A bicycle odometer (which measures distance traveled) is attached near the wh )5 b:o3uon1k vprkd3yeel hub and is designed for 27-inch wheels. What happens if yy)v3 pnokr o1k:u3b 5dou use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angul32tx.euxqd-kd4por1 ar velocity. Does a point on the rim have radial and/orq-xo1xk ut 2 p.4e3ddr 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 component of linear acceleration change?

Could a nonrigid body be described by a single value of thdaoxm h2l*)4ie angular velocity $\omega$ Explain.

Can a small force ever exert a gre1. ta a:s5yjrdater torque 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 to do a sit-up with your hands behind your head than d8i- pjn(;n lrwhen your arms are stretched out in fd(ln ipr n8;-jront of you? A diagram may help you to answer this.

A 21-speed bicycle has seven s+bufps ,q)s (iprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear q+s,)su (ipbfsprocket 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 r:a;7 ckugt d)being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advartug7;dak:)c ntageous.

Why do tightrope walkers (Fig. 8–3)b-.nen1(n koylq, ;syd ay;a5) carry a long, narrow beam?

If the net force on a system is zero,anm-4q7a1sr h l2 vqe( is the net torque also zero? If the net torque on a system is zero, is the net fohqasa- n (1lre74q2mvrce zero?

Two inclines have the same height but make differh20wulil*bqh66 7sm) 3zywqz ent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Expla lzw milzsqhby76* u 20h36wq)in.

Two solid spheres simultaneously start rolling (from rest)a.snr cmujt+z;s 04jp m13xf) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at t+xf cmp mjt.s nr)us31a4;0zj he bottom?

A sphere and a cylinder have the same radius and the sat9vv x4wm 4+wnme 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 kinetic energy at the bottot4w+ vnwx49m vm? Which has the greater rotational KE?

We claim that momentum and angular momentum are con7e ghl t2t2 /lgz+evl;served. Yet most moving or rotating objects ete2 h ltz v/lg;27ge+lventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equath0cbp9l:*)ih)ne61gc0 mm mnc q jn6tor, how would th)hmjh6gm:b *tcn0nm1e 0 i6ln9q c)pcis affect the length of the day?

Can the diver of Fig. 8–29 do a 3u 9rk+;fdmju somersault without having any initial rotation when she leaves the bfu+ru9 ;3 jmdkoard?

The moment of inertia of a rotating solid disk aboutkwpj )aa+r95b an axis through its center of ma5+p ajr)b 9wkass 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 mass in eachxb /2suiv .k+ i;+cnlr outstretched hand. If you suddenly drop the masses, ln.csxi ;r +/2iu k+bvwill your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identv. l5 lb4 swm4jg+q/tjc7wn c4ical and have the same mass. However, one is hollow and the other is solid. Describe an experim qt4sn/7v.lbw m 4wg+l 5c4jjcent to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In w9veel,5;n 9lrbzkvhsg0:(5 ivr yrh8hat direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential line l r9v r v,i(e0rbl:; ye5snhgh5z8v9kar 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 large frn*/ay4em zg (x k)q(voeely rotating turntable. What happens if yo xvngo( kme/za( q4y)*u walk toward the center?

A shortstop may leap into the air toa)x/4ya-9g 6 zkjan h omvud;9 catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you loo 9 h4xz)u/9 odjayng- kmaa6;vk quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, di sn/ bd62ds)siscuss why a helicopter must have more than one rotor (or propelldd)s i26ssnb/ er). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following anglesg77as/,n wka mm h;zqs/r1g.- lyzsi4 in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calo/j szgp+/ /,hkn5b(ni x1dt xculate, using the information inside the Frontp+ /,nk s1thjox nx 5zgdi//(b Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed atwh+ jhra(jaf01n l. x9gdg9;x the Moon, 380,000 km from Earth. The beam diverges at+j rjaxl9fn. 1hggd;h9wa0x ( 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 att.m94 md(6c gs*bi tye a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the mbeicm4 d(y*t.9gts6 blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball make1:ac ukhfu* i x8ek /hfo:v0gp,b*s8ns 15.0 revolutions, whch k,f vhu8:8a0i/o bp:nsg f x*ke1*uat is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revol)myv 2f6;w ceho,r g;s6)eyquutions do the wheels me 2,omgr;w q66y c hye;))vufsake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its9 w:3r+nllb8/a wnj aj angular vel ja9lanwl/ jbn wr:+83ocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (F8zsofo.iq r9 c 1dino;v/se f8-bz l5)z:eh9rig. 8–38). (a) What is the linear speed of a child seated 1.2 m from tclz9dozn:sqvheisr 8.rfz1 8;) f o5b-io/e9he 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 (a) in its orbit0. zhqbrk*4ncg 9w:q6 af5xvm6 rsnf7 around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear suvqu nf41(x53izf iy4mk (q 4npeed 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 celfm8cc4,(kfd 4g umu0/-ky qz ntrifuge rotate if a particle 7.0 cm from the axis of rotatiocum zc8y(qkm0/44 uf -fklg,dn is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about itsuptrub7ks 1e2el1s20esb)0 y +i(bw f center from 130 rpm to 280 rpm in 4.0 rwkp u+ se) (ys blefet7iu0s220b11bs. 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,p3127jk5kzmw q t5:qhpzkfg $R_1$ 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 themshcekw;wur /a*; 3aai3elves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 /i ;3haur* ekawa;w3crevolution 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)rqru34nrp:vq0kuxs -h8 x duk-7 2un,000 rpm in 220 s. Through how many revoluti n rk-08)k4 2uu:x r7hnxdvqpr3q uu-sons did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculaqg 6gfczb6ox 5 v(5homavua x+.5 (n2pte
(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 high0cdx9d:rhu.wo(nyy8 o aph++speed jets in a whirling “human centrifuge,” which takes 1.0 dyod9:o ahrhwcyp+xn u0+8(.min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360dgkzxq :7m(h : rpm in 6.5 s. How far wizxkdqg:7 (:hmll a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 8c/x* owezjs5mpx bl. 5/c3f;q 50rev/min It turns 1500 revolutions before it comes t;o/ws/ mcq*5 zx3c5 xpje flb.o 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 re:6dktyx o *)phduces its speed uniformly from 95km/h to 45km/h The tires have a diameter oy :)dhokx*6 ptf 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 qu+iryfffjh0 575e66w 02ehwc si qv,revolutions as the car reduces its speed uniformly from 95km/h to 45km/ ifivfusy567h 2cw5e6w ,e+0qq0 rfjhh The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on eaj g/c(ek9q ivhg3u d33ch pedal when climbing a hill. The peq (c3hj33g v dge9i/kudals 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 wi:zy, j*8 q.d 8wfokiygde. What is the magnitude of the torque yo :8 fi.8dgqj,kw*zy 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 ofst *0iry9b e1pb 1bq(h the wheel shown in Fig. 8–39. Assume that a friction torque ofqp0t (rby1*hsi 19bb e 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 om t9oph v-ib48f a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in v8i49-mo htpb 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 tightenin rd/3gscn.5,y sw-6iefb bj1y g to a torque of 38 ./w ,n bbr1c-eiyfy36g5sjs d $m \cdot N$ 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 rad h28u-rc lkfqt (nsy86ius 0.648 m when the axis of rotation is through its cenl y shqu(28 fcnktr68-ter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bi*(lpb p+fj6.ps xb 4c0pcif0 vcycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of p.b*px s 6p+fvcp b4l(00cifj the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a h /oc /wv3c0orjorizontal circle of radius 1.2 m. Calculatwoc//v30co r je
(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 5jxjzys s;c82rotating at constant angular speed (Fig. j8scz ;y52j xs8–42). 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 shown in Fig. 8nfb*o5hs2 y5dyn c m5,rakg)5–43 ky5 ), a* cs5o5ydgrhf2n5b mnabout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms wm508a* 2f zsysc17 (ahbch jtvhose 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 s;8k a;k0a uhu5y(ppecylindrical satellite spinning 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 reach 32 rpm in 5.0 min? 5ka8auu sp(;0;kyeph$\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass )v-knb13 zyob.kl wbak1 2y7tof 0.58l2)ytab kwnkk zv3- y1ob1.b 70 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 swin(k3bl(s * /nf92uvfyadi jws*gs 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 tangentixdky/wl8 ly ,si 5g*q1 u/n0lgqasv n5k5/(drally on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform 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, neglect/n l5k5*y n,us0wgvlad 1kxig5rs8y dqlq(/ /ing 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 an*2v sy4 vwz5wjq4 v/int erd89d is eventually brought uniwjiv4qdv z9y58r tw s24nev*/formly to 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 accelerates a 3.6-kg ball at f d2ia 5 q)6(iqoao p.2iz4mjhk)*rwl7 $m/s^2$ 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 forearm, whiaq j.g9n 7e*oh1m m,,4iz3opk pt4aaiy5d*v l ch rotates a 4dliyn,,jgzpom3 v4 1hk*eta q*.5oi7 pam9about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considere- a1ji jd(lz;vjq,s/ ud a long thin rod, as shown in Fig. 8–46 jv1u,a-qli d/j( s;zj.
(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 clfl ky ) wt;nnju +)yw)mw40m5onsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turns (rjy +vxvoc2sa,niw mpo)g- a.+3ip2 5cevolutions) and relcnx 2cy+,pp3)vviaojg+w m5 2s-.ia oeases 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 has a4vc1le (zz)p (jq br,i 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 torquanuy p)cqb- 2 3*g4ebse 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 r-( 85d1 .km(udkc-kjc sffxl fadius 9.0 cm rolls without slipping down a lane auj(1 cslk m.ffc k 8(k5-fxdd-t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the skppa c cp96c8 kg,h *ff8jizt8*qbg):um of two tf 6c88cg,:jtq9k *kpf gbpi8)ahzcp*erms,
(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 tdyle 2d-(7lz0te .j4 fq.k 1y/ginlz1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation 1l/ -yi(k t4elzelgn2 jzy0qf.d. t7drate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.8:9 gg gympf5okm 2-.oz0 kg starts from rest and rolls without slipping d .gm25 zgp9kf-ygo mo:own 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 iei-di6y8oo :2yqi )tflu-hp.j ;ofu/ts tip. It starts to fall and its lower end does not :o-j;pu.i8 tlhouy6 2dyf eif- )i/oq slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.22crj o4v o 5n1q7fm/mt10-kg ball rotating on the end of a thin string in a circlem452vm qjro 7 o1tfnc/ 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 2.8-kg uniform cylindrical grinding wh/j ticaroq2w1 ka:5mx01 wj .seel of radius 18 cm when roxt1 j51:raqo2/si.j 0awwmcktating 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 siu3 fmde tv3 3q)0elf/i8pa wa1de, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal a/8vm3ed l30)ipfu3 q ewa1tfposition, 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 redl(2z *w4 dwxvduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1v w2(l x*dzw4d.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 from8og,b)eu n.duq 9c*hq an initial rate of 1.0 rev every 2.0 s to a finalcg uode9hq ,q .nb)8u* 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 its center at a: qkz/g bq6 +n1mjm2bxzw.g, y 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. What is the frequencyqjz:xn, . zb+qbgg1 6m/myk2w of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a fi 9o-a j2otllr4 8uy.nggure 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 Eart--u mqi x;a t,jt-df0mgpsu((h
(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 momentm8.o6,5u a ldf vr ;k-iboynbs70cqr3 hh+-cu of inertia I is dropped onto an identical disk rotating at angucy+ 5b uqb o,lfo0uv- h ;a6nd7mr.ris-3h8kclar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at c(wz*5k;kqfmt8 qw)as6 rk6s2.4 $rev/s$ 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-gi,6x 0/ujyifud or 7.po-round platform of radius 3.0 m and f .0x7ru d y6,p/oijuimoment 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 rot-yxs8ra xu;1lc0xm:e 4ja 5 g3hggd(zating freely with an angular velocity of 0.80cyx: -s r 4g8 ggmxahx(1u 35edlajz; $rad/s$ 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, losing about half 11rac (sv,jxw its mass in the process, and winding up with a radiuj (x,wcrv11sa s 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 winbo0gcv0)fk8rygqrsv r)lu;t(kl)fd8 *7 g. y ds in excess 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 tbb8ioo (m3b ( l.wu1k$ 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, thatmr4 1f+gvv1 wv can rotate freely without friction. The moment of i1vv mv +1fr4gwnertia 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-rolymh z9yvrss*scw9e s85(;b 5und turntable that iess9v b;8( hmss5 lr ywzy59*cs mounted 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 odcs-sd6d21vk5 u* xrbhf lw1ei ..ru2f 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, Fdicv u61lbw5r2 2uxd1 f-krhsde. .s *ig. 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 h,op (*jju1lhEarth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbj1 hjpho,lu* (ital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 gth6wr-/ ajg;3r*ctnx ghs z+p0,q )am/$s^2$ 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.dpd04:tvyz/h ; vbg6k 20 m acquires a rotational rate of from rest over a 6.0-s inter0tvd g4:z pk hb;/v6ydval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and d fnh 1q2cs36wqiameter 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 strinc qn s3q62hf1wg, 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 speed of the rear wheedcm1m 5ig) ix8,oefeav 1( ju/l ($\omega_R$) 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 9*n/q2p 3qf p gq2almbmass 8.0 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 2 afqppgm blq29n/q3*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 lguc3)-pv a,cd ow-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, v -,cudp 3)gcauses 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 illustra wkzyh.w( m++zf*8h g4yey*urtes 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 2m 9;wig i diuk:r/m6xa horizontal 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 freely (i.e., nnwlh** *deil,*aoc*/c7qeu 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 gravity acts at the center of mass of the rodq* ic l*wae,*o 7en *d*/luchn, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. dkhv7ru 5.+agp)+qd cIt is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against whichcdug7.)ph a+5k+qdr v it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed *eohaw7 qc7 vi8/hg;sqa a*p.v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist anae8 v.qpwcasi*7qo;7 gha* / hd 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 0mx 81xz+y6b ht.50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above hi1b +8z xyhmx6ts 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 sot,:6 /5ag svs-ku cq6utsd d-l/lu 0kllid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular spessa /ctgl-5ld0t-v6lu d/6,u:uskq keds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which (4x5ixkh1 vo2 pa l)jwconsists of two cylindrical plates, of mass(v 42iw1h5)j xx pokal $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 hx1v*wqdvys+m8g5 4prolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h t dmv4q*gx 85 svhyw+p1hat the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, buty c mty+su1it o*.u8/4gcpow ov 3m8x( do not assume $r\ll R$ h =    (R-r)

  The tires of a car make /dq0ugclx/cl:cb:8 v85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tireslv/xcc :q:/g8dc 0blu 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