A bicycle odometer (which measures distance travelemx1(ovo.s 8 wrd) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inchmwox (1s r.v8o wheels?

Suppose a disk rotates at constant angular velocity. Does a pointmmv)qe378kpxa v/j-f on the rim have radie7axjm-vf/ kq8m)v3p al and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described byl/h 41e*ts7 mffp4ejv.4saxy a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explaini,-9hoyy voi0k; jn fbbasa2g- 4;ro ,.

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(hiedk; nj69h when your arms are stretchejei6 d(h;h9knd out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at theipgp;/wka w/.htuttl qsi *01ex02w. rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rea tppuhk2esw 0l* qa. . igtw;t1/0/xwir sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower53 ei59q92o;x q+ogj 4vnrfdt mlf;bo legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). 59n9 5;3+mogqjotf qxveo2libf4 dr ;On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a 5b8t86 gcpe -pnh,obllong, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the npa/no3srmr9k py5.z1 et torq3 ra1/9p.y5kmrs on pzue on a system is zero, is the net force zero?

Two inclines have the same height but make different angles witm,j8 mp hvmxf7y 612hhj.v tgi,4+drgh the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greaterhxp46gfm i2mvtg h7r,+h81m j.v y,d j? Explain.

Two solid spheres simsgg; gmy9p5 koc/z-2h ultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottomgh ;o2y/mskg-5pc9gz of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start h-4 e8sgcgfhr-ea,; y 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 g-eeahh,ry -84 gs;fckinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conx;fa(iry6 w w*served. Yet most moving or rotating objects eventually slow downxir6fw w* ya;( and stop. Explain.

If there were a great d sc.(ddvquc) d6a4f( migration of people toward the Earth’s equator, how would this affect the lensfd6d c(qd c(d u4.)vagth of the day?

Can the diver of Fig. 8–29 do a somersault without hav96fm b qm,6htning any initial rotation whe,n9b qtf6hmm 6n she leaves the board?

The moment of inertia of a rotating solid disk ipuqam / tz4zz( o7d.a3l(ha q54knx:about an axis through its center of massld t53z( z7:nh a( 4xkp4az q/.oamiuq 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 roxqn+d;u;x .1zytnktb8v)7vk tating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease,nk+ ;vdqyu.xxn 1tv)t b kz87; or stay the same? Explain.

Two spheres look identical and have the same qcq. )m zzy:4tmass. However, one is hollow and the other is solid. Describe an4) z mc:tzyq.q experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In whduh5codu6:mt/ 2n2l mat direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear accelerat 2mhnd2 6du:olt5cu/mion 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 freely rorjl,zjy /zud2 ,;ild 9tating turntable. What happens if , 9,2 ;jldirz d/yzjluyou walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As h0s f/38zg o5g qy;odpue throws the ball, the upper part of his body rotates. If you look quiopdzqgsof/;03g u8 y5ckly 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 angij;)ogiw ;mm.. ug,zj 1z p4bf: nowx0ular momentum, discuss why a helicopter must have more tg:pwojuoizw.0zimfb ; n1; ,4mj.g x)han one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles ih fjh k7k:d :*pxsx 2lzr)9:8ktsen3in 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 coincioa bfns nw,z1 czjob,wb1.p9k7, bv66 dence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sw, nzo6nw , vbfk, c.bs61j7z1bpbo a9un and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam divergpb,1vprm1g*x)c6 ug )f9 xhkkes at an an )rcm6vx1 kkp uh9x)* 1pbgfg,gle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the mlhled8 kzs 075otor is turned off during opera5 7l h0sk8lzedtion, 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 floor1 tp q j;t8 jiof2,0*)poothhv 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diamet 8h fo1h* qv pojo;,t2)t0ptijer?    m

  A bicycle with tires 68 cm in dinj)ks ev c0m,kp2i qr0p(a izc*96lm/ameter travels 8.0 km. How many revolutions do tme),j0kir 26i(p svmzq/a* cn 9pc0kl he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcua- 1z+7exvw(n ib stmodm7 9v7late its angula1 vomas+ b (m7 ztviwx7n97de-r velocity 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 make- qpxn bxrw6,fg tjw)/ulnb8-0 3vyh4s one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear sw,g0p 3xx) vwj8 4- n/rhut6fybnl-qbpeed 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 (a) in itp0h omu76h6 pls orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poiao75q yjs)j4 l(*vvrs nt
(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 7.0 cm from the q5p s 0ldax3s0o1-ei laxis of rotation is to experience an acceleration of 100,0005os3 el1q 0dxaipl-s0 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unifo6jcbuh 4au ysl yw:6 r4ta7x0.rmly about its center from 130 rpm to 280 rpm in 4.0 s.y7utrhxu.w byac6j46a: 0l s4 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 n-z4jv w7kvoi:jv2 b 8wvo.x:$R_1$ 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 s2 xg-kk5ke ,r/cc49a vjzc*g5a1xamwpacecraft 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 of as a cylindere 2m/4-,1xa5k kw 5c9rzav xcag* gjkc 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 q /wvn;sjyv0v6 0g uw:pr4b +ifrom rest to 15,000 rpm in 220 s. Through how many revolutions did it turn v6r p;w4v/+bs vg:j qu0yw0in in this time?    $rev$

  An automobile engine sloawd qi-9ay*thq-hg3 1n(jyp :ws down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirling :k+rcyp(2ct+)e bdu h“human centrifuge,” whichhe)u +bdk: ctyp2(cr+ takes 1.0 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 360 rpm in kwl6fy;suqcd ,9 + xt.6.5 s. How far will a poix dq9.wc;6yu tks+ ,lfnt on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when ivri6svfg*a9 4 t is running at 850rev/min It turns 1500 revolutions beforerfg69a v4*vi s it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolut 2lz 5jrz. l+x1+klzd66sn jshions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have ajxdsjzlz+ s l 65.hn6 rlk1+z2 diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed unifpk+ : y*5pud (mawtzz(ormly from 95km/h to 45km/h The tires have a diamet( :wa5+mpku d*tpzy(z er 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 weighrqz 8 tjivsq.8ry,b61gkbml ,- 8w:zft on each pedal when climbing a hill. The pedals rotate in a cil8 8mz rtyiqsbz-,gw rv k.8q,:fb61jrcle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is the 3lkfm)i,fl ec1kwd yv-:8j 2kmagnitude of the torque if the for k f:-lmekf,1 vl8i3j )dyc2wkce is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Asblkr46pcw+6 jfx hr )*sume that a friction torque of fh6kcr4rl+p6*) xwb j 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, arn, -scjogjnhh s yr1(85*ow v2e attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and dn gs*8osv, n(jhj ry 1cw5o-h2irection of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque zurp. 2 l1;o)b j(vmyaof 38zlv; 1my ).o2(jbapru $m \cdot N$ 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.c4;i(vzmgc/r1p eyipv9* 0di8-kg sphere of radius 0.648 m when the axis of rotation is through its center.ird v/c(*czp1m9ep0 gii4vy ;    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in de 8dlwqi vk*v4ubo5 3;iameter. The rim and tire have a comquvwv5l 8*be; i k3d4obined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the e1zby-w lk5x i3nd of a thin, light rod is rotated in a horizontal circle ofi z-y1l5wbk3 x 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 rotaxj(rp ey p bscc8.))u:ting at constant angular speed (Fig. 8–42). T cp8yxbrjc:).()p esu he 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 l-akt ld+vcu4c)4*1 0p wuzow xwgi/m4( gv 1sshown in Fig. 8–43 abo)4-i*wcv4 cd owlwl0u/azv4+gsk tpx1(1mg uut
(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 wi uz .sw/h. k7ux0ij0those total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct ra k,m y *2s3qxfbjns:*ste, 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 s:*sbsk y*2jqsf3nx, mteady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder withslxu)v-g zu/dh o -y+, a radius of 8.50 cm and a mass of 0.580 kg.ulg ox zu)yv,sh+/-d- 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 fro9t bl *dhe8v3am 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 wuq4m+ )zgl bh wz2+xuml0**brry-go-round and is able to acceguzw0 l*2 mx+z)hqmwb* +l4ub lerate 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, 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 o:x v7lz 5tp0pand is eventually brought uniformlvzl5 :70otp pxy 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 hpy3qlu,1zmz xp,71 qhgbv50+w.hy e–45 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, whicl bydsuex9g i -1*16kch rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to91u 1i 6cd- gbsxyle*k 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 3z)t5u fvgo;c.b1nq 7qmclmc**tza 6 rod, as shown in Fig. 8–46.1*v a3fl 7tg )z*ccuot cmbq6;mqz n5.
(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 massjo/7fqkh xf ht. d+)g;es, $m_1$ 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 turn9xp; z)j (ek5cllc, gvs (revolutions) and releases it at ,j xvkpgc9zl l(c;)5ea 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 momq,t:b7u r9;m, t0wrs9 lacjq aent 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 vi.oq p ;m1t76ulv )wow( u,titorque 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.remw92.b52vbswqvi0 s0h os2gnb 6m 53 kg and radius 9.0 cm rolls without slipping down a lane at 3.3iv5msh52q nv0.eww22rs9 s6b bmbg0 o $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as pny, b,3d egm92s e*ygd1mz,lthe sum of two termsd2 eynyg*mb91m ,g zels,,3pd,
(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. How much n/epn)gzd c /m)et work zm/n/ epg cd))is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mavd78+ r3*dxvych j: lbu/0rv qss 1.80 kg starts from rest and rolls without slipping down a 30./87+ rrdhd *yxv uv0: 3jbcvql0 $^{\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 itzl f+ibcktyr4+1 .qjk e q0l)/s lower end does not slip. What will be the speed of the upper end of the pole just before iqb.tlcki1qzylke 0)j/4 r+ f+t hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin sz v.wg2i8xq21qg, flhtring in q2livq1g8,whxz2. gfa circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel/ (iiyx d6o, msf*0law o6ios /(iwmlf0,a*xd yf radius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platforpy+hm7 ygck7i6u :dx.: apk(we;u7 fdm that is rotating at a rate of 1.3rev/s If he raises his arms to ad7d.(u f:awhpy+k i cmye;g6: 7 kuxp7 horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce w4 o*lcl2 b.c ra eh6juf//ri(her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If(l.aj2 6frc4r *hclw e/obi/u 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 rob-b )(wzjyh8s tation rate from an initial rate of 1.0 rev every 2bzwbjh ()sy8- .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 rotatinq +ifi qnk/*yirc*+-k3y5zdm g around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay stic/*ri d+nmkyzifky +q3q* i- c5ks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3.oj (outdvm,rh2222um 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 Eu/x 8x3lkgyt9arth
(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 cylindra ) ar2vk0k+8(lgytfc. u6t//ryn:z3k bs ssrical disk of moment of inertia I is dropped onto an identical disk rotating at angulg30tr:(.sbk+ / safk 8uy)rc nsv2a/tzlkyr6 ar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2vh8xh9kcv k28kw:* 7to2 cpqq.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 stacz az02jbg 17bnds at the center of a rotating merry-go-round platform of radius 3 7c0zajbgz 21b.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with ap7pdu eavb; a )9)m; .xgvsz7sn angular velocity of 07zgvba)9vs. ; ); apmsp7duxe.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 colhy3h*)-o uzvi lapses into a white dwarf, losing about half its mass in the process, and winding up with a razvhi u)-3 yho*dius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds inv0hv tpkx+ :1o97 tn)y+9ambpsnuht0 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 g4.i y pi2 88ss+zrkkupmsr 17$ 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, thatbv-s4z,o7 il6tothkzgs)vsw6bg;*3 can rotate freely without friction. The moment of 3bo;gv zikt-4)g 6,w67sv hszo*lbstinertia 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 edg-zjw 1irt.0.u4 gd7 vcdana u,n z4w(ce of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia v ( z4t,nrc 1 c-47 .n.ajagdid0zuwuwof 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 endhvdc) pkr0jah0rc0:5 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 behindh h r0jrvc0:)k 5pca0d 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 Ear9yw;- d3 r4jl n0luxhhth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angula ru43-;hhj x0ldnw9l yr momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fromwe-n ade+wr;2 o7.o t,bvbfz k*a/f n9 rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylbm mrre ix;f(v/9iq*/inder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivem9*m q ifr /ix(;rebv/red by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg awy .ypglg/w. u,cxy5zza* j8*nd diameter 0.075 *, wcyu ga*5zzx/8gw..yp lyjm, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the 8u5 mepxwgmoaj bbm*853 o s: d/6)uneangular 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 massq (qsdx+u1 to( 8.0 times as great, were rotating at a speed of 1.0 revolu+q(s(o d1xquttion 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 angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is forglg1t 6r0v y1n5c0:2kozk fqa it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total0kq k2ar gzvt 510fn:1yoc6g l mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “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 illustratenlv/xq 8w.euj33bx 0js 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 rollinc0d9k:6 vzt jl:ut w5: fx9*6ei kwkxhg on a 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 can7ozr r:w 4b;rxh,w1d/d.v(ywzuw k8n pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angu,vrzxyrr(7d:/4 .w;8 uwzkon w1b dwh lar 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 vab xaj+j8q so: t)mi77ertically on the floor, and we want to exert a horizontal fotxms 8:aibj 7q oa7j+)rce F at 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 making*d io2p ou/lg4d/n/ ks a turn with a radius The forces acting on t2/ ogdd*o//lisknu4p he 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 length 1.5 m and begins whirt/ 6sxqvxlpl(f23 y-:lh ygote,:;a bling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm 3bo p-:q:,6lh;v x/(asxt fgtyy2le lafter 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cy8v.)oso ruhq *wuv u3(+z7 jkat,ripu1n*j/ ilinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of3,8v(u7+iqhkoi just*j1r .ra v z pu/u* ow)n the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembl.fjx ;vy kn1+qy which consists of two cylindrical plates, of mas;+yxf1 n. jkvqs $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough trac- (xaijkq 5y ggd0k98nk of Fig. 8–58. What is the minimum value of the vertical heig n9qdk8kxgyg -5j(ai0 ht 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 not ahy;o4yoh mr+( ssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions yn; vh6z ye9q55hcd4.rle;: e4oz ij nas the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the anghy9cqjoyz evn5n5e ;r4.hliz ;:e d64 ular 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