A bicycle odometer (which measures distance travt zv mv76*qm* :v5oglxpf 24zikm6xo4eled) is attached near the wheel hub and is designed for 27-inch wheels. What happens i45vov6m4 q:6*im x 2xk*t plmzz7v fogf you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rimd 6,sry/z(y-. n vq8hunyix8r have radial and/or tangential acceleranxyq6v( /yd r,-8nisyhu8.rztion? 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 the aybg mpnrfgn1)mpi tic:504: : ngular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Expl4ndv1 miez0,ig5t pz. ain.

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/t;k74r7fqlfi pq /so sit-up with your hands behind your head than when your arms are stretched out in front;kf rfq/77lo4 tip/sq of you? A diagram may help you to answer this.

A 21-speed bicycle has 49f;kbe/hepoj( c9 ym;md2ei seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large jy2 ;eb(/deh9c;oi pkm4m e9ffront sprocket? Why?

Mammals that depend on being able to rue6-g mf 2j,twgn fast have slender lower legs with flesh and muscle concentrated high, close to the bodyw etg2-,fj g6m (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lon r;bf-:j7j;yt8c tthpu4b*rhg, narrow beam?

If the net force on a system is zero, is the net torque also zero? qo(ef,t0 3(cc tw 3d+dzo;rhcIf the net torque on a system is zeroctrwod, 33od( hq +cce(0fzt ;, is the net force zero?

Two inclines have the same height but make differengka5lm- h.hd8 thg r;*b2pz; et angles with the horizontal. The same steel ball is rolled down each incline. On which incline willtdba;rk5 -*hhg.zpe28 l;h mg the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an in1;+vmqecg ,q ccline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom o cg;1vmqeq,c+f 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 frixxeo0 jr 7qltl0271 bod 6h92ot :tahom rest at the top of an incline. Which reaches the bottom first? Which has the grq12xo7ao 92 ox 0 ret76lti:lh 0tjbdheater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet mostjey nt. mwmm 4e1l*uhm4y2: ,w x:v.kp moving or rotating objects eventually slow down and stop. Explain.p4nm*1w:,:umeh ym4lxktwejyv.2. m

If there were a great migro(9+kww k95zano k0cq d.ewo 4ation of people toward the Earth’s equator, how would this affect the length o+k (z409owk a.ec5wk n9wdq oof the day?

Can the diver of Fig. 8–29 do a somersault without having tcg)*s tdv8fax1x, 0nany initial rotation when shen)d c,xst8*v tfg0a1 x leaves the board?

The moment of inertia of a rotating solid-3- b8g 2yn)sxo;+hurzgu amf z*kg,w,h v0bw disk about an axis through its center of mass wym ggb8-hnwkz zxu+,v)r*3,bgh-f2 su; oa0is $\frac{1}{2}WR^2$ (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 holdin5soq*,f7xtj3v 1wm kva 1a/x:.wjcybg a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decreawcy7fxj11avs3 kxt/j . 5m*a,q:vw obse, or stay the same? Explain.

Two spheres look identical anla;az;h0mp / od have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine whm0oal a/ z;;phich is which.

The angular velocity of a ikjt qm1. c1a.wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at.ic 1.1ja tmkq the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rog.. ncu7ve 6 o uocop9acu1gfb);v-1dtating turntable. What happens if you walk toward the center?b.cono- ufegou vgc)ua 7.v 6c1 19;dp

A shortstop may leap into the air to catch a ballo73jeozi afkv)r28 )- v)ifbf and throw it quickly. As he throws the ball, the upper part of his body rotat))ofov-k 2j 87 )zrife aibvf3es. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservat kmj/pk)7 5rwibwk0v- ion of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more wa p)0ikjrv 5wmk7 bw-k/ys the second propeller can operate to keep the helicopter stable.

  Express the following anglesej dyle .();b/yv b 1gy t7u7wgl:ksh4 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 amafi b )3xy;xm57:srxca v:*.,lx sbfl:)kvqbyzing coincidence. Calculate, using the information inside the Front Cov ck 3svli;)xybq:rfy7:bs,x: a*fxvlx )5.mber, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. Tg d1 4i; :ek(ddi5abzrhe beam diverges at an( d5eiidaz1 4dg;rbk: angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 65 r +z6s-v/adec00 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleratio/vr+ a 6zdcse-n as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anoiu bsqxpwym iz02 hr0c/ ,0:;ither child. If the ball makes 15.0 revolutions, what is its diameter? xi,wh;yc:0 00m2p qursb iz/i   m

  A bicycle with tires 68 cm in diameter travels 8.0 km.4tcy,z 8c jji5kps:e . How many revolutions do the tj4 pec,i5k.:cy8sjz wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its abz;f9bqfw;c )v1mul dzn+ 9*q rb-m9gngular velocity in g 1dzn)b9 ;mflvw9; uz+mb9rf c*bq-q$rad/s$ $\omega$ =    $rad/sec$
(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 :ai(/8i- vmzlv4tq3f .gz, qwc5rj ezion in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m fvjzfqtievw ,la. g4rm8 zci :-/(zq53rom 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 orbit arof+vpcby:- ml4l87auq rfum: se,7 ozl+ *xm 8yund the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of 0s-nrda9*td y+i hu 2qa 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 a659eqcv 6pf wvjnw0o3 particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,0006eoqv96 0nvwf3j wc5p $g’s$?    $rpm$

  A 70-cm-diameter wheel acc x15e dw48jz7cg wges;elerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determine s7eg 8cj w4;wde 51zgx
(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 radiud)rnzt;7m wtp sfen5l5:lv 1kq: -2gps $R_1$ 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 Ap6 (6g:xmcfe3zu x8yefollo spacecraft put themselves into a slow rotation to distribute the Sun’s enxfe(68g zxumfy:6 3ce ergy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerw 9y,zsi 8eju8ates uniformly from rest to 15,000 rpm in 220 s. Through how many revolut9 zis88 ,yweujions did it turn in this time?    $rev$

  An automobile engine slows do1f1ur cvqwrn* b or:+0wn 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 yps tm0j+q3ej bd6f adog y +f7:*,dy(jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions +ysa7,(y6+tyedmq *:f3jdb 0ogd fjp 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 a /m5nq91xj.fd:1):*fb u lklyf ssf rsccelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How faxrf.y)fmjs l:kbunl*5sq f 1ds1:f 9/r will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned offpm(uu b+h*e+obx.(q xz1c (aw when it is running at 850rev/min It turns 1500 revolutionsou+u czq+(pha( . xx 1(bwbm*e before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its fbi:,nv-s3g hspeed uniformly from 95km/h to 45km/h The tires have a diametg:f, n-hvs3b ier 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 revolutie xkz.uqkl6x/sjt6 pgt- ah70m g(u3 +ons as the car reduces its speed uniformly from 95km/h to 45km/h.g(6 +0 ke7 g lmk3jauxquhzp -ttxs6/ 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 m63 ampf 41m6v- ef) zj co-;zwlk3ckuweight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 c m4omfake6v6-pj 1uz3m3z-c;w )c klfm.
(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 cm6ks+62 4clu bxvg 9jki wide. What is the magnitude of the torque if +9v u4 6ck ki62blsxjgthe 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 tor 3n;ig*mjo5yx que about the axle of the wheel shown in Fig. 8–39. Assume that a friction torqueg jymo ;3ix5*n 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 p, *2ykj4fvadwivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Cadak2v* wyf,4jlculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightenpw.fp1jbiw+ ,0;cl7drb x qr:ing to a torque of 38 .+,l jc;f0wb i:prq7 p bdxr1w$m \cdot N$ 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, c6rqtm/e ;ep radius 0.648 m when the axis of rotatirtm 6p e,ce;/qon is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.; ijwa;j72kh- qam ze(7 cm in diameter. The rim and tirha ;ak2w-q(jiz 7mej;e have a combined 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 end of a thin, light rod is ron0r5jidp6r/l / ota 7btated in a horizontal circle of radius 1.2 m. Calcul p6natolri0r7jd5 b//ate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angula4wq/v8xsbps cd +up*4r speed (Fig. 8–42). The friction force between her hands and thec */pv w+qbx4psus8d4 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. (a2vns m:lo (gs*xz+ .ihk9yv 8–43 ab 2z n+*9(svov x(l:i.ymahg skout
(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 pjel+ n,9,az n+t+npxfu h, 2lwhose 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 r*zsng.:t qzt/ ate, 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 n qz:gz/stt*.required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder wig 8a8/.jx/f)k syl 4xca l:ivtth a radius of 8.50 cm and a mass of 0.580 kg. Calcul/v tjk8.xcfyg /a x l4l8:asi)ate
(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 zbn0h+ 2zk;p ikwnd1)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 tangentially on a small hand-driven merry-gnaum e.7h)2zuo-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 unifou ahue. 2)7mnzrm 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 and is eventually broughlcf46ii5clx2kn ,edo9 g76b et uniformly to rest2lbl4 dci ie of5xe76k,6c 9ng 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 acaop fx92m/ma oy 4/v-4c*shjzcelerates 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, which f jlv/l b.8ftw8 k/z-yrotates abou. w8-yff8/z/bkl tjlvt 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 considered a long thin rod,wm5 n)pa xvn6/ as shown in Fig. 8–46/x5v )a6mnn pw.
(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 con-nu3) t.oxscpzo*zt0yn(5 cb sists 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 gv:+r(a) i8moh;mdk ifull turns (revolutions) and releases it at a speedv rh)mg+(k im:a;d8 io 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 of inertx )a oe)/h(q.opa mqv)h rwct,4k po24ia 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 developrnqa-0 wtf8r+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 roljxs0w,r0 4u f b z8*/zmeiod/uls without slipping down a lan /zums* d/8jozi0 r,w4f u0xbee at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect r-46xy1i/ qe xeaqs )k. ryi2wl/e qg*to the Sun as the sum of two terms, e/ew 1y)s *qqiari/. xl 26y-r gqk4ex
(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 mucjcmk-f.1baq hwq :9i8h net work is required to accelerate it from rest to:cqkm.81 iwjf 9h-bqa 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 roll t lidek2f1,m3s without slipping down flm i1e2,dk 3ta 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 ve cup.mlk05rg;2 ard m(rtically on its tip. It starts to fall and its lower end does not sli mcr20kar(u5pgdlm.; p. 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 rotating on the end of a thin0)on1cpn vvbkf p7m8,92c gth string in a circle of radius n8nc2f1ch mp7,)t9kvb0v pgo 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 cylindricabs )vcan(u9 i/(e k)onl grinding wheel of radius 18 (eous)b( kn n)cva/9icm 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 platfoge(5 e)zryl2 irm that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 52 i z(yreelg)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 onebe vd8bnko y8h03h *-l shown in Fig. 8–29) can reduce 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 1.5 s when in the tuck pk 3bln*o0-d8veyhh 8bosition, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initialxoi h3/0juw :nylef y0;o/7y y rate of 1.0 rev every 2.0 s to a final7 y3;l0wh0/xuojo nf:yyey /i 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 thro7(su:un.wlg h 2aci g-ugh 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 alu(7g.sn-:h agc 2iwus a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of cq du:t2bfxoc n2n 90 /p-9ghaa 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 angularkbl ,aopvtam)d2 h) -/ momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

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

A nonrotating cylindrical disk olx: iy6ut g83xf moment of inertia I is dropped onto an identical tgi 63y:8xxul disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 8q/ruqedmru .7 ;/mdk$rev/s$ 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 98 bs y,p,.hgsw 6vutzrotating merry-go-round platform of radius 3.0 m and moment of inertia 920 vht u.s 986wpz,byg,s $kg \cdot m^2$ 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 rotatin2addkp +17y amg freely with an angular velocity of 0.8akpy m+a7d1d 2 $rad/s$ 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 intzzks9 /a kl;6to a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its l 9zksa/zt6;k 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 in exceslmu 47uxbpmol9cpom 2* i19s9s 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 os5z j1d+gn e+.o ho*j$ 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 rmlh*1y7l q6 unest, that can rotate freely without friction. The moment of inertia of the persoln *myu6hl q71n 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-round tumhmk qz21y3+ tdl-q5urntable that is mounted on frictionless beari2 kqh 1-dqm+yl mz5t3ungs 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 groundg(0jf a 4:) qmhae uaeehk)g1+ with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping.ej):4a(u+aqgfeg)k1 e a0 mhh 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 side6s;0a eco+xd3/chma 3wyg 3 ro always faces the Earth. Determine the ratio of the Moon’s spin angular mom /a6oor3xa+csc3yh 3;dw m0eg entum (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 from r;ngz)pgi) h-rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acqo:-ed * f.nmp xn:m8pruires a rotational rate of from rest over a 6.0-s interval at constant angular acceleratio drx*mmen:-:n.o 8fppn. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindzdo qgc96 9z5vl *1xw.ak3xek rical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thinxg9z kvz9xk5*l36 o1.wdcqae 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 angular speed of rh hce7,) kp n.*hwii:y /p9edthe 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 a mo3p.h4jyp5i s great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron sth3o.ji py4m5 par 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 automo1 kfcte z56xs,bile is for it to use energy stored in a heavy rotating flywheel. Suppose such a 1ez,s 5cxtf6k car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “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 (+:nz sc vl+pban $H_2O$ 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) i cbkftc a0,;.is rolling 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 can pivot freely (i.e., we ignoru;rbaq 7 n w9-g(l //w8v2fqs5o ooojse 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 gravityf//(u8 -o9lq2w; vrsgo os7wabo5q nj acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, v6be. ww 8msf1ed-che85p:pw fhy6 q2and we want to exert a horizontal force F at its axle so that ew5p- e1 :8shc2 8qwdhb.wm6vfpey 6fit 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 roaiq zqg. 7g)ii 5ekxq+bg7 8mf.d is making a turn with a radius The forces acting on the cyclist +qq gxi qig7zie875bmf ).k.g 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 b+ 78kdi my kqqj 90i6ga2d,zsinto 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 torque required to achieve this feat, and where does the tor 2 kgqaz7qy 6kjbd,9+id0ms8i que come from?    $m \cdot N$

  Model a figure skater’s body as a solid-o1 b;atbz6- m)gftwu(yl2jg cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with o 6l fojw (gzm;1y-atg)b b2t-uutstretched arms, and with arms held tightly against her body.   

  You are designing a clu6k ) jd4ukm1q g j9.r-qv(wrwxtch assembly which consists of two cylindrical plates, of mass w(.qjwj 641 umrk dk-)9xqv gr$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 trago of0h; dw93bsbd:0 eck of Fig. 8–58. What is the minimum value of the verticaldbfbo9;00 w:hgoe3sd 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 d3 o8gx dl:jzna/gy3h;o not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revol vxg/n*2-jb jyutions as the car reduces its speed uniformly from 9 2-vyjjg*/ nxb0km/h to 60km/h The tires 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