A bicycle odometer (which measures distance travele9f,( -0+ bihjo uvgzcid) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use uh9c,j(0io bz -v+ gifit on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the ri7q e x /qaziiqb75 d5*zj20bcym have radial and/or tangential acceleration? If the disk’s angular velocity increases unjb0qb 5qz dax*e7zi2 c/7 5qyiiformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single vao3x 6w09mt jzslue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater 6gvc-,wu l5 rl 88qwartorque 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 difficultyah(/5l :lwl t to do a sit-up with your hands behind your head than when your arms are t lh:5/lyw(la stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at thp .5qetk51 3i,p bfwzse 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 tokp 1et5iqsb ,w f3z5p. pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with fley)9fpy i)c92vcged -f *1kol msh and muscle concentrat1)2 *lfogd99-efcmp) yci vyked 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–sris(qv ;j;1s35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the eaw7l14 7*yne udw6mr net torque on a system is zero, is the6 4wu7wmyeera 1n *l7d net force zero?

Two inclines have the same height but make different angles with the horizonta:ie()g7 b gnd3q* rmokl. The same steel ball is rolled down each inclinoe bgin*(:) 3qgdkm 7re. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (frol71bf350a ub2fcnxzm m 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? Whi b mcx0f51z 72l3afunbch has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius mgu,r5jni-*a h5 , bsmand the same mass. They start from rest at the top of an incline. Which reaches the botia5b-m 5 jhnu*s,m gr,tom 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 momen ef)enoata 6(l 1/z47ibgen81q1pjn wtum are conserved. Yet most moving or rotating objects eventually slow down and stop. Explain.n 1nt4g 61l8)aeef/owjp aq(i e1zn7b

If there were a great migration of people toward the Earth’s ev l p3(n: 9c+pnal-z;t zobh,do8pfj1quator, how would this affect the lpz+pc :t(z1 ;l3p8-b ohd,fnn9l jaovength of the day?

Can the diver of Fig. 8–29 do a somersa*z 6pxvs iv2.*x vju;yult without having any initial rotation whens2xivux v *.v pj;y6z* she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its center 4p3e gp,wl ,hoykk.2o of ma,e42hol,gk p o kwp3.yss 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 nai.7 x pprg,/holding a 2-kg mass in each outstretched hand. If you suddenly drop ther.agp xnp, 7i/ masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and haecj7zh j /rd b/lbj*zg m-m4vui6 989ave the same mass. However, one is hollow and the other is solid. Describe an experiment to detedz 4jm*a86zrh9gjm b- vb9 cje/ li7/urmine which is which.

The angular velocity i: dp5dl*u e1tof a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If thdde *p t:5ui1le 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 ./fey 6f*v 0,fr7pt zr, qnbtp4wam n-the edge of a large freely rotating turntable. What happens if yoa 4mpp6tf,z*f.bnt7wf y0 v qne/rr,-u walk toward the center?

A shortstop may leap into the air to catcyt.x in ,3w 8.jm5sfxkh a ball and throw it quickly. As he throws thet,kf5jsi ..nxy m3x8 w ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conse2zi(yq m5n; p6asc* uhrvation of angular momentum, discuss why a helicopter must h*sc2m zpa(;nh qi5yu6ave more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in/v kr0tq,4g nm 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 ofb+ wcy;hg:3a m an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as s3a;wg byhc +:meen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 kr iv 7:ucfs-h/2pr 8vrm from Earth. The beam diverges at rf-h/c8ui7 r2:spvvran angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a refqr8 5a+ twk;ate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 5fetqr;+8 kaws. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a leviw)b,mq /j r+rel floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its dii/qwr+brm ,) jameter?    m

  A bicycle with tires 68 cm in diameter travel nxxy8;o8 c.hsgq97le s 8.0 km. How many revolutions do the wheels makng8;l7yex.ho q9x8c s e?    $rev$

  (a) A grinding wheel 0.355nq kno yxgw0469f qq.*rhx. u m in diameter rotates at 2500 rpm. Calculate its angulax.94ok fynh*gqqr.wn x u0q6 5r velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete rev w0bbleu , fw2(v0caj ka4,ne)olution in 4.0 s (Fig. 8–38). (a) What is the linear speed( uelw042kb,bw0jacv, )anfe 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 8 m apa+4wks.tiyii,,f tr6-pEarth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sp+xh*zu+: 8 gw qtw)rs: phx:ymeed 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 rotatet)zf b4a(0gyw htk1 ;e if a particle 7.0 cm from the axis of rotatab 0t1kyz)gwh; (e ft4ion is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheej6gmcbv8w,ridu- e. ) l accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determj wcm6,r8. eu )bi-vgdine
(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 radiuxzd0yfnn+/ f3fzwn cw((*8 eds $R_1$ 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 sp lb0g i(h.a,;rskqpc +acecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation c slkq.0p;h+b(,iarg 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 tzz 6r7f-o-ktd o 15,000 rpm in 220 s. Through how ma76 tr-- kfzzdony revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Cu glthkefi lu-f hx2 yd.79:t;gwh0(.alculate
(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 hf18 a(zs p1dykighspeed jets in a whirling “human centr fsd1kz1py( a8ifuge,” 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 accelerate s**3 utjv;cbz1irn ,gs uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this tim ,v3i*1t*g;zjrncubs e?    m

  A cooling fan is turpj2 2md4cem3x+f s7sk5v.p xuned off when it is running at 850rev/min It turns 1500 revolutions before x 7pks5pm2vsfcux+em 4dj. 23 it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformlu4wz.z-2cw txu gw c+7y from 95km/h tuw4- z gc.uwc+2zx7 wto 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed unifpj9n(:a3 . u5rzdml*kb qj d+yormly from 95km/h to 45km/h The tires9mdr*l+ynu .jbd jp 3qkz:( 5a have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weig1ye 5 z a(p0g6;hfxs ,de(ciddht on each pedal when climbing a hill. The pedals rot epxeyh(61; icfs d0zad(,5dgate 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 the magk: r2d:m tay8gnitude of the torque if the force is exe : mgat:82yrkdrted
(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. 8–3wad3 c-) q+d0nn h-wdm9. Assume that a frict da--w3dnc0dmnq)w h +ion 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 ty amymb ;dk+2u hh/.8do the ends of a massless rod which pivots as shown hu2hk/a ym;ym8.b+ ddin 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 an engine require tightening to a tob15 pg37vpnd vrque of 38 p1d n5gb7p vv3$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m whe szt;izv d:7u(n the ax(vtu:7 sd ziz;is of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diametl0gx3z ,l(cyir 23bps er. The rim and tire have a combined mass of 1.25 kgrpg23z(l ybl0c i, x3s. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horize5 ,eb)9 rzd 078umtskkt ud;iontal circle of radius 1.2 m. u kds z8,70tdk;r95temu )ie bCalculate
(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 r8x. /4 lfiou)+a qttgcotating at constant angular speed (Fig. 8–42). The friction force between her hands and the claytagu)tq . oc4i+xf8l/ is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig w,7fb g6u0zaz5*22e( rx7wq hv tajre. 8–43 az7etverr2z07ua ,6(w qjfa wbgx h5 *2bout
(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 o4;,docsa0se3lp ks. i rkv3 )rxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, enginixhm)t mu37 o+eers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4. 7)imu+m3txoh 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 withpj uw)8cw26l e a radius of 8.50 cm and a mass of 0.580 kg. Calcul p6wjw ul8ce2)ate
(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 rest towe-dshg :( xa5 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-dr w- d18jhvs3jggwt gah 5+j;q+iven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform diskt8j+wgd-jh; 1g aj w5vgh sq3+ 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 ac t5iu4d/ 3fdvt 10,300 rpm is shut off and is eventually brought uniformly to rest by adt/d f43vci5 u frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 )z 6c68fb dz+ro/ cv-dy0f k1aorevv+ 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 the foy ha5zgt4uc bgv5g3 p(74d, fkrearm, which rotates about the elbow joiz4 t f53g,aypc(4uv k 75ggdhbnt under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8–ejw*jl oz9 fl-(: z8vk9e0eds46. we(fe0d8sjk:ez -9 9ovl *j lz
(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 h (ihls ;*aqm;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 resoq )soyog vx6)fz78e8t within four full turns (revolutions) and releases it at zey )oq)8o867 f sxovga 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 inertia oe8iavp kdd.ua(238wgf $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torqueia ul5h0rg1bgnys h39 qvsq6z09 d 81g 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 ie36x:b4 hm:tyv ar+kwithout slipping down a lane atkhirv+43:m 6xet y b:a 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respectr*ndvos--tw) : :jb.uo; duoy to the Sun as the sum of twon : *ojodrvs-y:)o;.bu- dtwu 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 mass of 16406ue 80xyy gj/z;y)v lg 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 i0x/jlvgez8)gyy 6yu; s a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80pase:i(cs.17d /b 4fncopbi ; kg starts from rest and rolls without slipping down a 30.0 snp( so;:p1 ic edbi4c./f7ab$^{\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. s1 u8nfp bt1flf*d :c2It starts to fall and its lower end does*: sn8f pdlcbt12f1fu 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 1qmer4v+f ,)sz,j ;hm5. 5rqcuxwc qnmomentum of a 0.210-kg ball rotating on the end of a thin string in a circle om f 5q5c wjqr c z1nr,ums.;4v)xh,q+ef radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum ofa 2*7,vd lcw s(6czsat s.v)zpr. zz(z a 2.8-kg uniform cylindrical grinding wheel of radius czaa.,()z sr6cd s7w*t(l vv 2zzp. zs18 cm when 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 akr8ej8 bm1/iot his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8k 8irjm1e8 b/o $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of in dh5ibf1)nwl1h 9 j1 ovqea)i8ertia by a factor of about 3.5 when changing from the straight position to the tuck posit1 5)l)1hiihaq e j8bnofd9w 1vion. 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 increasz rc9+7zi8t ( hey8yxbj7jt 0ae her spin rotation rate from an initial rate of 1.0 rev evea(zt7 jc8 txe87 ibr+0y h9yjzry 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axisyo h:e1c)z37 ;tjtiys 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 chunk of clay, appje1t;:s3yz tyo7i)c hroximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of aa y;g*aaookp* ( hw4f- 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 oj )zx0xj4 dx(vp3cw/p( hroex v86/fx f 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 fv .vwe92a)bccwq6q :lfby )9 identical disk rotating at angular sp ).v2c9: lb)6v qwf9wyeab fcqeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns4x*:e fivieyl:r *r7-lq 9tm zh 1eap5 at 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 4y parh1l,5a ih z74vpstands at the center of a rotating merry-go-round platform of i 14yh lr47vaah ,zp5pradius 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 freely with an angular ve6nevx7sshfm1k6wif.8s 3 tk r5l a8e4locity of 0.6 831wk mk h6is4ars7elsextf8f.n v5 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 ifsm7+qs45 wxwhite dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearestm fw+45qsxs7i 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 exceen3le-tv w 3+6/ubyzxhqqf( -ss 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 ozp:gsfnscw vd1, 3+ +e1uh 7hz5f+oy$ 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 freely withanbwf8 m43(q uout frictionf(4buw qn3m8 a. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.zb3 kk2t //uc;kuw *bi5-m diameter merry-go-round turntable that is mounted on f3/cw bku ikz2* /but;krictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground , h:vrkum(n( zwr5.hvwith 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, 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 o.nh(w zv:,km r h5r(vuf mass move?

  The Moon orbits the Earth such that the same side always faces tbb 6f zp 05pt+bi /+cvf;jnte:he Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In thc+ 5f/pijtfzt+ b6;b0bp ven: e latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at pco7rkp 9g/ 7dn*b8mka 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 b pq :7-zedp,t6rqj*acylinder of radius 0.20 m acquires a rotational rate of from rest over - dqa*pb,e: r jt7qpz6a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, ea6e ey,e7dfrh) i.zvd0ch 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 its 1.0-m-long string, i.eeid0vred6z,f7)y hf 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 wheczy rej.z:4fso:k f z:sa,g8- el ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as trc(33lf 3 pklm r2n67kvnwn2great, 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 would its rotation speed fkpv3 62rt ln r(nc7n3 lm23kwbe? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is foryot2iw,kci r,1 ma ;p7rouacf (4 )0vu 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 kgi 4aku2f(,u7c1;ow yrto r,i0 cmvp )a, 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 illustrateotko x,-wh -xv/f k,j(s 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=3.3all xmt2*jx8 pm.)n9 k $m/s$ 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 ak0qi p5p owfzk3;r7, (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angu3p 0 f,karqz75wi p;kolar 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 stanz5fu4z psfvv 8)w v48dding vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step asuwzff pz448v 58vv) dgainst which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist travelingzs 23cjf)qx p8 with speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and c)3x2f psq8zcj ycle 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 a9/zh;c:n/ak9s8mwwf5 4u wzraxhhi3o ,oz:g 0.50-kg rock into a sling of length 1.5 m and begins whirling the rock in a near/m8;3xrw w:gckwz9hhzs4ih5 ,a on9a /f: o zuly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her2c if;k .vk8ue arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for kf;ev8 2ku ci.a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly r no4tw2lf bdv.7rb:9which consists of two cylindrical plates, of ma.o:b flrbrw297d ntv4ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loo2w* rgjte5 m:lz syy4(ped rough track of Fig. 8–58. What is the minimum valu(s4 mzygwe:jry2tl* 5e of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dogjxtuyhrn-29pmh nf1cg+4+) not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly fr)i1 vp ;l qe7rdvm:9gkom 90km/h to 60km/h The tire7g 9r:e l1md iq;)vkvps 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