A bicycle odometer (which measures distance traveled) i1 knqbwxq(+z :xiqmiz9, ;g.ss attached near the wheel hub and is designed for 27-inch whe:mxiz z9qwk;+n1 qqg,(xi.sbels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angula)nzyzr(* f e3x.bw5wlr velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases wzfze3(wnxr. 5)w bl*y ould the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single valuekci(n /b2z,w3hj7k cp of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater tvt 5plo.u s22norque than a larger force? Explain.

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

Why is it more difficult to do a sit-up witk) gej fym96 l3hnk* 2/aeeiq0h your hands behind your head than when your arms are stretched out in friqy mn/he*0a3e26g9k)kfjel ont of you? A diagram may help you to answer this.

A 21-speed bicycle has se38hiw i+3ea(ma ceo +;hd-oliven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or3++ili(mi;3e d cwaeao h8-oh 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 loweg)b g5bg :u /mzt,0lvbr legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamicsg g/5lbtv),z0:b gmub , explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–3 4oj*+jv ub84;mp huk73sl8y2 khciwl5) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If thbkp1 b .ss+3p.krxiy,e net torque on a system is zero, is the net fs+bk p,.ib.psryk3x 1 orce zero?

Two inclines have the same height but make different andl8 byxgpi4xc h jlg;g .).01egles with the horizontal. The same steel ball is rolled down each incli.. 84;egd)jycl1g ix 0bxghlpne. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inz rar p330m5a fypba8/lo9h-zcline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy f-ab0pp a38rm9o yzr/ l 35zhaat the bottom?

A sphere and a cylinder have the same radius and the same mass. They start f002ud dye:kgo rom rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the g00oyd2u dk e:greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most mov,edqyrh76r0 nhyog ;5 ing or rotating g, e0;n7qh rrd65hyyoobjects eventually slow down and stop. Explain.

If there were a great migration of hn,tw *0y wmo3people toward the Earth’s equator, how would this affech, yw*mwt03no t the length of the day?

Can the diver of Fig. 8–w )(t.xzdu(ba 29 do a somersault without having any initial rotation when (a () d.xbwtzushe leaves the board?

The moment of inertia of a rotating solid disk about an axis throubueyyb2og*t7 2 ,v4 xzgh its center of m2t2gbey4vu ,b7 yz*o xass 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 sittpvfhk,mt .k 1cs.1 .pning on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop kp1. ,k1fvctmn.. s hpthe masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identicaq5r *;m n.u:m os/o knviz*gi8l and have the same mass. However, one is hollow and the other is solid. Describe an o; u:rqizkngsi m.*/nvmo*85 experiment to determine which is which.

The angular velocity of a wheel rotating on a horizsit)8d:v)h ixwk+t 4s wu8c(zontal axle points west. In what direction is the linear velocity of a point on w+t uk8zw8 v i) ):xtscs(dh4ithe 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 large-dfzewe)jnbu7 s4( )u freely rotating turntable. What happens if you walk toward the ju-w7 )4ub)(d fe zenscenter?

A shortstop may leap into the air to catch a ball and throw it quickly. As he 43 afs/b,5n wloyvug/ 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 opposite direction (Fig. 8–36g4l5uba n/,3s /vy fow). Explain.

On the basis of the law of conservation of angular momentum, discuss why a helicv 5wa:qb:vo-6za.rl copter must have more than ovabaocw zl5 -r:v6 :.qne rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following d-a,1n k;z 7w, fo3w 0wzwx3ks6qvp1sjxya ,eangles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, ujl7 siq kbz36a.2fl(mrpwc*xv ;1y9 fsing the information inside the Fropai2lc71 k6sf 3lxrq fw9z.* bjvm;(ynt Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Mu/ mda;r33bv, z t sl25dvh/csoon, 380,000 km from Earth. The beam diverges a/lmszstbad;/v 35v r3cu,2hd t an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turb gg,c (7cn-uxned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration a g,x7b ncu-c(gs the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chiu 3lk;+- qpivc5juiyh6 o0 yw.ld. If the ball makes 15.0i.h0;cwu 35 klqy- iyvopj6u+ revolutions, what is its diameter?    m

  A bicycle with tires 68 c9z7qoqrne ur z6719xym in diameter travels 8.0 km. How many revolutions do the wheel1neqrzuy ro679x7 z9q s make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm.*du ,qg/k)un2nvs4p a6)m ixe Calculate its angular velocity ini 4vmdq u k 2e*u)pn)g,x6ns/a $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 uc .(krpo/jgq 44o:mf07psni xwt), ws (Fig. 8–38). (a) What is the linear speed of a child smppwj47w f (rung0 4,) ti:coox/kqs.eated 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 orbit around the St 1.yjf:sn6bi6a 3tstun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spfe (w4u83543wbm )uwqf qikaap g97rj eed 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) mupk 84w/qpl 7/2pyyf iust a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,00y w7pk/yl2 4u8pi /qpf0 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelera+ v1itj-o79 e ufttw mn.cmv4ku.: e(etes uniformly about its center from 130 rpm to 280 rpm i1.ntoe k +v me t:tc. vemu4(u9fj-w7in 4.0 s. 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 radiu -a*ze4a7 59zfh ;myug*wvvqps $R_1$ 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, astqm4j7kr+0;aq :uqnhg; al 6b vronauts aboard 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 inj 4uvhbq;:arlq7mk 6g n+a;q0terval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 r-d -vc9 hg zxxnd6c2,rpm in 220 s. Through how many revolutions did it turn in this -hgx r czd,n6d 9-2cvxtime?    $rev$

  An automobile engine slo+k c(s.r u,y)xx,2avsshyh2sws 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 strjbv g1j 5.u6moesses of flying highspeed jets in a whirling “hu 1jgvu6ob5mj.man centrifuge,” which 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 diamete/ mehz 3tpvvb mbdxlls2 q 9*y2.+.y0u6/ ocbfr accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have tvvoc/3/p.ye f x90qbum t+ybld62.z2*lshb m raveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/m yp,e,oc 2j7ei. e6youin It turns 1500 revolutions befue7 o,c ,oye2p6yi.e jore 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 revjxa c:4nkf*r , kqw92dtt+9gaolutions as the car reduces its speed uniformly from 95km/h to 4a a9n2t*ckdkrt, 9 4q:jf+wgx5km/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

  The tires of a car make 65 revolutions as the car reduces its speed uniformly7mt2wwji t4n w81 ) dokj+4rsk from 95km/h to 45m tdww72jtn8kr4)k o+j w 1s4ikm/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 uh, g10ot*jwxpedal when climbing a hill. The pedals rotate iw0 t ,j1hx*uogn 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 forc4r5: m gman ;,7vg+io *bk 0jzkq8obzyqrdm;8e of 55 N on the end of a door 74 cm wide. What is the magnitude of the tj8y70 ag;+nb*r bz q5zo 8g :vq idrmm,kkmo4;orque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in 5kzde2+ sw3xlwi f7x /Fig. 8–39. Assume that a fricti wdxf7+ 3is wk2/xel5zon torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, j+0 1ad grh .sf1czcs-dt5kf1are 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 direction of the net f-.c5 t cz1k1asrgd+fh sd1 0jtorque on this system.

  The bolts on the cylinder head kccv4,k)+/ nqup*sk3 lkvv2oof an engine require tightening to a torque of 38k p32uv,cs+nvv4*lc oq kkk) / $m \cdot N$ 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 radius 0.648 m when theq((l1ma2mlonlj ; 8hp p ,ycrx*bx:x, axis of rotatio;yc1xpml ( 28: (xhramjnl,lbq,x*p on is through its center.    $kg \cdot m^2$

  Calculate the moment ofz-tq(e;w 52c1v v79c lytte rm inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ig5tl yec 91(m7wt2 -vrqvec zt;nored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotateddzhxri7xp /q 1o0t ;sxh+2 5qf in a horizontal circle of radisfxq/0;i7 1pro xz xh25 hdtq+us 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at const:qw)gt uxxayhs gx1- i;0tmk 11q))uaant angular speed u y )sht-1gtq:)xxgx1 mw qu)aki;a10(Fig. 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shvs9p g.w2r nlry)f 8p5own in Fig. 8–43 abr np wgryf2s.58)vlp9out
(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 oxygecqhr 3;xrz;t 8(z2va on atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinnh(.a+dxf e 2hz* cbuw3ing 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 w23z hfa e.dxc(+h*ubrequired 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 cylin+hb o vv 3f*ve2 cw1vpmpzd-+,der with a radius of 8.50 cm and a mass of 0.580 kg. Cac*+ hv3v +v w1odbvpfmp-2ez ,lculate
(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 afxrm3 cf(sp-* 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-go-*2tmbqysz59nh)6i*;zwb u )pb * xkacround 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 otheh;x9buyqw*2n pb6zk mc*t* b) si a5)zr 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 rotatvr icbemd95w(rko5 0g q(k25wing at 10,300 rpm is shut off and is eventually brought uniformly to rest bb205 (cv(o9gq5 krd wwemikr 5y 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. 8–41bz fzjst q4r0)d)b*g 5 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 acti2n)hk qw,xy,o9;u;gzpbs;opv x +nb*on of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 2hb v,y 9pxp);gz u* w+bonqxnks,;;o $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as showg/xvlk *r7-tk n in Fig. 8–46.x kgktlr *v/-7
(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 twou pwy0/jl,qz)d)dx 5r 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,en6n 7ujn dz* the hammer from rest within four full turns (revolutionnd76zne*,n u js) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a momelwgo8d8h8 0y2v +29ccyal /u )ideqmxnt 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 develd *rh 0 + 1/ei(revybau.yj jyxx*am:-ops 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 r1b x3r.c ss pq;1i6vmcadius 9.0 cm rolls without slipping down a lane31x6;cr.1cpmi ssqv b at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of t2vc scw 3f5vsu0d7n9)gegklq 0u b86awo terms,98su vb kc2gweu3fqd6 a)g0lv0c sn75
(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 .:a: tma7 rjyxm1 . fxknnq;g;fo4oh -uz 9vv;net work is required to accelerate qyvx;oh .o:zta9funmr a1 .n j7 f 4xk;:;g-mvit 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 cb eubyr j(g8*(m and mass 1.80 kg starts from rest and rolls without slipping down a 30e yr8((*bjbu g.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 v3p dcq1 srsl2*z: w*zzertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just befo1ldrss*p qz2 *z:cz3wre it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotator ,a)zd-hpr:z jq:2 qing on the end of a thin string in a circle of radius 1.10 m at an angular spepa zrj),o-r: q:zd hq2ed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the anguhc 9rdfuqrv* )7rus 7:lar momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rfr*d)v7r 9ush7:uq crpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate of 0p8sgnys5i/dh7),z zt lc yr.1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, th8lh ysr)dyzz. tcp, /i70gsn5e 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)uaizb;0s.uy 7ii/ 4i al:7rqx d0 (mal can reduce her moment of inertia by a factor of about 3.5 when changing from the ml7qba0uaia /l yi.;z0r4d i 7x:ui(sstraight 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 rotation rate from an a)xg (( co k ru1yicg.oot 992r-dzmm,sw+tx6initial rate of 1.0 rev every 2.0 s to a final rate of 36u1xa9(ksc gdo( or.rzyocw i)mmx g, 9-+tt2 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center 9gznvww3h7qxex h )juc 4 9./fat 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 ec7fzw)wj.qvu4xn /hx 9hg39the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figu4pw-/geor8gsuf7 p;t s1ikt+ re 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 ofs *8mq:lks 5:inx 5glv the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped or, 73qxzlaj:qv9d d-s3d.sbxnto an identical disk rotating at angularx9qd dldzx 3,qb7-3avsjrs:. speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2qai w(ibv0 f6t3z)k(.u touq2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands (j+jitm(8 pbek ;r ,aeat the center of a rotating merry-go-round platform of rie a( te km,;jb+8jr(padius 3.0 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 wity e 9: ;gw ,czvy-u)pn6exj3vhxa hy7j1tq )k,h an angular velocity of 0ejypze)ck3vxj,wgy a) hth x 6,u7q9:1n-;yv.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 even48lrwe/ r29gruu+ sjwzvv *fkl2m23ptually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what wovl w2lv 9gruwrkmrf +23/2pz4ej 8us*uld 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 wilj-w2.;.cihjq tv 4ykpi,+rq nds 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 zab (k;*qn r5f8as.ruv t7nw6 $ 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, that can rotate fre(y.dqcjj p9nz)o8/*h n 2uu28 hd/sv rw7xfuwely without friction. The moment of inertia of the persod2 r)x8/8uuj* f2 u w/nc 97hyoh(.pnzqswv djn 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 th(y jq;8wm5sg( 3nap uje edge of a 6.5-m diameter merry-go-round turqj(m(su8pa yn ;w5g3jntable that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of 5(6ao i z9upqathe rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of maapqi(o 9 6azu5ss move?

  The Moon orbits the Earth such that the same side always faces thduq kxx5p4i3 3e Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbpi3u 45kx dx3qiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rat qp1xgehs/b r5 5g-p-:qzf1q1uk t:fz e 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 acquireso;f 0(uyoya0 mw v +;z6skra.k a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered bz; +.ykfamkro6; av w0(so u0yy the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and dn6sy,l2u 1y :qpai:ftiameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released fro psyunl t,26a yiq1f::m rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angula,4dnlkipi i.zc;6 hpe0g .a/fr 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 masbo ;3hj0)g avbs 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the 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 momentu b3g)vh;o0 bajm.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy n0wdip l4ne ,+stored in a heavy rotating flywheel. Suppose such a car has a e4 ip n,0+wnldtotal 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 illustraxvqh *)z0 llyr1akf+)g 0ej2)rv 3lao tes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at u os)3dqvg; 00ev5bcnspeed 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 anl26sp.fqho af i/e4 hb9z9nd 7d length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horilo.f7d29fq6 h /henp as49zib zontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is s+c4drpm,0 ax ws.h bv5tanding vertically on the floor, and we want to exert a horizontal force F at its axle so that it will clix, h.spcmwab+5 dv r40mb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4 6foy:,4h7uk6lkohtdflz 4 p /.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycld4koh:4yl6kff6 7plt/, uzhoe 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 slinm0rhy zo,wn92lj,r , n9 /sigzg of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest tonn ,zo,9r/ys,gzh9mw2ji 0lr a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body ask fjrywf)t f9, j r(o:1/1bvwb8fqe6ge yq6*t a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate theqy1t fbw6:f r /89(jk f oj,yer*q1gvwftb6)e 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 cylindrical plates,6esr)j *03hu tlmr6a e of mass 6l e emra0jtu3 )*rsh6$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 t0-bv:nhp :x z9l3auzxj(vjf+cteb20 rack of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop w z2x:v:l(c-9j +tbx3bej 0pz0 ahf vunithout leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dcw8d 2*z .wgryo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed 8nvw6-d 84wqipq/obbh47v ty6 pgaw8 uniformly from 90km/h to 64 d4wo6vihwwaqb7tgbp6/ qyp8vn88 -0km/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