :ni7eem1b of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (wh :bnmoeewim1j:.hn7 .y?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated av1 3blu8skw-in a horizontal circle of radius 1.3 kuwa v81lb-s2 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 wh sab h/9c4besb0o,lniw . b.q:1imo)meel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and s mb/ac90bbnmi ) q.o 4bl:1,o.hwseithe clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of ogiruw ved (w:v.86,p the array of point objects shown in Fig. 8–43 aovw,( 8w er 6gpv:i.udbout
(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 t:w4xzfpssj0( :1b q azg;vx4 motal 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 sateawrmm; wj(moqp.2x . /llite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. mq morw(.j/pxm ;2 aw.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 radius of 8.50 cm and a mafb-t ate /+w hb+dyt7*ss of 0.580 kg. Calculate w/e-f7* tbbda htt+y+
(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, ai iy34, sjaajw*ds3z/ 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 :o7vr dry* 7wea 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 vro 7d ye7*w:ris 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 off and is rtouf j;72sa)xfd +n8 eventually brought uniformly to rest by a frictional torque of 1on)jrx2s ffd+8t ;au7 .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 accelergrnjes +,(0nc liuwlev7 r,m(n, (mg q:i +4cjates 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 soleloww*oii du+ vmo3n7;yu.hg5 h,x 2tl6y by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest u,w67 u3;o mvh o+d2giwth.io yx5n*lto 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 thgfg:nij+5f ltkj5s-p2 ekr rkc dn )w79-t+9k in rod, as shown in Fig. 8–46. 9wtn)k+kckekrfid -t57s2 f-rlpj9g g+5 n:j
(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 con-zu s(ubj) ujpg n-h0jp s*2t-sists 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 fou9 sqz e9(pteqzz*1q (lr full turns (revolutions) and releasessz9 zp1qq t9e *ezql(( 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 modu9t6r(j1cb 1n3vzfac 1q)amment 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 d r/jaxxhrm+zg (:v7)hgv *5llevelops 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 without slip ipa6eb takdvtz3v33l3:z4 z7 ping down a la7evbi:lt daz3 v34a3zpzt63 kne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect +ko*i.us 6e5ogl 0 0eloqly0xto the Sun as the sum of two ter5 o+xie6k ug00o.eslq oy0*llms,
(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 ma sajg 3-jma.d -dbu9o/ss of 1640 kg and a radius 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 is a solid cylindea 93gja.b /ujms-d o-dr.    J

  A sphere of radius 20.0 cm and mass 1.80 kg st2 5s15 -ecovpb,iw*vifx ha)larts from rest and rolls without slipping down a 30. f)*2-li pcv51s5 aibo,xvew h0 $^{\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 starts to fall and )6w 02uimhnui its lowum)h nw6i0i2uer end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210- we(dsj m*g2ts4, meiu/zm ,8tkg ball rotating on the end of a thin string in a circtz,4m se um28s *(iwm/j gdte,le of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum*(6a2f mcdny(2pdcb lgt,ko 7 of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 o*7ng abmcd,(ct 2(pf dk2l6yrpm?    $kg \cdot m^2$
(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 ./l nl8hwl 4n w+ha:ho(w(g vurotating at a rate of 1.3rev/ hwgv l/ l(:unw(h48wnhl+a.o s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in ).1o/jdtir m m ko2+:qyzo j9eFig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angulardqt2o 1moke) :zjrimy+j /o.9 speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can inc+dq wdrav* s5w(c39xcn:k *mdrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 sw dakvdc*(+n*dxs m3cw 5q:9r to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a ver11 ab6bs5fdxv tical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg ch 156bad 1bfvsxunk 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 momentuj-+k)8ry/w ;(m+ooisvc ega lm 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 momennxg/ c j77eo. -qz.0qiyp) tm7ytu-xhtum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an(xow 2lmzouo 45i-jc9l, eg b1 identical diskmbxw jl1e,oo (2gic 5z9lo4-u rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns mv 7 pqsdvnj;y,i)c) 0z ;j:tcat 2.4 $rev/s$ 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 ces 7w.sq hinvjkg+xl 1((p-j 6 cr2e3sdnter of a rotating merry-go-round platform of radius 3.0 m and moment ofjs(+j(d6k31.x v2 hnwerqs s-7cpg li 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 rotacpj 5 /e ;yz4v,,cffazting freely with an angular velocity of 0.f /;a4 czc,pezy5vfj, 8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white x4r+ qks1khma 1m3iv .dwarf, losing about half its mass in the process, and winding up with a radiusmq+31sha rk14m.kvxi 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)$\approx$    $rad/s$ (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 3q b*gnn,/ vmj*m5gplin 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 +yo:rydn ;l0l 5*yxw71i*iqo*nx0qlp u c3av$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freep73t;hx e m4(ws,gv blly without friction. The moment of in,7hb m xs ltv;g4(w3peertia 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 merryd h/rfr1 5)5np+dtb ry-go-round turntable that is mounted on frictionless bearings and has a mo 5 pf/yn)b rrd1d+5rhtment 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 ropk76ws,kzf.7a;rvj6 vyy/ftw2 c rn 1ye rolls on 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, holding onto it,v7c.a66y;z17y2v wrwjkrf t/ ,sfkny Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always lu2um+fbybs b3 -wu 4:faces the Earth. Determine the ratio of the Moon’s sb+yu3bmuwsb f:u4- 2lpin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at avar 4 ;g.anx:gkg u(;a; c+fuv5v (vfj 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 ,+slbh0y(zk:b3.1 hksb npz* :r u16nwzusi tacquires a rotational rate of from rest over a 6.0-s interval at constant ang kb:t nun,1bl3 s +uzyikb.1* hwhs6sz(pz0r: ular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made +if0kolp ;gqj zv26r .of two solid cylindrical disks, each 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 calculate the linear speed of the yo-yo when it reaches the end of itz;g0jq.rk 6olpi+v 2fs 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 wheelp 2bqs r25/ htvfhw6)i ($\omega_R$) 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 ofq/n dnmr15htygo9ggbl/p /8, our Sun, but with 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 process, what on8/lnm/pg 5g/d,brt ygq19 hwould its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution auikgwbozi 7.ah n+2kz4 n( / a9bi*chb.tomobile 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 travel 350 km without nniza7 khzii 29/ wbgbnoc .k +.ba(h*4eeding 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 illustrate;zt7 nzuou( v, rfp n242zc9ies 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 horizontal surface at speed v=li(*gy2vzs1mv3jh 09t-y/ v imgff 4j 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 fricti0 tn,c lmohp;4on) 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 accel,pl4mc 0;nht oeration 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, and w* :j/ s uik mpneb(m80jz,nif;e want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has he 08pi ;emfsmzn b,:kj*n(j iu/ight h, where h

  A bicyclist traveling with speed v=4djklyak 1inle9-d 0,1.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cyclld91d yk1el a-0j, nkie 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 r8;lgtr. 3sdz tock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, ;g l8s3rd tz.tand where does the torque come from?    $m \cdot N$

  Model a figure skatemr2tnr:w 0r*3jeaoy /r’s body as a solid cylinder and her arms as thin rods, making reasonable esr rw:oyj amt*e2/n30rtimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly whicg9m9 .uakv t .)ugzbxvcik*+,h consists of two cylindrical plates, of ma ,mbvkgk 9.i)z gvctu+*ux.9ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fig.u /5j9v6epqge 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 jq6 eg/ev59pu without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do n+2cg wy )cw4f6dtevr8ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutionsd+mbrl32imr: 4 dv5vif6l ;wv as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular 5mv 64r2vd:3 viw dfli mr;+lbacceleration 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