A bicycle odometer (which measures distance traveled) is at,9j2. yb8nfr f3tgr nftached near the wheel ft9n rfg b3njr.28,fy hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constanp 2js;ywe hv wj77c:7jt angular velocity. Does a point on the rim have radial and/or tangential v 2jhej:swp7 jw7; 7cyacceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of th tn-ubna*gt+ 1(g. nlte angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a ld*./rxxx 8c g;igk 2aoarger 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 hann3omkc-ji)w8ap3h:w ux; ab2ds behind your head than when your arms are stretched out in front of you? A diagram may help you to answnc8bawopxj i 2m:)uwk3a- 3;her this.

A 21-speed bicycle has seven sprocsmur: 70g 1m 1fyuqu+rkets at the rear wheel and three at the pedal cranks. In which :u 70r yr+s fum1mgq1ugear 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 run fa4b; hmpob+:whst have slender lower legs with flesh and muscle concentrated high, close to the bod;opb:bm w4h+hy (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8 a/7whk1gwv/a –35) carry a long, narrow beam?

If the net force on a system is zero, is m:y )fc. gz*nwp:3z0bluc vw(the net torque also zero? If the net torque on a system is zero, is the net for (gpzw:n: 0f )cb*z mclwy3uv.ce zero?

Two inclines have the same height but make differe5wnrn7t8sj- w,re /sm 6ap9y jnt angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at thersr-7nj j9te,58pa6 y ns mww/ bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incl7wwlabv r 4v8ez:xh9jc( x 66wine. 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 ener6bjw 7e96xvhrac4wv:(z 8 xlw gy at the bottom?

A sphere and a cylinder have the same ra,x a 4mivm j8kae w3a9ggq.n9/dius 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 t8gv43aim,.j 9q mnxgak/ae 9whe bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum ark)mfl;ktyj)r4;j d0yae/* ekt-h;e le conserved. Yet most moving or rotating objects eventually slow down and stop. E)rk;)kee/ m- ;tja4kyjtl e*0lh;ydf xplain.

If there were a greatq 4fot; qu,k(c.j:w0 - eegkcivk89pe migration of people toward the Earth’s equator, how would thkvcjpqf e.k(w-0:u9goe,8qe4i t;k cis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault, .ecp2uh x7 89ookxjue ,)ze.gma9hc without having any initial rotation when she leaveeo 7aczc .ho 9h2mu,ux j9,.) 8xgpeeks the board?

The moment of inerti6y -7coct 4qwpa of a rotating solid disk about an axis through its center of o6yt 4 -q7ccpwmass 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 ;f b a24u (x;jiai4bnamass in each outstretched hand. If you suddenly drop the masses, will your angular velocity j ;xa aiub2(n;4f ab4iincrease, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howevexqzic ,49 sqv ig rkb.s/o5e/,r, one is hollow and the other is solid. Describe an experiment to deter qs5 b,r4, kxoicvqe/zs9.g i/mine which is which.

The angular velocity of a wheel rotating on a horizontal axle point c jyqaycm3,4,s 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 tangentiacjcyq am 34,,yl 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 .bo(pjm 6gd5glarge freely rotating turntable. What ha o6gb (p.mj5dgppens if you walk toward the center?

A shortstop may leap into the air to catch a ball and th0. veig btex0;v4f2hc row it quickly. As he throws the ball, the upper part oft fg;hic.e 0420xvbev 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 lju3b-a7tynn05m2 z(jo g /r ilaw of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propellebn j 0(g37i5aumo2tyjzn-/lrr). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in rad/l/ r:t/rvpee ians: (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 amazin*(jy 4fsrt epiw,kt+ 4g coincidence. Calculate, using the information inside the Front Ciyftrj4 (*te+4k swp ,over, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,0:qom 3 dyz: t:-owrehc,p4et;00 km from Earth. The beam diverges at; ytd4 :o p:,zmhw3e-ercq:ot 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 tha;w*rt-n6 l bcsr 3 a:oz+zvx 6tjw2i4e motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as thezn;-j rav bsc3*i 62o+t alt:4rw6zxw blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m tc-s6hq0v nu+dl(5wv)hctm 4to another child. If the ball makes 15.0 revolutions, what is its diamelsc0-chdv (qht u w)v5tm n+46ter?    m

  A bicycle with tires 68 cm in diameter travels 8.6:+: gwuhp29iahr ho hwy.v+m0 km. How many revolutions do the wheels m uw9a2:hvwrh:p+ 6yhomi hg.+ake?    $rev$

  (a) A grinding wheel 0.35 m in diqdf*sv2q 6a /vameter rotates at 2500 rpm. Calculate its angular vel/6sdfq q v2*avocity 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 completd- oi7x4u1ofks hjc 5 ;aq44 -impng)se revolution in 4.0 s (Fig. 8–38). (a) Wg-spq5o4fcj;) hs- o7knudxi14mia4 hat is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity ofiud*u5bmtq v7) 0 gq.x 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 spee10nzwwt 10qm2 jdvl3l a+by6sd 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 wiv o9n;o- +ud7.0 cm from the axis of rotation is wi9 ;udn+oov-to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleratcvng4;( eipdky tq1cn4u*5k 5es uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determ ki4c5 4*nu denct5 kpy;1(vqgine
(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 radiuxt(tq;vkw yh42hh(r 6 s $R_1$ 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 the9x6+m0 ldq 0yx-ruas x Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start0x+ yx- x0dqsar6lu m9 of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Throum 3f8mzfmp..w gh how many revolutions d m3f.mmp8f.wz id it turn in this time?    $rev$

  An automobile enginevi9kd m 7t8c,t slows down from 4500 rpm 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 stresses5l3ck ic apckpwh)( tm3) -9zg of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final p5c93ilkt c c)w(3a-g)hmk zp 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 3wq.g l,*e lp*z60 rpm in 6.5 s. How far will a point on the edge of t*,*q we.p zgllhe wheel have traveled in this time?    m

  A cooling fan is turned off when it is ru3tze3bmx2:wm ah o .c7 u5ndj,nning at 850rev/min It turns 1500 revolutions z jxm o3e,dbaht 72:mwn5c .3ubefore 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 uniformly frk*bubcd, r 6*znj1s. uom 95km/h to 45km/h The tir,ud1bu *k 6zn.rcsj*b es 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 gm:-l; (rw04giprefuspeed uniformly from 95km/h to 45km/h The tires have a diam0 wm (r-l4:;uegr figpeter 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 h21 qkly;c/ nhg.xp zy*er weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 1p lyqn z/*2;xhygc1k .7 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 thewt zg3xax2i *-u.bh,k magn3 hub,twxa*2-gizk x.itude of the 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 t,ox )g-ghc+ethe axle of the wheel shown in Fig. 8–39. Assume that etg )xoc, +-gha friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attt3wq 4 ee:kezeik+y p1c4;fi*(d-rrhached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and 3 yp4wtefe1- +z4i;ri:kq*eh ck r(de then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightjtkg-x/ d)9k mening to a torque of 38 g- k9x)ktjdm /$m \cdot N$ 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 y40xdp)8vzb9z8 zusf 10.8-kg sphere of radius 0.648 m when the axis of rotation is throughsxv8uyp8f 4z zbz) 0d9 its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 6y-wtxq jrrt. v91g4b* 73 (t+tn xqrw8mqygc36.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The wtqj8t.yb 3r yxrgxt3v4q w(q rm1-9* gc7+n tmass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball .b7tv*;v-kljlz a,i /g m (jwhon the end of a thin, light rod is rotated in a horizontal7- l,mh(jb*jgl aw v/.v zki;t circle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotatin5q s.n68q,fk asy zd/v.4vcueg at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is e6avcqnvzys/qu5 .s ,df8 k. 41.5 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 poim: 46;gdlmup int objects shown in Fig. 8–43 about4i:du ;plg6 mm
(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 oxygen3hf5 sd.qind 6anmm-+ 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,k8xexcbu1 j4( 0k 6nf1 dzhs5g engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite hash sxb6nuz fec(j1dx8g0 k 541k 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 witf6fsf-yh d0 j1h a radius of 8.50 cm and a mass of 0.580 kgjfd0 1yff-hs 6. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, acc a/,/i7bb gujoelerating 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 / 4siozsmt4;)wvs g (fon a small hand-driven merry-go-round and is able to accelerate it from rest to a frequenzf(swi vo 4s/t)gm; s4cy of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, 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 andajre91si h6 nxrzsw( k1m.c2wq 0)5an is eventually brought uniformly to rest by a frictnns91 rajkzis50h.x m)qe21(w6w racional 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 accelerat,*lr wmrz4tfg;d 15aq7tcr6 ies 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 forv;w us:awkas (tp;n,2 earm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 1; p,wwntkua( av:;ss2 0 $m/s$ 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 cah,oi:,ixlx*he( imq2*hl mro m5iw 66 n be considered a long thin rod, as shown in Fig. 8–42lxermi* *l(hmhqx i56moi o,:w6hi ,6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses,b b8zq*vpjkktue8tc70,,l6- t c x ;gd $m_1$ 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 acce:m t8.ivmwqk- y2o .vilerates the hammer from rest within four full turns (revolvkmw m.8oiv.qt: i2- yutions) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor haybweit, //zkiv j2 ru9 8rt,.hs a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque oei 74pxs8 n:.vbe( mtsf 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 radiushb6 k5 nu,bnnf2s gm2;g1f xl5 9.0 cm rolls without slipping down a lane ab,f26n 5shbl2n;k5ng xfumg 1t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum ooq( c9aa6w okm0 ;hw0bf two terhq w;6ma0o9 kc0 woba(ms,
(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 k)0 n(t+v d 0cdijz1rl :+5zwcc6u;zg ockn0 cof 1640 kg and a radius of 7.50 m. How much net work is required to acce zdkw zj0;gt1++oc:)lck 6dcu( 00vn5 iznrcc lerate 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 20y/5wi ljh 2d7vca ( d8o3pmocmd-bwy1 .v4m4t .0 cm and mass 1.80 kg starts from rest and rolls without slippin b 8mjw-do 4/hdyca4.l1 7pwvt i5cmmyvd3o2 (g 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 verq xcxc 6ajua -00e.0kotically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just bef6xeq oaa .x0c0uk c0j-ore it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 4 70: eeyeho7ktf ze4w0.210-kg ball rotating on the end of a thin string in 7e7z4tekh0w:ey oef4 a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a t1a7bizytl5l 97k.hp) 3qfrb 2.8-kg uniform cylindrical grinding wheel of )7fk9zlltb3ph ti5 ay qr.71bradius 18 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 at his side, s v;sjsak .0f fp) ;s-rh*nzl1on a platform that is rotating at a rate of 1.3rev/s If he raises his arms ts )-vs0szsnffjhp a1;*;l .kro 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) can reduce her moment of inertiokqxtb n 80j83a by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s w nqx3o jkt088bhen in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increasel. 0z g(wc+f:rhj )t8+gf(hgiw( lljk her spin rotation rate from an initial rate of 1.0 rev evhhil0gz f.kt8j(fl(g ): g +wr(l+jcw ery 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 a f8fh b:+iz-yzfsc+ fwa3/za 4,hw0qyxis 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, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. qa+w8,z4zh0yfz3w:s h+/fabfy- ic fWhat is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momenm* /qq 9 1mqdp4yowr8mtum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Eartnc4hs3uw4 o2kh
(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 bhb i in6117 qrv z.c7.zrmcp8onto an identical b r68m 1rhc7i1z. zpc7.ivbnqdisk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 4en;c1hvww n / 9 br2sw7nk. yj/;jjq1ing.df$rev/s$ 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 bwxq 00 la 3(gnq4:e bdli4sl7of a rotating merry-go-round platform of r4 b 407ldgw3 q0lnaxe ibslq(:adius 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 mertzsy);s0ebtvhss ;2(z1f +en ry-go-round is rotating freely with an angular velocity of 0nyb ssftz z2t ss+eeh);;v(1 0.8 $rad/s$ 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 inte pk*5u3,w fqg ic a3b:z3l6mjo 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?(roundg fz3kj p*cwe3q3ub5a:,6il m 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 excess oz py(,a/ldm )tf 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 sd+9dcw2q )e;o*xg vrigx3sta8 ) +io$ 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 c1lr 7j+i6xlh q5*i-itue2er8onllz9 an rotate freely without friction. The moment of inerti2rxu5 6hnie-tr+ej1 lq *7lliliz9o 8 a 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 mer n9i oo0q43ldsry-go-round turntable that is mounsl 30ooin4dq9 ted on frictionless 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 with the end of the ro h,4v2f io;smype 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 sp,m2foiy; s v4hool’s center of mass move?

  The Moon orbits the Earv2sl: d* (dxwdth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular(vw ds l*:2ddx momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fro,4p2rl 2nop eim rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 a 7(gu*dq)dduae q37hm acquires a rotational rate of from rest over a 6.0-s interval at 3 ua ddguq(* deqh)77aconstant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyl3ih w( jmwh;g g(nhtt;b75t+.)squuf indrical 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,g);uit wb. wf5 h;(qtmu h+st(j7gnh 3 if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular l dmtt/ )+x1ly+ (notispeed 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.qhs29pu6hj ,r7 c8lau0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass hu728, aqp9lc rs6hujin 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 olik* z6,l bpwsey 6,4stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “sp* kloi , z64ewlp,y6sbinup.”
(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 illustratexf8pys v170aps 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.dt9l rw kovy. */wz45n3 $m/s$ 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 pivo4da771q0iip. ycigg or8ic 8nt freely (i.e., we ignore friction) about a hinge attrngp i7 i07 .y1id84iccgoq a8ached 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 gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on ty+jj;qagdno oe .,k61 he floor, and we want to exert a horizontal force F at its axle so that it will climb,gyjo1on kq.+6 jde;a a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road iv r 43foazwz80s making a turn with a radius The forces acting on the cyclist and cycle are the normal force8fvr 0z3o4 zaw $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 length 1.5 m and begige)s6e rqd-q 1ns whirling the rock in a nearly horizontal circle above his 1-6dqg )ereqs 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 her arms as thi4r1ii; w kqfx4n 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 ir iqf 4kx4;1wheld tightly against her body.   

  You are designing a clutch asse) 2wlpz. mdlavi-u3i 3mbly which consists of two cylindrical plates, oful lda-w3 z32 iim)v.p mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped ro-*no5l8hr0 s/c4he0:g xanp puzutr4ugh track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point s / nzp n4cp g xtl5u*0ouar:8r-ehh40of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ashsl5n--fhgjw2u lu/y 6r;1y4rd j / iysume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutionp /8h7pkho t5e*c kcs7s as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) k75s/h*ekop8pt c7 ch 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