A bicycle odometer (which measures distancehl0 1a)) d gbfb86c:1nn9i,zcz ensa v traveled) is attached near the wheel hub n,0sa : bnd8gn zcbz hc69elvf11ia) )and 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 a u).wlh7 oeedrlje*,43h jt.yngular velocity. Does a point on the rim have radial and/or tanghe)de74 rj, .oylt j *wl3h.euential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be dx i9 pp5.ze:rvhhlr -;escribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain.u 1 f(u uxrx9w/2 raphhzm57 b;imi.i.

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 difficusi,il.+p .iz: rh jeae+ ycx)+lt to do a sit-up with your hands behind your head than when your arms are str s+zrx:j+e eh)c,ip.l ia+. yietched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal 5d b fzf,f,unf* gy):xt3j+ bc ur.vc7cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or anc.,c*,r:db+ buu375 yf fz vgt f)fjx large front sprocket? Why?

Mammals that depend on being able to run fastovlp,zk 0 s4xj)qe e.wie7z7 +kmf)o*v4 obn) have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–j)vkw,7f oxe4n) bizqz o s*vmlek7p4e) +0o.34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry jl0m3wzgic-63l*gl 09hms a ga long, narrow beam?

If the net force on a system is zero, is the net torque also zeromh d(ra hq39x(? If the net torque on a system im rqh3(xha9d (s zero, is the net force zero?

Two inclines have the same height but0;zg.mtck7 ei make different angles with the horizontal. The same steel ball is rolled kzei0;t7g.m cdown each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down .kwer4p 8eqnnfwy2)n.q:q f: an incline. One n.fwqwpr4 n2kqe fy8 e.nq)::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 at the bottom?

A sphere and a cylinder(wbr:tf i ,.3rb0fwb w have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which .i3 w f,br:rwfw(b0 bthas the greater rotational KE?

We claim that momentum and angular-z.see6w rctbm 5 r)l* momentum are conserved. Yet most moving or rotating objects eventually slow dowt5 *mzcl -brws.6e re)n and stop. Explain.

If there were a great migration of people toward kq z7( y+lqs7qm;-cpj the Earth’s equator, how would this affect the length of tqjm qyzl7;-7cp qsk+( he day?

Can the diver of Fig. 8dz eodmyl;o:2 i33 p3k–29 do a somersault without having any initial rotation wh:m3 d2o dp yl;eo33izken she leaves the board?

The moment of inertia of a rotating solid disk aboutf3 we*pp;g f2 (uec3qf an axis through its center of3gu(efw*e2 pcpf; q3f 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 on a rotating stool holdgs0)(ht zve 7;qznm:ving a 2-kg mass in each outstretchev:) m( ;stgv7 hzqezn0d hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the dpd3tsv8h4 5rsame mass. However, one is hollow and the other is solid. Describe an experiment to determine which is4ts8vd 3hp5dr which.

The angular velocity of a wheel rotql/2ruv/1m hyating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of th1 l2v/uh /myqre wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Whmweiz+zj2u 3(zi hn/ .at happens if you wa z2i/uhze nzw. ijm3+(lk toward the center?

A shortstop may leap into the air to catch a7m 7cy0k.su rv- xo)p. zoceh0 ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look qm-7ce phu0z)v7 srcoxko y.0.uickly 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 angutu-gqoa 292/unr-zw jlar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to 2 u/z -nwj9goratu2 -qkeep the helicopter stable.

  Express the following angles in razn,2vr4nl:hg2jd7vea u3jqd3 i x k5, dians: (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. Cazvtxc9vh+8j **2sm m2r bmc-qlculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moonvt* xqmbhc*j9v 2mm r8 s-2cz+, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from 7rrc:uwi +f(z (chaa: Earth. The beam diverges at an w (zh7r:cc a:(+uraifangle $\theta$ (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 f 8siz5-e:xoyxb*.dr rate of 6500 rpm. When the motor is turned off during operation b:izx s5 -o.yefdr8*x, 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 )wmf d8ep))fx level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its dimw x)df 8e)p)fameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many 8y+)* nvjzcg6j 1sb uqg vq:mzeg1-u4revolutions do the wheel s j6gbev4u:n)cgv- q8q+y1 g *zzujm1s make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2tz e:,;ymb, bj500 rpm. Calculate its angularz:m , ;yebjb,t velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round mak 4 tfaq;ofg6jjw)m k:9es one complete revolution in 4.0 s (Fig. 8–38). (a) What is the wf)4tfaj ;k gm6j9qo :linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit aroundb x3tm+hr(jkcps9a ;8 the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point:)otupr5;e xbt*, gd rcuc;a2
(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 rni )mp)j(8g(ou p pr4totate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of( )pujp)(i4 tpom8 rgn 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center j,z 7thdx,k3mfrom 130 rpm to 280 rpm in 4.0 s. Determihjm k,, z3x7dtne
(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 radiux29+cq7h)tu 0j-3ihyqnxy u hs $R_1$ 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, astronauby*-mxfusyx,l 6 5 3z(0l klj yrniq,2ts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they a *j l6yrmf3ky0uqlixl(b-yxnz2, s5 ,ccelerated 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 s. Throughyr:a4-3+fv-xd gz -my,oegg s how many revoluti4vr:- sz3oga yxe+m-dfg-y ,gons did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpmtelroe *2 nm/- 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 stw* 60stnvpo+nz :v-3lo :nha dresses of flying highspeed jets in a whirling “human centrifuge,” whi:-h60p: 3zt*wl+ no andsv onvch takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 3 phh1y26k j)bsj;-hrwo1u/)k www-v uc33nmf 60 rpm in 6.5 s. How far wi2n1skj1b)o3v - 6hw/mpw rfuk ;h- jwuch) w3yll a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 revzbeqi 69w y+1r b8sq +1(opazhuo2gi 5olutioii1qewh r1o(2p8gb+z sy zo5+a 6ub9 qns before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65qb gct/h 7;fqf fkm7q18e0f ibb)x3 q, revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires bkc0 ex )fb8qhftim1q37f ;/ qq,b7fghave 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 7z2br5-(vuiq llde74feejj( 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires fjl7 b((eedi 2 jzve4-r7lqu5 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 pedal wheho rxhs 5e,9(bwsg8u /n climbing a hill. The pedals rotate in a circle oe(g8 bxows/ ,u9sh5hr f 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 n 7 hy,;yukzd6*kx3ic8taw4x end of a door 74 cm wide. What is the magnitude of the torque w6k z ckn48yhix,u;d37x* a tyif 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 Fi4/*sj95 r( xph) iyv .ck3ew zqn9t/pkrv)njyg. 8–39. Assume that a friction5y3qc/kn*ieh 9 sp4)v(x 9vkwz.) jt/rj nr py torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached qwecc6r*x 7x 6to the ends of a massless rod which pivots as shown in Fig.cx6*w e7xc 6qr 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 cylinder head of an engivy qr r7a+.5hj2qtgm sef;8h7w33 uyane require tightening to a torque of 3887qr g e 7qs.rhf ;thm 5a2yujyw3va+3 $m \cdot N$ 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 oe 67e2j xt p9ca2f ); wcw.0dcaztjqf6f a 10.8-kg sphere of radius 0.648 m when the axis of rotation is throughjc2ft)9. zda7 wjwe;q fc6t0 p2cxe6a its center.    $kg \cdot m^2$

  Calculate the momentjhi 15 quis;x0 of 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 ; uhx1 q0s5ijiignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated mvn-8 o 1nf2. pos:3y*umyg y7ai*yatin a horizontal circle of radius 1.2 t g*8 osvym-p1 3ny. *u2miy fny:aao7m. 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 cons1k0kqhl+*9k tk, + mnuf co1xktant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 Nk*19 ntk,fckkl homk0u1x + +q 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 ar(zb1i +0obc.ok)wk cr-bq+wi ray of point objects shown in Fig. 8–43okcw-+1b+cobzi i)qk 0( wb. r about
(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 o3a7i8cm+qhrd(x 7a wff 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 cylindrical satellite spi7a mte8-d9 kzqnning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg a t 9zk87adqme-nd a 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 cylinder with a radius of 8.50 coq 2/qb a2c m4r9u.cnsm and a mass of 0.580 kgc/.mc4 9 qsq22abnrou . Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating i yd6ap*1kr+k3d 9y imn ib1s2ot 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- nsxa,0,qlme.*od 0re4ku 6yxround and is able to accelerate it from rest to a freq lxk4 rd .anq0 eomux*,0y6es,uency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually broughw7wt9i5 8:oefsn l fj7t uniformly to rest by a frictional e7fw:isjlnto 5 7wf 89torque 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 accd:*,; ipnosv*sce:shs :7tkp5wfe( b elerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, whicduek mw+v 1b+*ht;ie: h rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest++mv1de ;*ueh ib:tkw to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Figj6 b7qs* blu.a7zw7x f. 8–4 af7w 7zbub.sjlq*x6 76.
(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 consiqjja r- m22:ciqc6 (ijhg*c 3msts of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turns (revur2 nqxl.yeg3: g3ll ;u:*is.wesk- oolutions) and releases irg lu.yns x2g lok.l*:- :3 w3uqsie;et 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 moment of inertia odx +d5l7j;q q fe.yjq)*shq3of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine 17dcd/hl3ov)g 5xj f 19 :eij5bfvasq develops 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 without slippingm+pk*phgsp 8 8 down a lanep*m88p+p gskh at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with +. b *.wcbewhorespect to the Sun as the sum of twoo+ b. *wb.wceh terms,
(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 7.52kcu 4b o37bgh2hjg6u 0 m. How much net work is required to accelerate it fromh gugk3 b264u2jbc7o h rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 k/mrfkemvf: q g.5)df: g starts from rest and rolls without slipping down a 30fk/rm:fm f5d: qv.e)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 vertically on its tip. It startsgiz( vut.g4 uupq4 ,e* to fall and its lower end does not slip. What will be the speed of the u uegp,ivtg*zq4 u4.(upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end ofrth iqn /- q(v+i+29ck3cergu1z li8z a thin string in a circle of radius 1qkq9ziz - cet icr2/vgh 8in +3l+1(ur.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 unihlpgps9v ve*u(/ x357etjl 4mform cylindrical grinding wheel of res3plugj/(t9mlh v v*xe 75p4 adius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotatino,1d -z(cqtwu8da/z/ qievzqik1v/1 g at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.cd zdvqw v1/(/z kuz,a1 qqi/ -o1ti8e8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can red)0yn,/e(ob fvm7jqd kuce 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 poj,(d/) kqvn y7of0bemsition, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increaa : 6ru0oy c.bg3ks9inse her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of niy rkgcb063:o9sa. u 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 3tg0fvf2 (ikz2u4 asenpol * :around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat *:(itsz oel 2af3upk4f2 gv0n disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular moment+x7 ; j-lp4qk7x zjxbrum 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 of t8 s7hs3zr0mz 8h hs+ojhe 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 ou1;oxnv r 8ie.f inertia I is dropped onto an identical disk rotating at anguu1onv erx .i8;lar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 tw s5u er51j5ng qd80s$rev/s$ 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 rotating merry-go-round 8zf(* o ; 9uxrr,0vcyun(gocaplatform of radius 3.0 ,;c yf ru 8a(vgoxu0zr n(9*ocm 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 rot9m3 np-d3uy5a0b 5xri3+lgtsr z( ta bating freely with an angular velocity of 0.8 b iab +3g590t-3(uapmy 5 stzl3nrrxd$rad/s$ 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 colla tz.q/ty8zf9i8m6b kv4qn zk:cyu1e7pses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuq qy:z81y 96ikf4zntzub8/vkte7 c . mming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 120 sy8+k6 l0i krcr1ra*d$km/h$ 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 x,0vu6 weh-o b$ 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 0zkmoq7 qe,(zcan rotate freely without friction. The moment of inertia of the personqe z, k(o0z7mq 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 pers 0gw)k 9l*qiuhon stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of il*h)w0 q i9kgunertia 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 the kvk;8 tp/dx8jil ,r-lrope lying on the top edge of the spool. A perst r 8kix,; /l8-pvldjkon 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 mass move?

  The Moon orbits the Earth such thalnx3v* ;lpu4 ot 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 to*l34 pv;lxnuhe Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rato ( 68hdmnb+4g.cjei ue 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 shappu (wv 0ojkw, w(wr:-se of a uniform cylinder of radius 0.20 m acquires a rotational rate of p(w w (vorw -:s0,ujwkfrom rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.0a), * c0ufh,p/bqkeqgc+n xv; 50 kg 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* 0cbexgp,,+/qv uhkc;fa )nq 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 angula7uf1 rmo;df r+-ptf-pv1/fau r 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 masqr1jz)e7s3 j ps 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-quarterqsjrep7 ) 13jzs 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 energ(7h i4+h/*bzb ttxouhq+ mfl2y stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1ztbfo*/ u7h+h bm+(qlx24h i t400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an5dd(6m wug 0247a xn2vz seltv $H_2O$ 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 rolli39u mkleb ,eq-0ypb5b ng on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignor f:anfu-)z-bd j x;q9me friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleratioq; )xamznbff du -:9j-n of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verticall(kl1q6 g o9(3q zvmb )olt*qqky on the floor, and we want to exert a horizontal force F at its axle so that it will climb a stgvz* 61otl qo3lqqkmqk) ( b(9ep against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn wiedc1/ txxrs 9f8 1 r;aq54frtyth a radius The forces acting on the cyclistfr5c/eyrxt fxstd; a4 9q118r 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 0h+ lpi;j rs++djh2fpvj:mv52 .50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torh sf;pv2h2r:jm+++dlji5jp v que come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rod 0qvmga: p*qzba0dv,8tepw4 /s, making reasonable estimates for the dimensions. Then calculate the ratio of t0qv/zwm :qp,8 p *aegv4bt0 dahe 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 cyl 79kdpjvm-pe 4indrical plates, of madp kpv7j9 4e-mss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rx/i1xe1 vplj 3adius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the verpx/l3i 11v xjetical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ay4s+ ns5;w0xz*kzc) qp7 gq wxk7 w9ccssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car redt:ldr,mrk,7 c km2il. uces its speed uniformly from 90km/r,c7.k rkt,mi ld 2:mlh to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s