A bicycle odometer (whichbf7 ,p0py aaz4hz 9kq7d,8fwh measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happe7709pw,pyh4z ,8z fafhdqkb ans if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular ve vj ,cjyq;(ubu .x0l0rlocity. 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 tangentjv0; .yr0,q( ucx lbjuial 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 the angula 4yrb-;vs1bj vr velocity $\omega$ Explain.

Can a small force ever exert a greater tore sb hgaw/6jij/u*hy+ ,8 f;pique 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 zdr- *oyo4jx8jic. e-a sit-up with your hands behind your head than when your arms are stretched.ozy* xeid- jr oj-c48 out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven spq)v99h 3ogxu rrockets 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 ugvq9 rhox)39 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 fast have slender lower wt)g2lw .e xpu)jqw 8v qu8f26legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dyn qju.u2x t8 l)g8e6v2wpfwq)w amics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry/7ymj;xtc2 ep ; s3rjv a long, narrow beam?

If the net force on a system is zero, is the net torque alsz:wtipse*1/w, pj;e2* wpiv)4e aelso zero? If the net torque on a s4wsit,s ae 2ve ) e:we*zjw1 ppi;l*p/ystem is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizont(ugrk *5 qraz( duq:t.al. Tuzgrr ( a:u kqt*(.d5qhe 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)(;mqgs /;afn3 yp :e.qj;-axx soqs4p 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 energj;( 3p.qemapsy 4s;;fx sa :og /qqx-ny at the bottom?

A sphere and a cylinder have the same radius 7e 1;u6rczbpn and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the6n;bc u1 zp7er greater rotational KE?

We claim that momentum and angular momentum are conservedh./qv:14 7ts ha1ol8p bp* *htu offri. Yet most moving or rotating objects eventually slow down and s s4h1 i:qptlo*vpto7 u r.fhh*af1b8 /top. Explain.

If there were a great migration of people tph-16i6gvftx 9j 0ih yoward the Earth’s equator, how would this affect the l 6-yhg pjii9xth6f01vength of the day?

Can the diver of Fig. 8–29 do a somersault without havizdnkn2qsw,2b /3mkz* ng any initial rotation when she leaves the boszd2 /2 3kkbnwqn,*zm ard?

The moment of inertia of a rotating solid disk about an axis throyeqzh4 9 zadfh*1b ./gugh its center 4h f*ebh 1yzg9dqz. a/of mass 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 rotatixcc,5utm kjv /rqh3 vqd4o518ng stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your h 1m4cr58t3,xj vvc 5/dqukq oangular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and ux-*76ult68yzw- jnr7khc: zvw vn *w the other is solid. Describe an experiment to determine which i:u n -ylj xww7*-66*znk hruw7tvcvz8s which.

The angular velocity of a whht zlj5* eb; jw7)c:bdeel rotating on a horizontal axle points west. In what direction is the linear velocity zbj*j57w;lc)btd : eh of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a largexg2gzasvw b a8 qg*vo*q-2w6* freely rotating turntable. What hwo2 sg v- *aq6zxbv*ggq*2 8waappens if you walk toward the center?

A shortstop may leap into the air to catch a ba,sc0jqx1 x5ff4ign)a(8(q fd:yjb gwll and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the ocgfq5jbdyq )w8 isx,njf gx14a( (f :0pposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentumn;1ixx:ln -nu r.ckr,vh7 9ypxo hr/4, discuss why a helicopter must have more than one rotor (or pr 7- 1 u;:r4.xk9 ho/xxnihn,pcnrrvy lopeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (aqbzne 1 ,7(cwa:mc xu)) 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. Calculate, using thi5ydb, o 9v9t9f45sskr kp 2f5/ ny4 ki+dirfse information inside the Front Cover, the angular diameters (in radians) of the Sun and bdf sidni +k4v929kfs5/y fi54,rsk 9ypr5 ot the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 38k )/l)pa ho:feu)h ek:0,000 km from Earth. The beam diverges at an angleae))ek/:phuo h) kl:f $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotj.rrz :dm;u )w- e8wiwate at a rate of 6500 rpm. When the motor is turned off during operation, ;mw -)zwjrudr :iew8. the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the balr gr( a*l2b.cue i 0yy(pr ,ct*cv)1fkl makes 15.0 revolutions, what is its diracvu0f2y.1kei* lt)ccrrb( *, (gypameter?    m

  A bicycle with tires 68 cmt(;f6ql lc q .1ykl+ul in diameter travels 8.0 km. How many revolutions do the+fll.ul c(lq;1kt6y q wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates (:cnirz gi6jb *ogichk m+52.at 2500 rpm. Calculate its angular vei (k mozng +ig*r 6b52ch.c:ijlocity 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 revolut+ cayjr-6fgrn0 g- jlel6+ d,cion in 4.0 s (Fig. 8–38). (a) What is the linear speed oj-r-g lag jfyecc+6r6,l+ nd0f 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 (a) in its o7/p mb av+l2dl:p,yoh rbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spn0vd0012k5zf*;rva (usy yzl q,dpzmeed 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 7.: zxmvgm)i s8nq*5bp173+/d6llpstdw gpmm60 cm from the axis of rotation is to experience )6 pm 8d/zbdt6+g* qls37lpp mn5sgwmi 1m:v xan acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates ueb)p0(np - ;bdx)svftniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determined bvp) 0(te-;sf)xnb p
(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 b.4wd.xl qu;:;f xk z71pqijg$R_1$ 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 l3ax15q1 vvr oa58w ctaboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought oft3q1w vvc51 oaax8rl5 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 uniforml6eaffpt / 5 +bd:54i gdhqqq; 1:roooay from rest to 15,000 rpm in 220 s. Through how m o fo: qi+ 4;agq6q:d/bhep5 5d1oatfrany revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm x3lqp sjj1 (5gin 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses b- iv-v4n wb jpt:*m4rof flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 comipw vbb44-:rjt* v n-mplete 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 3606w rk*iakl;e0d0abf s88 (4t9 h hsu4rotfo)i rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this time? t(;esb il rit )wko 46o rhdhfuka89*0a84 0sf    m

  A cooling fan is turned off when it is running at 850re,+w c9+julxz+ dqhy4z v/min It turns 1500 revolutions before it comes to a stop., zy+j94dq+chu +w lxz
(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 uniformloc( x sxpad5h )dtk* iof8,+j8y from 95km/h to 45km/h The tires have a doo + 8jit5akfs*c8(x) xdp,hdiameter 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 8c. bs1o ur2 zvv zaavj0p 21p1jn/g2aits speed uniformly from 95km/h to 2aau zpvrp/b . ag081ov 2nj1zv2cjs 145km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal whe 687o3: rufmfsv/ )c z4;dek:z 3zwhb*hbhnvxn climbing a hill. The pedalsb)/vnx e;k783bsc 4*zvo: uz:hhf 6drh 3wfmz 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 56f-k88 5vmzeto1cjx 8ne.v7r ky oxf uim1-2l5 N on the end of a door 74 cm wide. What is the magnitude of the torque if the f. k-6cn7eeum 1xvy18ro off8tv-mz ikl58 jx2orce 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 wheelq4m ff )n 6u6be9a9d rng.com1 shown in Fig. 8–39. Assume r9m mno).f96nd6fug b1 4aq ecthat 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 the ends of a massless rod which o;c 5n3g 6z)vux2+9kh b ddvgjpivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net to 3gz+9co2)56jd vb ;vn ugdxkhrque on this system.

  The bolts on the cylinder head of an engine require tightening to a torp h) m* 3xmelx htcr2z-od+s95que of 38 m5h th)zx3prlex 9m 2*so-d+ c$m \cdot N$ 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-kgi0,icjy;5:po 2u .llsdbt/gi ):e qts sphere of radius 0.648 m when the axis of rop t yuitc/ogdi,ebl .0: q)jsli:2; s5tation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim h c2p4lj 7:gaband tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?)aljgp2 c47:h b.    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in ,hgbc lkdcqoa 8:*y y6n5a4u z3(q a:ja horizontal circle of radiu(ukd:: ha cj8,zgaao 6b4yqn3 5cq*lys 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 potbtz4:9jlk brvbpo*ux ;yw(+ + ter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N+4u x+vby9bj p*tzork :lb;w ( 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. 8–43z 7k3b.mzcj. f.kf7. 3zb zcjm about
(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 whose total masyxb91tl9z 4k/b e. bhgs 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 sax: y g4iend+nu-zv- /t9)heivtellite 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 sagyt9uneix+ )dhn z-vv/-4ie :tellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a unih r) 9tvypjgjxv3:i86 .1ygqfform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Cyfrj1 :96v38y hq j .gi)txgvpalculate
(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 sw+a2ybc92gl eygh j65td dk6 h0ings a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven me) *ihcg7 oikyc 1t*0hosy iu-;rry-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 p0ckyhth -is i**c1o7 ;o )guyiroduce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brought;wm ; yvlm8qy*h(j) cn uniformly to rest by nvwyhy)q8mlj(mc ; ; *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 yzu-i+h ok w(,ym- udlg3 jh(: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 so) vcggga9lmz;jq1 /p9lely by the action of the forearm, which rotates about the l9 )zmj1q9a;vcgpgg/ 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 b.hn48swbta9 x5epuoej: 4 nw6lade can be considered a long thin rod, as shown in Fig. 8–4654n:xuj49t6 o.p8asene w w hb.
(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 o4zij 4uy0 jm39rhxc-g f 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 tuxo:e p0l6 ul7rrns (revolutions) and releases it at 7 lpol 0:erxu6a 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 moment eq 6 *4/iwlxanof 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 develop4zlaf3; u buoffdj5y7it4uq 62. 9byy s a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls9ydivtaw 7a0 mp+ p2+o without slipping down a lane at 3.3vp+0o2 wdp ty+a97ma i $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun asl1bgc ls i6g -x+zd8(s the sum of two term(gblzixcssd16g- l+8 s,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. Howe5inht;/yqbhh r/ w7rwdh9( / much net work is required to acqnh yhhwt (dr//brw h9;ei57/celerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withou*k- *x+dtswf(jf :jdg t slippingf+(:xjf-wtkgsd **j d down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and z k1px4 dc18ydits 8 d1zxdycp14klower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball r, kw8h44f a7dg6rc orsotating on the end of a thin string in a circle of radius 1.10 m at an angul arkc8fg4,4 w d7ohrs6ar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uje 5k,tm. ;spvniform cylindrical grinding wheel of radi,st;j m ve5k.pus 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, han9t1f/r baj1b gds at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig.r 1gbat19fbj/ 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 in:bt1i9e,g+heq im) cm ertia by a factor of about 3.5 when changiebm: i19+) imetc, hqgng from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin r8, uqy7c/v5dxhu wc-zotation rate from an initial rate of 1.0 rev evy5chu,-d /qx7 c8uwzvery 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its cent - o 7ebx)-br*9dofkb j+aq4awer at a frequency of 1.5rev/s The wheel can be considered a uniform disk of massak49o- ) we ao+jrq7f*bdxbb - 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 momek,yqvik3 l8,k ntum 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+80 (eho/ivlhyhv9l r 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 an vyoc+* gup)zt *n(glq 8vj 9/ridentical disk rotating at angular speed)c/uro gqv lj(*n g*zpv9yt+8 $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 5 ucl;8xj qcy9 2i(dzp$rev/s$ 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 sta- 3b/aozp z() a1ypjpsnds at the center of a rotating merry-go-round platform of radius 3.0 m and moment o)/p-ao zjbs azp3y1p(f 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 froe:u2tr: n) xweely with an angular velocity of 0.)u wrt:: xe2on8 $rad/s$ 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 dwarlp,s m 4ife,6rf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass r6i,lsem,f p4 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 invol-s+i /l1 r2vg6td yfh8fqnq.mve winds 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 iu74dmugx(4iqb5us22mrc ,v$ 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, initiallh/b,,pzqegk k2 y3op fd:is1/y at rest, that can rotate freely without friction. The moment of s:k,zpkfo/g e 2b q1ihy3,pd/inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m dvnqw,/1:ekx1wc( +9 q ga xvbtiameter merry-go-round turntable that is mounted on frictionless bearbwc9,(1qvang1 t:eq vw/ + kxxings 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 the8c-lho6r l* ggmvo-1ma+jpk ,4,jr r v: ay*cf rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holdinc-o**pvlrca -r jh g 1ja8vm6:r+fgk,yl, 4mog onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwa;2w 2kkxck tfq+g)p e/-o bw.xys faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axixw .)b 2+;efkk/ox p2twq-gcks) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at f d3w s8 tr3 xeg9+a4o4sp.jyta 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 unif z :r7tkry4jw2qb2:f8hh 3wdt.vth8uorm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceler t r88d2bvt3qhj2hfry:tz u k7w4w:h.ation. 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 a4em bxh94sej*nd diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of eh9s bxj *4e4mmass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angularrp+*q0ftfp/o1*akg m speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were roy(y s-os *,5zwj f;ei+ti(hyltating 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 te5(sfy-s* ;i+twy(lh ,z i ojyimes, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-p p -ps.3g0gsufollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 3gg. p03s-pufs 50 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 illustrates an zp0-my/p p1rb -mf/n j$H_2O$ 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 horizo s xs6)cs+c6tnntal 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.rf fa:h15gzxo st-4a a0/,nca, 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 rod, as shown. [Hint: See fg:so14anhf /0za,ata-xcr5 Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on thdwp7y *kigv 9pu (rev394-;frsjxt(m e floor, and we want to exert a horizontal force F at its axlsr9w-vf epdpk* x3;t(jyvg7iu( 4rm9e 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 fla .y lx5qdp9s,0zzyqd; t road is making a turn with a radius The forces acting on the cyclist and cycle are the normalqyyd;z5xls. d z09,pq 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 into0ex-,c js 2 v2 ,mxh*ghresov9t-wk)e a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the r)2ch*jk0, wom-e9veth 2x vxs,ge- storque 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 thin roddox ika:b rw-si9a0c v9*s- 3es, making reasonable estimates for tvsb0 *i:3xo-9wsrik-cdaa9 ehe dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two ufhnkcg c7*0g i;j.+ zv+-c eccylindrical plates, of i e;-j+z + hccgc*nu.7fgk0v cmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 all(xzc)b1-ykpla *+ cdong the looped rough track of Fig. 8–58. What is the minimum value of the verticlzkc+ *b)x dpl (y1-caal height h that 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, but do nbobghzrmyd,8*/x:i 9 ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 8y ;ol:;zd 8vdyd6o*gc5 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wa:z*gy8 do ;dvl6;cy ods 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