A bicycle odometer (which measures distance traveled) is rled02k n3z1 z0 m *+mw(yp8dfs b1mfaattached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bikm00sw2 1d8en+fb3m fm(zp ryz*dla 1 cycle with 24-inch wheels?

Suppose a disk rotates at constant angular veloci fwlcx5clz43bvk3vp:4:.4 o2u, r)dlyjvf city. Does 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 the magn)vckzp od3ll cw yucv2 i:3:xv.4j r,flf 45b4itude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of t0k:bgn1vvb* jjn.7l phe angular velocity $\omega$ Explain.

Can a small force ever exert a greater tt5or- xmbha 0:orque than a larger force? Explain.

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

Why is it more difficult qmkkyr6rmmdoccm7z) k2bq)g:p5 6,7 to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagra7r2yqzmrkpb ,om dcm6)5)ck: gk6q m7 m may help you to answer this.

A 21-speed bicycle has 8pdi.. sgmqc* seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprodp8q c. *gism.cket? Why?

Mammals that depend on being able to run fast havv (s(,ayqhz+-t qv j5qe 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 distribjqz y+ qa-h,q((vstv5ution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, l3f( / t 3;vgdhh-i9u :pgtya bxhn6b;narrow beam?

If the net force on a system is zero, is the net torque also ze9 :(ppifb3rc qro? If the net torque on a system is zero, i p ibf9:3qc(prs the net force zero?

Two inclines have the same height but make different angles with the horizoth)(p e4p2r2mb(svj antal. Tpb4r (2e(sjtah2v p)m he same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. O)dw v1 y .fcd2vwr:fn0ne sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the inclinew0cf2.:d n1r)fv vwy d first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radi aqdqv9h r:)msdao 6eb72pa42 us and the same mass. They start from rest at the top of an in qsh6 4b)d rv 2pqoadm2:ae79acline. Which reaches 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 8b kg8n ;rgc/yksipuc * tktw.)0vb6 8are conserved. Yet most moving or rotating objects eventually slow down and stcp0 r8kbk;bg*t)c vgn i.w8k u/86yts op. Explain.

If there were a great migration of people towa rgj+z 95u:0j hq v8ypn c5rx7+pb,ptqboms;4rd the Earth’s equator, how would this affect the length of the day?cmrg 8j7 5osbz+x0t 49v,qnuhy:pbj +q pp ;r5

Can the diver of Fig. 8–29 dov9kcad:r. c; e a somersault without having any initial rotation when she leaves9d:rkca ;ve c. the board?

The moment of inertia of a rotating solid disk aboutvm j quss6k0*-;ic+ lx an axis through its center of mass*k ;q+6jv l0si-xc sum 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 fp 7fab/-7xt oa rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explainx tf7baof-/p7.

Two spheres look identical and have the same mass. However, one is holldain ,07rk xw(ow and the other is solid. Describe an experiment to determine which is whiw 7( dra,nxki0ch.

The angular velocity of a wheel rotating on a horizo . tomave/ 84izump9q:4g ,qclntal 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 decreasine4l mmugq/a,ziv qo tc:8.p49 g?

Suppose you are standing on thmgyoz g /m9r;9e edge of a large freely rotating turntable. What happens if you wzmor9g; /g my9alk toward the center?

A shortstop may leap i6x tbaok.7 )j zqcpgbuc,)6 )fnto the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly )66x)qc,7ac zbu .t jfgok )bpyou will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuit fob8 5+oin:ss why a helicopter must have more than one rotor (or prbo o+fin8:5tiopeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radianstu8irt ,m(z 0t: (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 Ecwvqh )s; fo gz8sbz1(,6xrq7arth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diamf8( 6;zw7s)qzoqgv 1hrbc,xseters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km fro7t9 +rjt )/x5on hxknuv;z nx,m Earth. The beam diverges a t ;koxnr5x7 un t,j9n+z/x)vht 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 ofb4 mv6nh8,py7cw g tdbnwii 3 5ci8i)sa-, w/y 6500 rpm. When the motor is turned off during operation, tcanp-y gbm3i5s,wv4d,yb 8i)c78iwti /hw 6 n he 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 another child. If thd+zt8.i uzl6;- sbeuve ball makes 15.0 revolutie-dz v.zs6+ ;iut8bl uons, what is its diameter?    m

  A bicycle with tires 68 cm 4h h8z 0ak(phwin diameter travels 8.0 km. How many revolutions do tkhph(a z8h4w0he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diamet zay)(xgf 3b*der rotates at 2500 rpm. Calculate its angular velocity i b)zygx fd(3*an $rad/s$ $\omega$ =    $rad/sec$
(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 onyhvb 1;c- c(dme complete revolution in 4.0 s (Fig. 8–38). (a) What is cdbc1h;(-yvm the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (+ i (,ors a6 qshjerw 7scx7+i8b6mb(ja) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sp4;e9 zchx aj 5x2bnpdjxq* 2d/w zj-ux;yw47zeed 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) mu 6 m8k00uutmnkst a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an accele mutm86kn0u 0kration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 28 fsha /1gys)+c0 rpm in 4.0 s. D1) y+gsfhc /asetermine
(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 radiusbkv*le:h0cgt/iuj,/e o fz 9- $R_1$ 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 spacecraft put th57x xm7d, cknpemselves into a slow rotation to distribute t x,5mndpk7c x7he 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 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 rpmpg.l9dtyp*-h/gm57. wc siz m in 220 s. Through how many revolutions did it turn in tm7p-z5i9wd/.ly*m s g.tpghc his time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in w3 p+e( 9 cp*7ddsppqp2.5 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 o(i7 ow +9mhwostresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through oi(m 9 +7oowwh20 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 ulpb7 4*cimwp 5o3-dt360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in thi7otcm5-*pp3wdu bi 4ls time?    m

  A cooling fan is turned off when it is running at 82h t6u+-eq 6qv-jzwo hfo3:wt50rev/min It turns 1500 revolutions before itv jotqu to2 f 3qw--e+6w:zhh6 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 bnw6kgms42*vwtgh47omroe/vjh ,*0 reduces its speed uniformly from 95km/h to 45km/h The tires have a diam0sg wrew*t7v44hn6mmho *ov j2 /gk,beter 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 revolutionsb7shx) e/ vl)d as the car reduces its speed uniformly fr /d vhlb)7sex)om 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts hf1ru )kt8yd*g4+j 7xp kdo 1sall her weight on each pedal when climbing a hi) u t4p+1xgksjf7o yddk8h1*r ll. The 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(zu *e v,4pqjvv+f7 qm of 55 N on the end of a door 74 cm wide. What is the magniqvv z+fpe7jv(q,m* u 4tude 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 the axle of the wheel shown in Fig; z/ 0w7r( bxl3 kj(ciyow)ojk. 8–39. Assume thajc xo;0y7k )(3jlwz wk/rboi(t a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to tv+t 7x/lesu++w+ glu hhe ends of a massless rod which pivots as shown in Fig. 8–40. Initialuvgt+hlu + +/e+xsl7 wly the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a6rzsoua8p:a8r o 4;blx b5x.w torque of 38w8;:8 a alr6rbpox. zx ob5su4 $m \cdot N$ 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 101i;hlksx h f3l oh9(f(.8-kg sphere of radius 0.648 m when the axis of rotation is 9klxl fh3h iho ;(sf1(through its center.    $kg \cdot m^2$

  Calculate the moment of inc;mh0g l61mtr* usa2+wxh z1kds vhf1d 93/xd ertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 t0 1 *dx611dmu/z9rwghhvsd 2lkcsf3+ xha;mkg. 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 in56ofkukb;qxn*tm e/z y,+ 7tl a horizontal ;7* l k6qt xuob+zfn5e/m,tkycircle 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 rotating at cor/-( 9(spnyy1tlzl gt nstant angular speed (Fig. tlzsny1rg / p-(ty9(l8–42). The friction force between 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 point objects tz.5n(2 u*ggafpcn 9mo7nwp2 shown in Fig. 8–43 ag2*oaf pw 2 7guzn 9(cnmp5tn.bout
(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 a+x9 7k/(g catf5px, ta. sxzdh,5fozu toms 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 spinnirh, 9z/rlvt 36z9 ikbzng at the correct rate, engineers 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 rezh,r9viz6t3 lzr9 b/k ach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of 0. tyr;nmqfm;q,1ictb 1+/ rb:h5801ri ,ynm rqqm1tfb; th:b;/+ c kg. 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, accelerating it fromn,wr 3-yqvnk; 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-go-round and is anz-- 8ls0 xh, tpgw,imble 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 kg, and two children (each lwx- zsgi8-0np t,m h,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 n 6/:naf 7 dwsxns4 )6q0oynzjrpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.2 jd7sn nq no :zfx46a)s6n0 y/w$m \cdot N$ 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 acc7 8w04 wemtnjjelerates 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 ba 7w 2 zc6xz i;8huf3l5beu1yis thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the tuy2zh5 1bcb ;3lau 7 xe6wzi8friceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown inc t(qdz dv)r19 Fig. 8–46crd1(v q9)zdt.
(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 ljzf5z.eqx-8c2 un 0x 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 restg m3cj9p :cb5gk6sqml3.g* um within four full turns (revolutions) and releases mjcl9*p.:3m kg 36g5 mbsucqgit at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a momentq -o sgsncg h0+lzhh ;8i6i.o: 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 of(8du3oeufa xtq: dhuhg / dv967le-c( 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 rolluju03rhke k1mfl 4hh u-pq5n/n:/y7hs without slipping down a lane at 3.fuul hh:/7 ekrnp305/1mq jn uy4 -hhk3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect towjm/: ,kg 6brm the Sun as the sum of trjk/mwm b6: ,gwo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 164:s) bs6setue/f j3er t)p :h:l0 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a / l)6tpu:ftbssers)e3e: h:j solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls f4/h 9p(dorc,;-*p djc t fdxywithout slipping down a 30.0 -fh f dd 49c (;/tyjrppc,*oxd$^{\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 ve0p6wbkw(r j n-rtically on its tip. It starts to fall and its lower end does not slip. What b k6wjr-w0(pnwill be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball r 1z nnl 1pbr5spu+6h+totating on the end of a thin string in a circle of radius 1.10 m at an angula6phn l us 5p1nr+bz+1tr speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2bsm+ 6p)l lzc5.8-kg uniform cylindrical grinding wheel of radius 16zcs mlb)l+ 5p8 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, on a platform that is r1*r oe86mzn9 far yj9motating at a rate of 1.3rev/s If he raises his arms to a horrz91aj m*yno remf68 9izontal 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 momentvu.23f db)g xtv o:9wr of inertia by a factor of about 3.5 when changing from the straight position totf or u9vb.dx3 gwv:)2 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 rqr 5j2sfm cw.7 n. *ym;sezz7increase her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a fisqmz 7rcn.ey z; 2 m7j*fr.s5wnal 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 rota7xlay5pvg,;fm *e y2b ting around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considereb7fa2 evypx;*gy 5l m,d 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. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a fkj52vqkj3ugpu(. g/ r igure 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 ut/ +rf nw.venn8/ou /momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

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

A nonrotating cylindrical disk of moment of inertia I is dropped onto 8 m *n5liowz)erisg2: an identical disk rotating at angulariwo:8ln*2 z)egrms 5i speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 /bnl -w8lds ye8+lf8f$rev/s$ 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 cenn07da( o0neycl4*qy hter of a rotating merry-go-round platform of radius 3.0 m a07(cyl4dnney qh0o a *nd moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular6osn j/eo,wp- velocity of 0., ooewjp- 6sn/8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a r*vxy3iqx84 g 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 wougy4i8q *xx3 vrld 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 exces6 rdw 8k/qnz fb5w:;7uzu9n0/tw g dows 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 isz8) )5gkc/cykj) dhbe4eu-$ 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, initrg; 6(e t0m frf9ign3vk5r3 vfially at rest, that can rotate freely without friction. tv nkrfr;3m60 9gfeg r v(3fi5The 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 edgek)3 .tne i:ra (f,jwdhy;ob.q:mrjl; of a 6.5-m diameter merry-go-round turntable that is mounted onay rj k;q:w(:mltn;)o ,i.3f. drjeb h 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 ro()owjw4(x6izbk k4pills on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, 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 centerbj46(iz wx4 ()wkpik o of mass move?

  The Moon orbits the Earth such that the sf ( ad6l/vx- r7kz6pk(;imquj8:qc maame side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angulaam7q6(r j f/az( vli;ckq 8mkd-xup:6r momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate o(.x7u/m e 22edsyimvl k0ddb6f 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 9zgoez4kx g)0x.p f5lxt :vz1the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest ovg.lx x 1kzo4zvf0z9 )pxget5:er a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of +5ro y4*wm4 ffptwb)5s 995o,3p jjvv llwlyotwo solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is y45 ,*vtf vw3lw j 5po)4j9fmp95r ywo+sbo llreleased from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is tpxmba2dch19a; 7wl+ fxu t(/ ehe 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 2ghkm e.nm,vn6x2u +h8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no an6m.v22 gnu ,hx+hke nmgular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is ffxdchj.;i6, q5;qywr0 y 6yuy or it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheeld fqy y6ywu. x6;r;q, yc50hji “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 illustrates an 7x)5ndq:bike$H_2O$ 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 surfaceojd ,r+y6 6qgw 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 pivo ubhxek.s.:wio+y a:(t freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is wuo.bsi. :(hxe+y ka: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 the ba) kfloj4i9+ floor, and we want to exert a horizontal force F atb)jio49a+kl f its axle so that it will climb 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 is maki 9zuk++4pw9.hei biqjfpz9u+ 6eips,ng a turn with a radius The forces acting opbifu6 hj+pi wku9,q 9 szpi e+4+ze.9n the 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 .e w9o4jsn 7yt od3006ai grq0xvt g9rlength 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 req0vjo0 tr36 7s.9rqi tgnw490aegyodxuired 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 thin r61 cvk;k)kko,h p4uqt ods, 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 tightly against her bo 6cqkpk;k t4uv ),k1hody.   

  You are designing a clutch assembly which consists of tw a 4q(8kq w6ih0pal5fwo cylindrical plates, ofa( a6 540hi8fwqplqw k 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 aa;4 58t(kapcs7xqp ztd,ofn0long the looped rough 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 hi ocpk,57qs pa4fxt(at8z;n0dghest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not rs*wa y7tif,5assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speegt+i(*u77 wf o x l,hbi;z9utzd uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wa*t,bo+zix fhlzu wut( 7i97g; s 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