A bicycle odometer (which measures distance traveled) isz 7wwfnz;x5c0 attached near the wheel hub and 7f cw;nzz50wx is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a pointu vwsnsc 0lr.8hhl(kq: /(rw6 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 would the magnitude o:s rk hh/vw8r(.qncwl 0lu6 s(f either component of linear acceleration change?

Could a nonrigid body be described b.ctjnd 455dhd y a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explainwnig;th(/ x(ad 3 5din.

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 yo6sgabj,)fy4sa1c3y our head than when your arms are stretched out in front of you? A diagram may help you ao,gb6f)aj3syycs4 1 to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thr808gl; p*-l darqnc pmee at the pedal cranks. In which dnp8mq 0ag;plc-l *8r 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 front sprocket? Why?

Mammals that depend on being able to run fast have slender lowepu-fg .zq2cu3 :jh5oj r legs with flesh and muscle 2jq p:f.j 53cuhz-ugoconcentrated high, close to the body (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 lodk3c + zjk2jf+ng, narrow beam?

If the net force on a system is zero, isb 1iwx al 4rh301at0+3mehpbl the net torque also zero? If the net torque on a itmhhl1laa 034px 1eb30bw + rsystem is zero, is the net force zero?

Two inclines have the same height but make d1/safipc 1*mesxy 0 0oifferent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the/ i* mscapyf0x1o1e s0 ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down a-q,j 0utmo)uyn incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the inclit,) 0q umoyu-jne first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have) gy*3/ jgvwfdrm1m3 g; k3sih the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Whic/kif31*s3gvhymwm3d; rjg g)h has the greater 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 mome/1gmu)8pat u nntum are conserved. Yet most moving or rotating objects eventuaunt p1ug8/)am lly slow down and stop. Explain.

If there were a great migration of peoplmoqrx90vka ej19zfur eatgf w;0 t520mq9s3 +e toward the Earth’s equator, how would thi 59w r ;9mq0vfum+0 3oajqxkast r 0gezt29e1fs affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without hxod:xr7z:qg a .vx) .maving any initial rotation when she leaves the boarax.qm:v x )g.rdz:x7od?

The moment of inertia of a rotatun j k:bh6(0h42om dlqing solid disk about an axis through its center ofmdo4q:j6 (h hlkbnu20 mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting onj6*vm;gfn.y0e hzd m7 a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity incrynj; hdfme0*v 6gm. 7zease, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is h+gn;o/ -2il3s0nqfwp gz 4lde ollow and the other is solid. Describe an experiment to determine which- nil0eo2/ngf +3pz gq;wdl 4s is which.

The angular velocity of a wheel rotating on a horizontal axle poin(a*opdk s,f e9ts 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 the top of the wheel. Is the angul, e9fo*ak(pdsar speed increasing or decreasing?

Suppose you are standing on the etb:yox 3b 5xdjc m3d;8dge of a large freely rotating turntable. What happens if you walk to3:t 5xddjbomby cx8 3;ward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. A/g2fz(nvy /ao xzd+onv ).* les he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate i)xf 2/d. oo(vvzngne* +y laz/n the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservimdu* u54xk -zation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stableku5d xuz4i- m*.

  Express the following angles in radians: (a) s7lu;c6m jjevtj98s/ 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 coincidenc t f gr)h7yq7p0a4r8hde. Calculate, using the information inside the Front Cover, the angular dia4ryfhdgapr7t q)7 08hmeters (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. Ths(c1f(l ;vz sae beam diverges a(zas1(lfv; s ct 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 0l+e t+b;k3l ll fluv6the motor is turned off durit l+fbkle6uvl+3;0l l ng operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anot 6gub*si;+rli( 3i jonher child. If the ball makes 15.0 iis3*jrgbui (6o;+ln revolutions, what is its diameter?    m

  A bicycle with tires 68 cwzcq:f5e7 k7 hm in diameter travels 8.0 km. How many revolutions do the wheels make? 5qhzwk77: f ce   $rev$

  (a) A grinding wheel 0.35 m in diamete g0kq5ab f7qd9(b5cx*wpyr m*r rotates at 2500 rpm. Calculate its angular velo7qqfb 5k5grc d0(* m*9ywbp xacity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38). ( f;enm1xxf lz 5: cqzxrfd+d:,,h /vs3a) What is the linear speed:1dx3slznv+ dxzf5/rcf,; hqxem :, f 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) h6lu+cxo f9: hin its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pointi : g7g(+hev did98 fivmepg1:ywi56:kk1cwk
(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 1wv,tgb- * n*gf;rc- sf/nxy ca centrifuge rotate if a particle 7.0 cm from the axis of rotation b n-ycrvg- * *1nsw, gxt;cff/is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel q 05tjad, 9wf:r1 ouushy *.sr isw03bux x5.faccelerates uniformly about its center from 130 rpm to 280 rpmtq f. 1i f0uusbx :. 09r3hs5s5wrjx*a,douwy in 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 radiuiie byh55w*3 6wayoy6 s $R_1$ 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 spacecrauqyf,6hbx 9gsvxkf.xx 7y p*5b(: p;t ft put themselves into a slow rotation to distribute the Sun’s energy evenugt9:pp5xxhys*bvx.f fx ;7b q,( y 6kly. 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 accelerates uniformly from rest to 15,000 rpm in 220 sh 1wvmbt-gyhj, 81w4a . Through how many revolutions did it turn in thibwj8myah ,g w41hv- t1s time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculatuldwh*v-5 flf0h./ ux n9sgx)s.f0y a, xqt b+e
(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 p;y0gw/s-dbr h5 -smoin a whirling “human centrifuge,” which takes 1.0 mingowdsyrhm;/5-0bs- p 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 4 a6ggh4:qwn- j;tyq5 n6bduxuniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in tw;g6najy u-5x qng4:q6 tdhb4his time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 r/+6a( 41fjnotksbqze wp- -ca evolutions before4cpnbft+j/ -ksw(-ozaq 1 ae 6 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 /t ruxs8nduoim62 9m; the car reduces its speed uniformly from 95km/h to r8;d i2 m/xtmus9o6un45km/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 it s3u:qn10f8at aih ):j pot:eys speed uniformly from 95km/h to 45km/h The tires have a diameter :p fn:a 01oatyhs3e iu) jt8:qof 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-, hp:pan m 9 n9vx)*dj0pudla on each pedal when climbing a hill. The pedals rotate in a cir:dlhj v0nxanp9 -u,p9m *ap)d cle 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 n-u*kp w x wx)k7.k2 wn2q,qpqthe magnitude of the torque if the force is exertk-u)xnqk. kw ww2q q,pn7px*2ed
(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 wheez7+)ifay b*nje wt:fi e4dqga 3, rj.-l shown in Fig. 8–39. Assume that a friction torque of4 .zf*+,id jtyer7:wb)a n-aie fgqj3 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass mnoqb6+4 18eu1 +espyronh u, q, are 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 direct,u1obs +h1y6 8n4qq po+uen erion of the net torque on this system.

  The bolts on the cylinder hqrms;sw.jc/6 ead of an engine require tightening to a torque of 38/m6j s;rc s.qw $m \cdot N$ 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.klfi:u ,1je w-,8talb648 m when the axis of rotation is th:1bjkewtf,uai-l ,8 lrough its center.    $kg \cdot m^2$

  Calculate the moment of inert5 /0 ( b+ zjvuvv4.jrtvnwkf7oia 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 .nw+47kz0rvjvutb/fv(j ov 5 can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in asj zvvfjdj-w*zl x7 swr-54tk 3;--g a horizontal circle of ra k;-vxz-3 -zf7*g-wssjv4 ra5 wjdjltdius 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’s2epi.n .aw(man wzo0q6ng+ ) d wheel rotating at constant angular speed (Fig. 8–42). The friction fora e.a60wzgwp o+nn 2d.mi) n(qce 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 point6fpcwwd9 bt4nnhe o+l g,+;e * objects shown in Fig. 8–43 dbgwfl +94ph ,wee6n+nto;*cabout
(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-dj-fk z..jq a oxygen 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 spinning at the corr) h)vff j(r3 fdxu51r4qmeea6ect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and fra3e 6vqj)f15 ( uh dmfx)4rea radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cyl;hgt9b e)dt )zinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculat zgeb);)t9tdh e
(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 from rest to y r:6/e1ga3ny tjp 3izy8tn-m 8sh.ls3 $rev/s$ 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 f+1o+ ulz lr0g9uy,py merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, n+1y 9gz0uyrfpu ll,+oeglecting 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 eventualvy ikb6)0cx -9b n6 rwvy 4z3vyk)hs.bly brought uniformly to rest by a frictivxkvkbw 6) -43v0y ci 6n .zb)9bysryhonal 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–ikn)1m e)/:t u/v fthu9 ;2i*lh mhmco45 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 sole63m m,1ghmd fwly by the action of the forearm, which rotates about the elbow joint under the action of the tric3d,6fm hm1gm weps 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 consideref2) 0r;k 6mdnpc zr,ygd a long thin rod, as shown in Fig. 8–4m nr f2drpzykc)0, g;66.
(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 ofyaw1xw;. p;6re*pa nbdh* 9ms 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 4yj7tyoy x*,wgg3g5 ithe hammer from rest within four full turns (revolutions) and relg, gyy4 5jt*yi3xgw 7oeases 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 mome;eo19 2(c chj wpo q18hdk,kx4fahh k)nt 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 developstn2v0f sqv +w/ 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 7hn- 7iz5j kot-.3 kg and radius 9.0 cm rolls without slipping down a lane 5t zjn-i7ho-k at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Su0 vo g hu080pmb5roper ho:.7pn as the sum of two hp.0m oope7or0gh b05 r:8puvterms,
(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 1640v s*+)-xjye jc kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rcs exj)-jy+v *otation 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 rhl *zyfs086f ig pgz:1est and rolls without slipping dow1gh l s8z6gf :zp0yif*n a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertical twe/x9mjm-c:a;xz9 p2(nqkvs 4(nxy+ k n *qcly on its tip. It starts to fall and its lower end does not slip. What will be the speem(xn/p+4 c(mxv2qsy9 czk t;we9*-kanxj q:n d 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 rotati53o-oi w; elupng on the end of a thin string in a circle of r3i woeup;l 5-oadius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momeni2 j5c2e .xak)8 hioettum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotaeiok.jth e c2a8i5 x2)ting 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 platform that is rotating at v7pi7ef9sn4 h a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48,9ph77f 4nevsi 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 8koflu y+h+g:mg7ks qyef9(0mx .7 nr 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 position, what is her : u.e0l(7kf8y+kf7 9r ynsm hgo+ gmxqangular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase heg0p;/ 15sw(g.bk l ceq nicueme j)p+7r spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate k+ncq l15c e;0/b 7pej.ui)semgwgp( 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 axifww -yllm3c ).63 vspms 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 sticksmlw3 -wp fy)3v.6lmc s to it?    $rev/s$

  (a) What is the angular momen8tim; an co08)rxstvb,9yef+tum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momem+yic288s ucpntum 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 of mom7d)e+*fk edn4rh8axgq(xg 5d1vm *atent of inertia I is dropped onto an identical disk rotating at angular sa8dkedxn)mq7a+ *ve4 dgrt *(1h5 xfgpeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns(lgtstgf038x at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the.9h ;mfxhgvk; center of a rotating merry-go-round platform of radius 3.0 m and mome; ;g9 xhmf.kvhnt 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-e2eq lw0usal7.mzk7 7 go-round is rotating freely with an angular velocity of 0.eqm klzw777e.ul02a s8 $rad/s$ 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 int1bv ,gb5 4f/bfll u4bbo a white dwarf, losing about half its mass in the v5f fbl1u,4/g b 4bblbprocess, and winding up with a radius 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 involb1. e6ihmet8d fme -s6ve winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass 7e0ecb,0,.yu le: z9 pjaqdrr$ 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 ro.+h+q3w8b d3g (jyf. jbxuzuy tate freely without friction. The momen+gy.x.8u(wz +j jy3hdbfq3but of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m18u 0e :nhiz xj,;fzya diameter merry-go-round turntabuzen , fix:80jha zy;1le 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 wi4aw - )pk-zb7ohw lh1ith 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. What length of rope unwinds from the s4za)-owhik pw b1-7hlpool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such ty3 f nc)c.u1mphat the same side always faces the 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 orb)u c3mf y.cn1piting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a o-a58 gli 7rrj64dqyhcail3.zc(4 yn wi57 rl 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 a19p*e i+-fpiar ct :yd uniform cylinder 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 de:if*aey+9- pcr i1pd tlivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 k +)sifqxqx :qe+x:sd6 g and diameter 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 th):i+ s6:dqxxsfxq+e q e 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 l 1lhyt j)pi)/is the angular 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, 4f3x:o3qg9 nhskcopr :y v-s -but with mass 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, whatvx-ocy : sf hps- 3:3korg9qn4 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 fo zx2,l jj*may5*u)b bqr it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to zl)aqj b25m*u j,x*y btravel 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 illustrateslq3am.r-bxn omh0tlz /-u9 3dc e.r.j 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 h odv-ih;i6tq) orizontal 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 lv+ob/ l4+5s s/7pznncr9em tyength L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, ac 5s7s+/ /9nrpz4+ bvyemltnos in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is staq u:uo( 2sya*fz z06ilnding vertically on the floor, and we want to exert a horizontal force F at its axle so that it willzq(o l0s2fu: *a6izuy 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 a8ls y.*6k eeau flat road is making a turn with a radius The forces actin *sykleea6 .u8g on the 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 lengu m g /jsf.d5 y/qzbliyk)r)41th 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 requir gi4/kfyy zjml .ru/q)s5bd)1ed to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arm- *kyagdwt 0mccs2 tpem *27*+zyc2vps as thin rods, making reasonable estimates for the dimensions. Then calculate the ramcc0t a2 y -2*c2p+ypkt *s7z* gemvwdtio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assemblfhm3 db9*bq1.f ,kqje y which consists of two cylindrical plates, of3q1j, *bkbme.f 9hfqd mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alona4f d,i j)pel1g the looped rough track 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 pi)l4ea fj d1,highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but0m-6.hdjb2n usvr j .a do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions aazyw1h 6qxld-:wx x6po6ub/7ww k8:a7ju fs,s the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0d7fzhwsw6 7uqjxwwa uo:,xp1lxk8:6- a6 y/b.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