A bicycle odometer (which measures dist 2u 6he5sbdhq3lqap3l.02hpj ance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use bu q2pq3j 36pl ahe0 hd.5h2slit on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doesqxjwc/ d8z8acc. ue7u vxq53wuxp9.x53ee(s a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the p. e9qx w3e (svzeq7dj up/c x8.wuacx355c8uxoint 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 angular velocity x6 dy5 howcm:0$\omega$ Explain.

Can a small force ever exert a greatx7yh6z.( h oefer torque 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 sitzkglvupe2 (k-a)o+h ;qnr/2d-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you tohqkn2v( azok e)grd; 2l+ p/u- answer this.

A 21-speed bicycle has seven sprockets at the rear wza,1)- 8m bbvccw5 a ns6.ppkucax,3wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket?3 czwbn,acp1s)-c6w pa5au ,xm .8vkb 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 lower le08hnu726j aj r+3zmbr tyd a5ygs with flesh and muscle concentrated high, close to 8yz my+r6aduntbrj 0j35a7 2h the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers ( j9cvc rcb5n,g/omi+i(rbe+((k7 wx rFig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is thew-h*//kt zy bhc)p- yjulh;5 f net torque also zero? If the net torq-yct *l /-y hb khz/)w5;puhfjue on a system is zero, is the net force zero?

Two inclines have the same height but make different angles with9p-i(-boo xa c the horizontal. The same steel ball is rolled down each inclibpca-o io9 (x-ne. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (frv d77nf h2-xvvom rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the gvh xn7dv 7v-2freater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the sambkpo 7(l b;2x5*kvn rge radius and the same mass. They start from rest at the t(pxo7kg *v5;brnk lb 2op of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and5tlefje8 -3.llbqq4p0olj; 7aovx ;d angular momentum are conserved. Yet most moving or rotating ftj 03q;eob ae4v8o-ldpl7;.xqljl 5objects eventually slow down and stop. Explain.

If there were a great migration of people toward tn y9dpl sj//y7he Earth’s equator, how would this affj nyplys9//7d ect the length of the day?

Can the diver of Fig. 8–29 do a 8j 0p6btitfzg,zk,ny+fu1 5 asomersault without having any initial rotation when sh ,1pfb+z,z0ty8fu k5atg 6nj ie leaves the board?

The moment of inertia of a rotating solid disk about an o+, w/fbb:x f6cye2g yyp4el) axis through its center of f)ef4 :ly bec6 pyyo/wgx,2+bmass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each o 7tnlu8 un6l7yp gmz+.utstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay7lp7.+gtz uu8nlmn y 6 the same? Explain.

Two spheres look identical and have the same m)doj*wil( s2o 4r/n phass. However, one is hollow and the other is solid. Describe an experiment to detdw(nj) oiro4pl*h 2 s/ermine which is which.

The angular velocity of a wheel rotating on a horizuew m5 aj:6huw s/x62yontal 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, dwu6sj526w:e u h /yxamescribe 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 freely r:t-,/try*p.* i jz: kbqi axheotating turntable. What happens if you walk towary.,zt* j h t*i:e:p-bi/xkqard the center?

A shortstop may leap into the air to catch a ball awqlt+h/w*h8 hensk/jmwff .)mw (67pnd throw it quickly. As he throws the ball, the upper part of his body rota7.lp6et mnsk/wfqh8hw*ww ( fjm) /+htes. 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 conservation of angular momentum, discuzt0w dp:o54 72p b (/u:calg2 bybtlxdss why a helicopter must have m/ bdxp:zw5 l otapb : t2l(d0ycg47ub2ore than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: 3 gtgbye o+0-k(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 bec u9:qmr+vp 1qk/xzg 7ok)( 9zh+) oybnrcs.z kause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angulab/qu k+ rrx(s7zk1zy)v)p:9 h k9 oqnc+moz.gr diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the d f jjgfj 60622v3;fars8lgixMoon, 380,000 km from Earth. The beam diverges at0 l j ;jgsd3jf2a86f 2v6igxfr 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. W 1 ) . dpuw ,uozp9rcpxzyy;agzf;i x18;eqe47hen the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as thr zdpc; g)1xwe;;upuqp,azo zfe87i 1 y4yx9.e blades slow down?    $rad/s^2$

  A child rolls a ball on a level flocxzcgj) mqee *)or ).njp;:u 8or 3.5 m to another child. If the ball makes 15.0 revoluticmpr8):xzngq) jjoue; ce * ).ons, what is its diameter?    m

  A bicycle with tires 68 cm in diameterj94rkn c7 1gb/:rlic /:yvbqd travels 8.0 km. How many revolutions do theyrbi: k7n9/v:jg q1dr lbc /4c wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates aqh9 ti9lp ymag5z:ibe;vd-;9:6 si.my/giswt 2500 rpm. Calculate its angular veygt:h;ge. 96i s ql-z m wya;:ms9iv9i 5d/biplocity 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 szq ma1zyiuw ay2s4 z0g1 oqf)0(a,l +y (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 y,mz0(f1wi zas)ay024 qo g1y u lzq+am 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 .t 19jnd gbj7fSun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe wzl27qk,flev ,jpyaqs k7tf )4c o1 z7iw+v)8ed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate ipscl f )n2vuw::a)ek8 f a particle 7.0 cm from the axis of rotation is :2ak pelnuw) ):vc sf8to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accel;otg3fob7. 3yb d;rxa+ kq/iferates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determine frxok;.oq73b/yf ; gt+i ab d3
(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 radiuwb9k,ttzap5 jje6mz. 5un 14fs $R_1$ 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 Ap5ygj5g 2o izc2ollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft cj25z ggy52oican 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 acceler0 m h4h8;guppzhh) /xzates uniformly from rest to 15,000 rpm in 220 s. Through how many revoluh 4 hh;/)0mhg ppzxuz8tions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in c9nkf 0a nm*+bgk);eb2.5 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 fk j2yj 4f*bt(;og ,haglying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaoayg;*fjg b,(2 jk4h tching 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 uni; exk)u-epzj t+1xa 6zformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled jtz6pk+; u)x- az1e exin this time?    m

  A cooling fan is turned omg(a9x7 *jc:kn4n boiff when it is running at 850rev/min It turns 1500 revolutions before it comes to a stop. bm7kg :*oax4c9 jni( n
(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 redyk0m8 s6d 7ockuces its speed uniformly from 95km/h to 45km/h Th8m d7cksk0y o6e 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 vynapwyn 5t;io 2y -*7as the car reduces its speed uniformly from 95km/h to 45km/h T o75;a*y-ypn ny2 tiwvhe 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 we zg7(zz)gh rt5ight on each pedal when climbing a hill. The pedals rotahg t)5gzr7zz(te in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 ceyiso3wmtr5, dqcu 43iaxq0,6 w4 x y+m wide. What is the magnitude of the torque if the force is ex,yx5o 04meqx3r w sit3 au4i6qwy dc+,erted
(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 thevibh,8ce/z9a ml5ipp, t6+b2 a o:d ys axle of the wheel shown in Fig. 8–39. Assume that a frictiona8o2ima6 i vz+9 b5p,sp/:t ylbcehd, torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, ab rq,z*j:t1q r+,ejqp re attached to the ends of a massless rod which pivots asqzpe *j:r, q+br1 qj,t 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 torque on this system.

  The bolts on the cylz)mq p e0 sy)e-5vwt6uinder head of an engine require tightening to a torque of 38 e5y6m-w u)vz0 ) setqp$m \cdot N$ 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 thf: 2 5/ +x svbzgk7z:vqwcvx8qe axis of rotation is through its center. 7:cg/xzvwfs2qvxzv k b+8 : q5   $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.789 0cu eqfkgqw m)ihe217u i+j cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (wc1heqi7 feu80i9+wu)q gm jk2hy?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod ac7h-k/++jil zv9*rkw4qi d zis rotated in a horizontal circlek h+viw-/ 7ilzckdq+ j4azr*9 of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rota(k ov8jn)cm:-nj 2cglting at constant angular speed (Fi8okcj )lnj-n cm(: 2gvg. 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 ii8t -43 -pzhinynm5k point objects shown in Fig. 8–43 about -mhi8ky4-t p znin5 3i
(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 consist3 /:owcitni9ss of two 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 cylindriyuo kwze4 f3v6nb2-8ccal satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required stno- u e3vc 2b48zyf6kweady 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 with a radius of 8.50 cm and a ioc9fzz3r9h.viln 8uf) x(u 0mass of 0.580 kg. Calculau9.zfzhnl 9)uvo30(x c i irf8te
(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 b;m6x.sdvgk/qnhbj )x6z g/ .j )sjxt- at, 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 o* yyfds5+tb. on a small hand-driven 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, a* f .ysydt5bo+nd 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 broug5lt1-fopwr-ln:m:*a nfd v/p ht uniformly to rest by ft n :v-pf/nl :ra1d-p*5lw oma frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-khc3fga 46 2ylyg 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,; 8 2-zp40pwonbjxka e which rotates about the elbow joint under the action p-zeko0 4;npxb8 jwa2 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 thi un ed480ngz-in rod, as shown in Fig. 8–46. e i8u-gnn0 4zd
(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 5 1zy1:fxyvy htwo masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full tm2s*eh7 x9g x2sb7 gdburns (revolutions) and releas bxb2g g7ed2ms79*xh ses 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 hahqagluk:2 h+:o ygl60k;dd, bs a moment 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 devectj3wmox).t4dm;qby : ae, ( ilops 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 rolls wi.k(/r2wa i- h +yud(skw h4frgthout slipping down a lane k r g+a2dr khw 4(yf(i./uws-hat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic e0rh c(o.bnlxnh: x4 9gnergy of the Earth with respect to the Sun as the sum of two terms:h 0(gro nbxnhc94l.x,
(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 radiuea7jrpy/y9 ms 2ihk 7/s of 7.50 m. How much net work is required to accelerate it from rest t a7/ 2pis97y keyjmhr/o 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 gnv3 dy8qsz /*z,1yyd and rolls without slipping v /dzd g,13nsq*zyyy8 down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tizt0ihb9 2om7y5yhn3l)z 8*/ cps t srsp. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use con 3ost)50 /7zhyp*c2 th9 ss 8ynzblrimservation of energy.]    $m/s$

  What is the angular momentum of a 0fn b3 y):kyp4z8mb*.qbr.t:g p isvg,.210-kg ball rotating on the end of a thin string y sq,4gy*i.) krbm b f:n3tv8.g ppbz:in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grpae7-zkl )z+ mnvib9:inding wheel of radius7:9 -) zanmzibk ep+lv 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his si/5btlx )oh xwfh 2+,nzde, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotat,znxhfb/w 2hl+xt o 5)ion decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one tsx0 2dtbndl zb13mt/5rm2- rshown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 whenmr3t 5 l zd/rdx-10mbts2bn2t 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 angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.9eb+jx yf v*k:e9no;s 0 rev every 2.0 so;e*fkxy9 sjv:9nbe + to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a verti iy28 ef:gkxw4 us8/lpcal axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the8 4eygw uplsxfi/k2:8 frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular mombh0mcxw b/n0 ucp. t4.entum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum k0jb/c -ubf27affzi; 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 moment of inertia I21i x59e:f* gp.bwat4gg j k+z v:feyc is dropped onto an identical disk rotatb twe yif 5:fgz*pg1.ea 2v+j:9kcx4ging at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns y0;; wi5ymvp xat 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 center of a rotatin g utw84l yid+wo+du)zwk(t yph00- s.g merry-go-round platform of radius 3.0 m and m)dyyh0wgwul i+wzou80 s( tp-4tk d .+oment 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 rotatinuag p i4-,orh9ga87s hg freely with an angular velocity of 0.uosr87h gapa9h4ig- , 8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about halfb-xf4udc;a (s2nad z; u4 l/wb 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 would the Sun’s new rotation rat4c uz;b(ldb; a4dfsw/ n2au-xe 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 excessv z6y7vr 0v ;0veg1dgpat;g+l 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 q64 gnv3)pua-jr mru :$ 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 plat8l l2)cd-h)i5ap ioqnform, initially at rest, that can rotate freely without friction. The moment of inertia of thelp52h q) li8doan-)ci person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter mes,dw9(h*ka,6tc0gir: aczel 0/ sz d orry-go-round turntable that is mountza60 /,iasogc d,:ed tks9*z ( 0lrchwed 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 e0.5y+ nykyl hxnd 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 withoy .+yl50 xknyhut slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwaye7s9o1bouue 0s 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 cas 7ouose0e ub19e, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rlhs sk+2k* ajy4 lpd2/ate 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 ofvmhp tp i *drb;2jd08/ radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleratb mv/ ;ih8*p2d0tjrdpion. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eachepkn)3jp8uo u,, pc q5 of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solidoenpqc3,jup pk, 8)u5 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 spe6 hyb+1 di(u* lbagce1ed 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 okuvmye 25ras*h 6(y/Sun, 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 gr*v 52k/ msy(ra6oy hueavitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stqhrz , nez qbnpyu f8edut7,)h.7p7n *6y l07zored 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 anyzh 8b7eenf,uyh7 ) z uqdntppr6.,*qz 0l77nd 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 an h9h /l+fd 5)oosog0ip $H_2O$ 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 sua5zoh( m 0ov.drface 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 ignore frictio8 + y2 k5c:p7ud),xpapmm onhcn) about a hinge attaccy:o,2a dncmp h5k7+8mp)xup hed to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is stbay 9 ..b/dofc8 tx(2;aywpy tanding vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (F yxtyoa.tby; 98/d2pawb .cf(ig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn with a9 /ijcpqyo8ky:.-tx z radius The forces acting on the cic qjox8z-pk9y:t / y.yclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins c xh r/93ce.rkg74 yfrwhirling 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 torqh7/rckcr9yx fe.g4 3r ue required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and c*i o9 ;o:bnxgher arms as thin rods, making reasonable estimates for the dimensions. Then calculate the rxbcgio9 *:n;oatio 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 cylindc,q)-jjz at4wrical plates, of ma,zqt 4j- ac)jwss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 track of Figadd/ 5/m4s xe(ddu1 3xox.o xc. 8–58. What is the minimum value of the vertical height h that the marble must 1oe(auxmd/ x./oc3d xdsdx54drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not d,e m4lvdym1dy2241sgym ux .assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly .s)jxd16sdhv-;bui exh 9 :t tfrom 90km/h to 60km/h x:-hv)91 .i tebs dtx6;dhs juThe 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