A bicycle odometer (which measures +*, czs)-u dddmsldx-- 6njoudistance traveled) is attached near the wheel hub and is designed for 2* +---s)zs uld6,j oxucnmd dd7-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Do 7lji5hyl.dx1y(fs ;oes a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would t h.is xdjyo;(ly175l fhe magnitude of either component of linear acceleration change?

Could a nonrigid body be describyz0z e0wzy.myr vm*, *ed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a l /xw o5 q) :pucah79 cs0jpfdf.d0y9 nxkyjl.,arger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind your head tha:)v5z yzynk;s n when your armny5z;zy s )vk:s 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 xuol) dg9h .(epedal cranks. I(ud) l ho.9egxn 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? Why?

Mammals that depend on being able to r7gw:b(wez(lqd 3. f ziun fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this diz w b( .3w:e(fl7gdqzistribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carryhr n ;bt:.va7(c)c zedv-q71bpzt2c a a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? Ifqfnpq+xckht;23/d exv+xs 5n.) h u2q the net torque on a /q+ k3)ds2the5xq .ph+v uxc;nxn 2fqsystem is zero, is the net force zero?

Two inclines have the same height but make different/1w 7jk:7qch rp :)as.9mfxxoyzotm5 angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the sq wp:)oo:ka 7h r z5yf/1sjtxm x97m.cpeed of the ball at the bottom be greater? Explain.

Two solid spheres sicl y px5v2wiwt ; erc)u700e.smultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic ene ce7swv20 plx0eurw c;5) ity.rgy at the bottom?

A sphere and a cylinder have thetf97 i+fbtr64xpz:tw5:dkw h same radius and the same mass. They start from rest at the top of an incline. Which reax4:t+wz fripwthb:6597fktd ches 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 momentum are conserved. Yetc0f dk-0mu sbv5m6(jl most moving or rotating objects eventually slof 5bv s0(cj0mmdukl6-w down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how 9aak:b4cd 8k lwould this affect the length of ta cl 8:4kd9abkhe day?

Can the diver of Fig. 8–29 do a somerscb0 rf4 ,bsiyy ;ux;wnrx1jag 4-)e2 vault without having any initial rotation whei,uxjx)nyv4 -r;arbeyw4s;021g fb cn she leaves the board?

The moment of inertia of a rotating solid disk abojme8/ co2(ra)kxu ,l kut an axis through its center of xeoual)k(rc2j/m 8 , kmass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg m0)fm9/b lf8hg f 8.i0 1hoobd emi(bzfass in each outstretched hand. If you suddenly drop the masses, will your angular velo0mogihh.ff 8/eo8 bd)9i1z(lbbm 0ffcity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is holl :7qbqs9kp+ luow and the other is solid. Describe an experiment to dete9 klu7bp+ :sqqrmine which is which.

The angular velocity of a 2 v9cj)u4m/7i v(jejx) d rwlzwheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasin7u)/9() 4jzd2l jixwvj cve rmg?

Suppose you are standing on the edge of a large fr8pf8,/ aksin1,r3vejk tqi :k eely rotating turntable. What h er8kna v3/f p8,1kq ,itki:jsappens if you walk toward the center?

A shortstop may leap into the air to cakj nt9dcb. 28wp1(gbhtch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hipsn1. thdk9j8 w2bp cbg( and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conse9 p w:)rs.r1mivshq8grvation of angular momentum, discuss why a helicopter must have more thansv.8)qh gi:r s9pmwr 1 one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles i/lz8r+x vlg3b4no /sgn 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 becaus00 sx/ g 8dharj-tyoo4e of an amazing coincidence. Calculate, using the information inside the Fro4x- s /r8tgaj00 dyhoont Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direa2,wqai-(pa, +gaxz 5wp 0n lacted at the Moon, 380,000 km from Earth. The beam diverges at anaiwwpz2l0,ap a g,ax5(+q- 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 a rate of 6500 rpm. When-wnl.qk+-o su the motor is turned off during operation, the blades slow to r+ s-o n.qwu-lkest 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 anothuw*3 5ohidx w4er child. If the ball makes 15.0 revolutioix35ww hu *d4ons, what is its diameter?    m

  A bicycle with tires 68 cm in diameter traveldkr3 t/x5k.qoi3al rc zf55t, s 8.0 km. How many revolutions do the wheel3tlockr3i.z55f,r k/xt d 5qa s make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 250jyukq 8 iet 28.93 se undo3fi.gze((m0 rpm. Calculate its angulare3 e.f8i9 dng(e(uyuo.tq2i3 s8 jkmz 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 ck 5oya4iw, uw4r8yvbrw)x(mxi r- *u-omplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child sea 5*y w(u,-4urxm4rbrkioa v-)wix8ywted 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 Euv.ysw lrsk6lyboc ( (:e1i:(arth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear shta1iel9c 850cf0 nl+s 8 lzmrpeed 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 femo):z/kn0x: nt*x2w.oj z b7.0 cm from the axis of rotation is to expe0/n.*wmfn:t)z:zkxoo eb2 jxrience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accele71p*.xorwl -cjes4g s rates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Deto w14lsr*7.j sc -pgxeermine
(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 pt77swygqjyzx 8: ,3g$R_1$ 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 Ap-f*s cff1az8hollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The sp8hfz-1 fc*fas acecraft 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 r. 5 iop4ahvf-tklgl+)/ s) junpm in 220 s. Through how many revolutions did it turhjlf.4igsuo/plk5)n+)- at v n in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcud7p- 1;ts9licoo4aag late
(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 e,ctbv ik/6:-1r :ijk* otcaj jets in a whirling “hu/vacretikjo6,* k :tcb-:i1jman centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to e2:k*sba34dlw8+zrn9lgq vzn)zx 1 m360 rpm in 6.5 s. How far will a point on the edge of the whee q *):m zng+8a4z2slwndbz39 klxrev 1l have traveled in this time?    m

  A cooling fan is turned off when 9sngw.:2 q0cci za(9fv-q msoit is running at 850rev/min It turns 1500 revolutionsqfwqon g9-. (i c2s0va:9cmzs before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make g5dyt6k/arn3 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/hnkrytd/g 63 5a 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 e(v,kytsib-) /0 chmk as the car reduces its speed uniformly from 95km/h to 45/eck )b-(t0ivmsk, hykm/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

  A 55-kg person riding a bike puts all her weight onxuoma6c;e/r4b s)0-w gwk3ul each pedal when climbing a hill. Thxouw3ue0 mg;4l/krs -ca 6) wbe pedals 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 cr /kdj h;wxejt6ymbct*50+csip 5. w4sbz)- on the end of a door 74 cm wide. What is the magnitude of the torque if the forccr).p-ew wcmcb4 jd is 5t;hjzb y+t*650xsk/e 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 shown in Fig. 8–39. A rvt.k30mr +v-xosg(x ssume that a friction torque t.(r ro3mv +0xs-vgk xof 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are atti w3yz8-r)enm8d6mknb. thz 8ached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially thm3zn. 8)idn8kwe-6r mzy8bthe 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 require1 fjcxr ht*e/tut()-n tightening to a torque of 38 t j* n)u r/xh-te1fc(t$m \cdot N$ 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 of3r o ozs50c5zbw.m ,l7yh k*zw inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its centerzyo.7 bzor0sh ,3z5 lc w*kmw5.    $kg \cdot m^2$

  Calculate the moment of;sx/jfy( a 53swhwml4 inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass m s4(hls xjw;a3f/yw5 of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horivv-sum/q u.;. 4oqn ydzontal circle of radius 1.2 m. Calcu-v v4duuyo ; sm.qqn./late
(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 rotatingigm2c svbp+ v;6yp l27 at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N tot7 6p+imcv2pgvl2y;bs al.
(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 cka-2g 4,m kwwd(xv1s in Fig. 8–43 a2, w wsk1-kvgx4( adcmbout
(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 whosem-tzd f 21 xzidph7,hm/op;1w 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 spa1rvif t mp89 6wy+rh-inning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–w+v rp f 8y6aitr9mh-144. 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 radv :k+t+g wtpes)sxl6, ius of 8.50 cm and a mass of 0.580 kg. Calcu,s+)kp t vgxl:ew +s6tlate
(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, accelg)ie0qi e+ a(jerating 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 apo6wnan8w2 ;d small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in p6nw2 donw8;a 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, 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 i kkdy*iog3 ;g 7y3ji0zn4fp6 ms eventually brought uniformly tok*76m0i kigyog3 3 zf;dyj4p n 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. u5(gqiulq/k 98–45 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 thrown9 :am9 l:brbjn solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The br: nmlj9b9 b:aall 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 bego pl8*eyu3j85:tlfi7+y ih(w8yxlg considered a long thin rod, as shown in Fig. 8–46. : gl8lloti3+yj 7y 5hf8pw(*y ieg8xu
(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 oft +t33 wafya umo2sa+ -zc76fv 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 fourf8ms*,wkoevt ) rlw 7kwim/13 full turns (revolutions) and releases it at a speed of 2s il *twf1 w8owke/37vrmm,) k8 $m/s$ 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 inx/t o)o.qe1 8fgtokios 2i6 e,ertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torqu/w 720om6 hnyn4d ucyce of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 908 4en2(s ozb i6kwgh4ubrflkg and radius 9.0 cm rolls without slipping down a lanb440w6sg (9h2in bzlfk8re o ue at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the 8 x mmqs(wd49j)tusnk2yz+ q5 )qu1gjEarth with respect to the Sun as the sum of two termsw k 5y(qqn9djs ug8)u+s1mqxt4zmj) 2,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50bas6j7oo5q , )i 1lluk m. How much net work is required to acce l6,51abku7 )qisjloolerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls without sd7vc:jxv w 1.olippin od.v7 1w:jvcxg down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced verticv37b o m3: wg7zormq:cally on its tip. It starts to fall and its lower end does not slip. What will be the speedq:7crvmz3o7mo b3 w: g 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-k6 7wyuo9 i 05nvxf/pb//w akslg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angw 70a5 ywlkf /6/ni/vsopbx 9uular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-h-k ;iyuzutv- ts2n21kg uniform cylindrical grinding wheel of radius 18 cm when rotzkv-ui u yhn;21-t ts2ating 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 hisud+2i(gr h7c .o+aggwyiis s8. 0e 8wk side, on a platform that is rotating at a rate of 1.3rev/s If he r0o+uk7s.g.w8ii8wysa(h i2 rcgd e g+aises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) c ieh)i +wzqrhh, oqd z(3d9q4*an reduce her moment of inertia by a factor of about 3.5 when changing from the straight positionz*+3ozwrdid)qh qhq(,e4 ih9 to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an init-7uxfxupi)9 wnbryrmd ,1p8 h5 bu9/pial rate of 1.0 rev every 2.0 whx7r urbp8y)9 xn5u1mu-p9 b/pifd,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 rotatin(s vv- hw2 v3v38zrcgbg around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 rz8 gv (3h cw32vb-vsvkg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skad7j 6v:f3d)7utgea1ue2tp bywbol58ter 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 zujy+30wwnt 6nuw )akt(5,f qthe 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 iner9fhp; fg0n; gz:*y1jkskkel2tia I is dropped onto an identical disk *9gylsk k fn e2hz jp;0fg;:k1rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsqbl-.cs)vuq/ 5+0t n yf 5wnid 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 stands at the center of a rotating merry-go-ro4q; h(bkomjc6v(qb r;und platform of ra(q b6vk;rboh4(cj; mqdius 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 merrzq p97;qbclf ;x;i:i hy-go-round is rotating freely with an angular velocity of 0.;z;qxh:iiqf 9c p ;b7l8 $rad/s$ 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 collapses9m+ gz/ xnbj1r into a white 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 nearr+z9bg / mnx1jest 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 e,k q4/6l ;psffdyf+ou xcess 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 tcf1sr3 cjl a,8w8s8u$ 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 h/5.w::ld2osjyje2wsl qsa3q: xa t4 skp2i ( rest, that can rotate freely without friction. The maew25(i/l.o t:: a qhx ys34js:p wds2lqjs2 koment 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 rotnx1/snl5ai ;: .hiof a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a1nxi.hol :/insr5 ;a t 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 i uky6 l4* iu7;jfogv9the 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, Fig. 8–50. The spool rolls behind olf9v*j; u7k 4gii6uy 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 faces gr3fvhn 0jw67the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital wh 7fjv03 rg6nangular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 8 cbzb+pm q k/+86l g.zsggg4u1 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 0t,v9b82s pba- hkpej3.20 m acquires a rotational rate of from rest over a 6.0-s interval at 89pb kj v2spbh,t3 -aeconstant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindricu 43*p1e.hxhs/xc xaral disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric)1prs./ a*4 exhxcxh u3 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, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is th9v) py5n*glt qgf/x(q e 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 with mass 8.0 t :bpap6rc scu,+ 0xpj8imes 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 would its rotation speedpa:r+us bxpcj,80c 6p 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-pol*s* mh (if i1bia 3zeggz4(nd1lution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, am3gi4b g*n1fa* (se i i(zhdz1nd 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 illustratx 5a+sc9a,zywba8l2jqf1;v x es 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 1t fi 07 3w3owzrt1*x(i ffq;ym99avoru*jsuv=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 lengkyh,, 4o9+tsl d ;badlth L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod isdhos+d k; ,ay9bltl4 , 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 ha1bowktd ;/n e:s radius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against w deo/wn;b:t1k hich it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a fd75 4i)t9saj wq/ xdvjlat road is making a turn with a radius The forces acting on thtji j/xswqd )a79v4 5de 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-kgaa18a :h (zc3j.z;a1gghp plv rock 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 required3 cpag vh:(8z1h .aajg1 alzp; 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 thakv xi9* ui+nyx+r/q4in rods, making reasonable estimates for the dimensions. Then calculate ayq ++/n ux* 9vxiki4rthe 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 which consists of two cyds+:teo6jo;2 7fp k/ q,3ln lf9spnnglindrical plates, of maspn;7lfl ofn sopn:3tq,kg/e29+j 6sds $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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(w*c 6nlrdlb jz4m6u;gp 0kb+ of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is kr6p(*jwncumb;0 4+6 lblzg d to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do n1z.lskr96m dm ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformluad; hj)g m()ay from 90km/h to a) d(j;mg) auh60km/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