A bicycle odometer (which measures distance traveled) is attached near :s:an)k)vf dt. *w3xs2bdj*suhv3 th the wheel hub and is designed for 27-inch wheels. What happensfxbutsn *:3.v a3djw hd2)) skhs:v t* if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on tkp/izl53n m3i he rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either cz5i3npi 3lk/ momponent of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocibm10qck5( q tbty $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explxt2av xt( vn, 9,lj4iu8l6 v:fpp5tjzain.

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 yo;3 e9cdh9fv65 dxzb *asp ;megur head than when your arms are stretched out in front of you? A diagram may helpfgcpa;d svhd *3;5zb9m x6ee9 you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thr72y;c sl,8jvgkkir un j+gj w7x7k,6eee 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 pedal, a small front sprocketvju7xj,ijw; k2e7k6+nk g8lsy7, crg or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower leg:1 ;iv,cmifwygc p+ b-s with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass-;p1gcfmcy wb: + ,ivi is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a l/ex 0k 7..qwwqonmvm4ong, narrow beam?

If the net force on a system is zero, is the net 8tv2 bci tf1gm/ f(b9dtorque also zero? If the net torque on a sysmcbfbgv9/ t28fid 1(t tem is zero, is the net force zero?

Two inclines have the same height but make different angleezy- fp/c m27js with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the2/7cfpj zye -m bottom be greater? Explain.

Two solid spheres simultaneously start )n48 hung a,ft. zlr2orolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which fl n4,tr nz)a82 .guhoreaches 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 h90 jx0e1j lhnrave the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total xr1eh0n 90ljj kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum ar7pw.1zqn mg6m/ m3+m kuf pj+qe conserved. Yet most moving or rotating objects eventually slow downgpm16nkqjpmf/m+m 3 +q7z .wu and stop. Explain.

If there were a great migration of people toward the Earth’s equvxoexp ; ;sa17ator, how would this affect the length of t7 evsx1 ;;paoxhe day?

Can the diver of Fig. 8–29 do a somersault without having any (v, 9w)wd uts.7ycxnqv.kam*initial rotation when she leaves the boar)xmwv7a* qny.,s vk .ctuwd (9d?

The moment of inertia of a rotating solid disk about an axis through its center hrtpcqea63x+,-j4q hof massxr, e6ch-4+a jqp3 qht 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 stot*: i7pg n5rkxol holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity in5*t xn 7gpi:rkcrease, decrease, or stay the same? Explain.

Two spheres look identical and have the same ma:5qk -l(cx w aj6rlaq9ss. However, one is hollow and the other is solid. Dq-wj(x:a a5r c9ql6lk escribe an experiment to determine which is which.

The angular velocity of a wheel rotating ondgg+xug tjj+h2hyu 3;lu.82 e a horizontal axle points west. In what direction is the linear velhu;t j2de 2 yju.83h +guglgx+ocity 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 decreasing?

Suppose you are standing on the edge of a (elid:k8 *tv4 w+y nfularge freely rotating turntable. What happens if you walk toward the center:fi8 tn4l*we( v+duyk?

A shortstop may leap into the air to catch a ball and throw it quick.v 7id. 2nfnsts- un;ely. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.te .un-nd.v2i;sf7s n

On the basis of the law of conservation of angular momentum, discuu(rm.u9noi7i9 zn. y iss why a helicopter must have more than one rotor (or propeller). Discuss one (ouzi.99iyrm uni 7n .or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians:i) w 2f6ga7*jgt wj 8ieet7np, (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 6rxr/ke f 8eze0cx:/z.mzn (ocoincidence. Calculate, using the informati 8fn(6x0:. kemzrzecorzx/ /eon inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direc n6fzc;yfs 0q/ted at the Moon, 380,000 km from Earth. The beam diverges at an ann 0; cfzsyf6/qgle $\theta$ (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 rgpknqize .)1a )f:i4rate of 6500 rpm. When the motor is turned off during ok4p ) a:zf)1.rgiiqn eperation, the blades 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 anotpc/c-k mc8xh 1qrr )*pher child. If the ball makes 15.0 revolutions, what is its diamerhk1- 8) c cprc*/pxqmter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do tec8-4 nxyhtd t*5,ovw )juj:r he wheels mak4u8,5et wn -hr)ct vd ojj:yx*e?    $rev$

  (a) A grinding wheel 7 zv ,tkh 7j 3;c qvymz.bw92r*t7iavl0.35 m in diameter rotates at 2500 rpm. Calculate its angular ve9vwvvctlty 3 .qarh2777;*jkz,bmizlocity 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 completjbv 1/l, zd0jj8m jq-re revolution in 4.0 s (Fig. 8–38). (a) What is tj / 1jbjm-,8lrzjv 0qdhe 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 tc g :b4j7ki3h3owmi 3g65acq velocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed219g5edznf mi+gp b,z 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 particlep 4,.:yhir ymc 7.0 cm from the axis of rotation isimrp:yy.4 ,hc to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerav s1vs4;0tbm vtes uniformly about its center from 130 rpm to 280 rpvvstvs14 b;m 0m in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius dnubiww ul .5b;00d1s $R_1$ 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 spacn dk/3(ji6cud ecraft put themselves into a slow rotation to distribute the Sun’s energy e k i6jc/dd3nu(venly. 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 unifo4hv- pkf(dbb)rmly from rest to 15,000 rpm in 220 s. Through how many redfhv( 4-)bk pbvolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 vc95kb4k:bqy.9c l ryq)b fc4rpm to 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 a whirlin(acou3019k 77f o1hcu k 6bykkhkt ;hxg “human centrifuge,” which takes 1.yot161k7x(k9k3bku;hah 7fco ch u 0k0 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+*yfvg+5,q4 kho blloen7p0lz6r5dj 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the f0pzeh*o+dl4qr6lk 5v+7j o,b gy5l nwheel have traveled in this time?    m

  A cooling fan is turned off whenfv13c8 xj qkx: it is running at 850rev/min It turns 1500 revolutions bxjkx:q f3c18 vefore 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 makm,dh;c9 whire0m rt z8yh 3:h/e 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter o /h mrmh:cz wt;hd3eh iy0r,98f 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 red d7kfpti.w-m7 uces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 fik .77mpdtw- 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 we(/ui/h0( rl l wep1*8 qktancvight on each pedal when climbing a hill. The pedaik8lt h(vru( p1nce/ qwa/0l*ls rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is the mavpy,x t-)-ap bt kole0k:*t-wgnitude of th-kt)0o pta:*ktl--, wexb py ve torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–395 vdtwp8;/2ya8wwikf . Assume that a vtw;ipad/2 5fwk8 8y wfriction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of .x f*4+zs; t2ghn/nmabv0tp va massless rod which pivots as shown in Fig. 8–40. Initzphxs*.vt t2/vm+gbf4 n n;0aially 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+q9pzfsfv;o 19 6bpw9zj ;mgq to a torque of 38 o;m+91pvp6 qfzw9s; gqzjfb 9$m \cdot N$ 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 g yzie*+alg8.m when the axis of rotation ia .e *lg8zg+yis through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The 4zl*) 5:b(z3civfmv5msr-di u9kx xu rim and t -4 ( zimvm b9 li:c*xz)r5vfduk5xs3uire have a combined mass 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 rdv r/wtvok(sg; *1 *qz)nh7v bod is rotated in a horizontal circle of radius 1)v;vzqd1vg kt(*obwh 7sr */n .2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s whet3+g ha1/hoa rel rotating at constant angular speed (Fig. 8–42). The friction force beht rgoh1aa3/ +tween her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of polgrf+ffm4rfw we/ r,j j-d11( int objects shown in Fig. 8–43 aborf(jrl4j+,f1fww ge1/ -rdf mut
(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 con c*j9:ak76k v2ix1df1 p*xt m dhdg3yasists of two oxygen 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 corr 9 jhipajca/8;ect rate, engin9;pajai 8/jc heers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylind5t6xdrr)( uxg dbmv i0x;t(g 3er with a radius of 8.50 cm and a mass of 0.580 kg)rgx(0rt6xi;5uxm btdv 3(dg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swingo-:x7ogu 9nok bm(t;ls a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-goc+5r ks ; xzn;x+2fe(.jtk esc-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kgxz ;k ;sne .2f r5x+tcs(eckj+, 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 10,3)z/dynvbdnbx q l5*m1 q0gypy2 ce)h:6p8o8z00 rpm is shut off and is eventually brought uniformly tb ym5:pbvc2dynh gz1x)q/ly oz8 pn* 06d8eq)o rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accele0 zkb mlnr qo:z.2x5yte//z p5pdb)l, rates 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 a3er. +wev/ng sbn6qv -ction 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 3v. grnq/ n-swb 6+veerest 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 ro)bo/d+opbeo- 66c1bue/ bvcgd, as shown in Fig. 8–46p/eo)+ bcgou/6e- bbcb6dv o1.
(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 consistsq hn qodf+ipz3( (6kv5 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 hammerf+ 582tq;0o ;ozwopg+ceb eyx from rest within four full turns (revolutions) and releases it at a speed ofoczw 8fb2pq+;go x05eoy+e;t 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 oip,ntgr:v69kig:; zqb /, cedf 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 * gjcol3b p7be:guf0 )develops 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 (0f8d/mp skmywithout slipping down a lane at dpsymm/(k 8f0 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to t50 w9ir lttl4(65n.o / glr,cvrf;jhwr*xosi he Sun as the sum of two terms,w590 oo gxil 4fscj/v*6rtni; ,r( h5rw.tlrl
(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 13n( pj40 u3qxbqvpphe(uh/yixvf 5- ,640 kg and a radius of 7.50 m. How much net work is requirup3(ep n/hqf(x0ybu , 4v v-35phqixj ed to accelerate 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 mnx /3twz( (qu6 ve7cihclii 04ass 1.80 kg starts from rest and rolls without slippin wuq0(/36tz xei cl4c7 i(hving 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 vertically on its tip. It starts to fall and itsyp 31izmw2b xi1djd0 q.xuzdc)hd* 1 / lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of jw30. xip ddcbyu 12xh1dqz*1idz m/ )energy.]    $m/s$

  What is the angular momentum wpz5.u qtkrp( 8a c,jb)fqu2+of a 0.210-kg ball rotating on the end of a thin string in a circle of radius(zqr.u+p j )ktb2pqwauf5,c 8 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.:j r;.hpdfl m(8-kg uniform cylindrical grinding wheel of radius 18 cm whe(hm;plf.j:dr n rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at hignwhl;)c 6o 2b;(sbzms side, on a platform that is rotating at a rate of 1.3rev/s If he rai h (6l2o;;bgnmzw cbs)ses 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 ins kehu ,m*ry/k(gx5ehpb1x5- Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight po1su xkhp eexgy 5*5/(kh,bm -rsition 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 initial rarwzc h1 .pin7b9(j/ bvte of 1.0 revhv7.cb( niwj1pbz 9 /r every 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 axis through tjl57 c:l*avp 5x o(sbva57gd its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass opaab*5(57 v5lsx d:jlgv7 tc5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure fek8/q:k+ y57td) 5g wziq/1 zqkdkiuskater 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 momencftampydd59w3k 5 0t2 ue/as. tum 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 izj19 jv. pvp68ho02jo jg.fpw8tg8tm nertia I is dropped onto an identical disk rotating at angular spj opov.8mgh906tzj8.8 w gfjp2vj1t peed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a oue67 r)hosbl,))q uht 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-round pla:1,lh.-fcflx1vgj(e k(-lsa 3 mmuactform of radius 3.0 m and moment of inertia- gxmffk-sulc. chaelj:1l(3 1m(,av 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 ffkwb 5ba+;n ba7/cxr *reely with an angular velocity of 0.8 ak7x/r5 ;abbf +w* cnb$rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, 799m 4jgcd ssulosing about half its mass in the process, and windinug7 sj9m94sdcg 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 nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excew)ym-ifjegcvl ,8 *o k,*1(a+aiya kfss 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 ut6c0dqv 7hs4:qn x:k $ 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 withour p9za fo7hq(f -ek3(it friction. T9p( -kfi7afhzoe3q( r he 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 6.5-m diameter merry-go-round mhbqo* as).tpty -ss,: /c6n kturntable that is mounted on frictionless bearings and has a moment of inertia of y /,oqn -kb:c.tps6ahm* t)s s 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 ropoezcw2n0 y j53e rolls on the ground with the end of the rope lying on the top edge of the spool. Aj3w nc2o05y ez 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 of mass move?

  The Moon orbits the Eart)x77o ;tae6; ntxit uah such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (t x6 )o7;7tx;nu etaiaabout 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 aq0x/b h1dsk o.t a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m axxh;fl,rf7v n b t/v,;cquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque d7;v bx,, rfhlnfv;t/ xelivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid.pxy,fnv h,(t 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 its 1.0-m-long string, if it is released fro v yx(ft,.hn,pm rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular 2l/u+ hcx+ny jgwp3 07ylu(m7-izuizspeed 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, bu/yv*z0egapdm6, e n n e9ki3aupfh,++ t with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gra,+u 3nhz 6e iapvp /e+yng,df09am* kevitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stoo,t nn n4s+9te/kcgrs93c2z :frc+y ored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses:o9tnn+tcf9 so c/g4rs+nr zy,2 kc3e a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustratv9x5gb4.ie( qzsw) /rej v6jj 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 horizontz jpej0 2a(5sjal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freh or -6v 7r,jhyknvukq,/4,knely (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 angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of graviuo-4kr,/ njk vh6 ,hv,qk7rnyty 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 flol h6,fjh8lj;zxp27 uzor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it r 8,7;26jplhzhuxzlf jests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=41 ami1tqjm/i ua:1ow5.2m/s on a flat road is making a turn with a radius The forces acting on the1a:aq i i1mou/5wtmj1 cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of len7g xy,xer6sz3 gth 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a z,yr x 6s37xegrate 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 3m;5ckus fcbgspy /c4022;hk hzwpm 3and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tig zc422yg ;mkw/0hu3pkfcs mb5ch;3pshtly against her body.   

  You are designing a clutch fgd wepge 0n36(trl+(assembly which consists of two cylindrical plates, of mass e60+ e l3p(tn wf(dgrg$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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. 8j p):tjmahct4(e/ )zj –58. What is the minimum value of the vertical height h that the marble must drop if it is to reachpj)/zj4h atc e()t:mj the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, b,oafwoin5x)wpb s slk,1h d0x ,,ab6 ,ut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 . l,(dj9d edqzzq9puce)b2*hrevolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tire2(qzdh.9d) ,dbclp *9 jeuzqes 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