A bicycle odometer (which measurerkqh9 pxed5.4/2jkg3d6 kb k js distance traveled) is attached near the wheel h6 35ke jrgd9dj2k q/x.hp4k bkub 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 angular30 gtof-sz w; zd(o9rj velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does theo0rs39-d;w zj(fztog point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular vel:e m-dm(cr u 8dwbl8d ljf qw//*da37vocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain.rcl6pu5sz.+ n5/i08lk*ya ew n: dyg y

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 youd:6 hto/ 0ydkuvk :svspr;+fprj v*81r head than when your arms are stretched out in front of you? A diagram may help you t dv:*tj v/rso6uh+ds kvpy1 p0:;krf 8o answer this.

A 21-speed bicycle has seven st3m88 iv q2vj6gf :iuuprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder giuv88t2q6j 3f i :umvto pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with flesdqwjr.e5y xgx5l3 2;r h and muscle concentrated high, close to th5 ;qrjye3x d2gxr5w.l e 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) ca;*nppkzcw p4a,pq8 d.rry a long, narrow beam?

If the net force on a system is zero, is the net torque/em swtvn ;j-y;.t q2c also zero? If the net torque on a system is zero, is the net force .tjmw 2 -qtvn;c /s;yezero?

Two inclines have the same height but make different angles with typsmyv.l 561d(ti qw7he horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom.yl ti67vp1s5dw(q ym be greater? Explain.

Two solid spheres simultaneousl1scazmo m fag(2(b: 9qy start rolling (from rest) down an incline. One sphere has twice the radius and twice:a(9samq 2o cm(b1zgf 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 have 1,j/ mp0p dngml4p )uuthe 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 totmd4/lgnmu)j0,up pp1al kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving o9gm5ny:m qf12zm amc*ctb( kq9o +fq.r rotating objects eventually slow dgb(: mq+tfmykm9m c* f5nqz.2qo a9 c1own and stop. Explain.

If there were a great migration of people toward thec; zp r(wkh3nc1y iv::lt wt;8 Earth’s equator, how would this affect the length ov: : rky;lhp138;twzcw(intcf the day?

Can the diver of Fig. 8–29 do a somersault without having any ini*0 lycj +su7.y ti,fa d8lfhr)tial rotation when she leaves the bo l lahfrcidj) 7yf.sy+0t*8u,ard?

The moment of inertia of a rotating solid disk about an axis through its 5d-;f,q u:y bpzgtk-ncenter of myn;- u,p5k gft-zbq :dass 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 stfrp j: qiy3(:qool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocyqf:pij : 3rq(ity increase, decrease, or stay the same? Explain.

Two spheres look identical and have 56 tvwt1a1;x;p whkd fthe same mass. However, one is hollow and the othet;va;fx6h 5d1 tkww1 pr is solid. Describe an experiment to determine which is which.

The angular velocity of a 7 ydnd.f llf-)wheel rotating on a horizontal axle points west. In what direction is the li)- 7dfnydl lf.near 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 angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable;gc 8l0t v y/kl09cusw. What happen0c;sy90tvwc llg8k/u s if you walk toward the center?

A shortstop may leap inv ky1 9gxhi 1t-c5rynexdk/u7.:t1 hy to the air to catch a ball and throw it quickly. As he throws uxv kt:/1tdi.1hh g7yy9yr-xk cn5e 1the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservatio-beck /thlr :91dg /urn 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 helicoptekg //u 9d- rrel:tbch1r stable.

  Express the following angles in radians: (a) 30 k9v ;1zj:es30v4aje ar )fm cm$^{\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 coincid.h05s 4mxn11sqx-f .cxb/ bkw8 nbmihence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on ck- q5.mbifsb4mhx 8sw1 1h/0.b xxn nEarth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, ufcwxgxe(po 2l zp.s 4ri (:r9fx.5h6380,000 km from Earth. The beam diverges at an angle g4rrf f. opisx9:e 25pwxcx(ul(h6z.$\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blen6 9obq9si6 9dnyoe 0pader rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleratid96oq 9e 06o9aspyni bon as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. Ifv946du.yxurnlf0u d - the ball makes 15.0 revolutionl d96-4vu0 yxdnu.u rfs, what is its diameter?    m

  A bicycle with tires 68 cm in diameter :, -k :4oqstt/2hqtqjdj zbj;travels 8.0 km. How many revolutions do the w,s/zk2qhdot4b j:jj q t; :q-theels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter +z)p urml.x/p rotates at 2500 rpm. Calculate its angula /mlr. z+px)pur 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 makes one complete revolution in 4.0 su68oy8 fuk s)k (Fig. 8–38). (a) Wk o8uy86f)uskhat is the 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 onazi6 fp8rx :jza i(+:f the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofmz ;a khpp y1bd8.q/yu+c4sa,i1fw 2qnr;9 ux 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 if a particle 7.0 cm from the axcrw8 l5rxy 37gda-hsum:9x(lis of rotation is to experience an acceleration of 100r(9-d rmhxlx:s3 uw87gyc5la,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about kxg4 j 1 oom;a2.wbxy3its center from 130 rpm to 280 rpmag.x;o2 b1jw3 omkx4y 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 radius8yh, lbbc9(1d3nh an *jjr2d 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 q o,:wgj4d-py spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the sgdy-: j,o4w qptart 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 unif/s q shx+ iee*71j3j1gaaq- ajormly from rest to 15,000 rpm in 220 s. Through how manag 7-aa +e/ jhs1sxjqqe3i*1jy revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calprl,o 8rj, 1mbqy /x9 *2gvqzoi2awk4culate
(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 rb27kf y)q7k gm j/ ank+7rdx3flying highspeed jets in a whirling “human centrifuge,” which takes g)jkd /+n7xbka7k3rm 27fyrq 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 360 rpm in )0q ga-hocy wjz+ )hh36.5 s. How far will a point on the edge of the wheelhc 0)h wzaqho+gy3)-j have traveled in this time?    m

  A cooling fan is turned off whw7yr n w6ro/w1chci/7 en it is running at 850rev/min It turns 1500 revolutions before it comes to 7o1/rww6h7 /rycnwci 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 revolutmp (w; s)lu9n 0 )90ixnxw6gnsvlwrv )ions as the car reduces its speed uniformly from 95km/h to 45km/h T69rnuiv())xn9 sx0 m0);vl gw slnwpwhe 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 s(vmrq s,j11m the car reduces its speed uniformly from 95km/1 v1qsm(,r msjh to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal27or: 0gha h 8 ttl7;sv;d7mnt c+jfxs when climbing a hill. The pedals rotate in a circle of ra+2st87c:oh glmxrt h ts0nv;fjda7 ;7dius 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 ofu j/h9y69o tmx 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if t9/x 9omu6jhy 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 it4 , pg9jazv1n9t yzq-n Fig. 8–39. Assume that a friction torquj9gn1v t9y4,tzqzpa - e of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of maar:4ph7 fervu jxwj- 5+1vo1ass m, 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 direction of the net torque on thisarxr u1 p1fj-j+w7v4:o5ae hv system.

  The bolts on the cylinder head of an engine require tightenin/c 7ubpyt6;kq g to a torque of 38p7 yu6 qc/kbt; $m \cdot N$ 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 b p v(s;6( kimvift) tu9(obmlq76en7 10.8-kg sphere of radius 0.648 m when the axis of rotation is through i6mv(o(m ef9q(lpi7; bt 67t)vsi un bkts center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyc* )3gtyyo) 6d75h3mslpmr h3z dkr fi(le wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kgrs *y3k 3r(imhzoyl7 )t3m hdg5pf6)d. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on tiub ;bb+kv,5khe end of a thin, light rod is rotated in a horizontal circle of bb+ki; vkb,5uradius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular speebm6nmtaa0)w1 ujn/i *d aw a 6)n0jnu1tmmib*/(Fig. 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point obj8e-qhu;7y,ymo lz z1uga* q4oects shown in Fig. 8–43 abouu7u za -mlqoz h1g8, ;qyo4ye*t
(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 y 5jd5a)oe2. jsa.zgboxygen 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 correct rate, engiiay44;(q er8kh vao e4neers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius o;4 e aekryvi4q8 o4h(af 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 uniforz/560 rqgqd*wb bw cr3m cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calcur6 3grwbqw5bc zdq* /0late
(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 bdd;(up m.h,lp;hy2qgv3w u29lkacz 1 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 on a small hand-drivengkoc0 jkr:v+,fm)n-v y0. ykp merry-go-round and is able to accelerate it from rest to a frequency of 15-,kc:0)ryjkg.pvf kyo0n + vm 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 isczb-g+jp)yw h7 c,a0o shut off and is eventually brought uniformly to rest bg7+ypw0) ch,jcz -ao by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball3pra csk6hj65 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 bs.6t ddrx y38thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. Ttd8b r.x s3yd6he ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8–tmm+a bu lu9j3 6v,p-q46,-pqluta +mmuj vb3 96.
(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 two ma5 )pm)tr hk5uzsses, $m_1$ 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 f, s, k( usf1 s wqtxa0(dd53dgl-nc) 7tpcyq/aull turns (revolutions) and releases it at a c0ad)c,su57(tdx a3dnl 1pq -sy/w,tgf q(ksspeed 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 has2le zg;k- 6ft,o2.t;i zuwn bjiuq9: j 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 develops a torque og 2fm. vomu.n;f 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 9.09 w,end.8jmnj cm rolls without slipping down a lane at 3.39e dmwnj. ,jn8 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic/8 aq(qfkx,j9ge5 o nh energy of the Earth with respect to the Sun as the sum of two terms, /h( 8gn5k,eajfox 9qq
(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 an b9agh*v3yjk65ii o. mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume b .*o6iyvg nk ijh3a95it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls xqnt(tw04*cu without slx(tn 0t cw4q*uipping 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 itlf3zoljuguv 7m/nj)kl39( 72knp.5k/l rrtf s tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the kr kfrk)lj2 t7v7/u un3o3 /m lz p9f5(gj.llnpole 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 r*k2z15 f n-e iz63g jnkkvec3of a thin string in a circle of radius 1.10 m at an angular speed of 10.j5e1 3zck*gz 3nnek-62 vri kf4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular m1qj2dbzeb9br/6 tsi 0e/m*xn omentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm?/9ibzee6b* xr 1t dbmqsnj/02    $kg \cdot m^2$
(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 platfo8v t9, zhz- apnyuqll2(o(2fz rm that is rotating at a rate of 1.3rev/s If he raises his armsqy (f2 ul, vpt8-lon2z9ahz( z to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can rediv 0zivs. :* j7 5g);svcafxwsuce her moment of inertia by a factor of about 3.5 when changing frosv ;cj0x5wig7 z*fsa) i vs:.vm 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 hl , r4ib3 xl/l t0;2l)muwaibver spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of ,lubmrv0 ial/wl4bx)i; 3 t2l 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 a ,y6vjf,d8 ju 5yeip.ay6h;dn ;mjq5txis 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 ;e fhjqp;,6 adi5d8 j,vmy5y6.ju nyta 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 sticks to it?    $rev/s$

  (a) What is the angular momentum of a figurefhwu*9 ag da8y+mo::n 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 momentm.(c *u)k1: c.m 5zar1bjvgrgjbrb r-um 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 I is dropped onto an id0et-j kx) i:v2tdi :taentical disk rotating ttikdj: )-e2xvi :0atat angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at llkr+ 2go.o qn4g 7l9h2.4 $rev/s$ 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 rotating merry-go-round platu l/vq-l58ewqf- q0dzform of radius 3.0 mz/8l- q5-wqe luvd qf0 and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with 7tf 5 qe5fb -8jm(j8oht k,dfwan angular velocity of 0.8 ote7 -mhb q wfdf,(5t85fk jj8$rad/s$ 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 zvfmeb i4yy 14,qti.usby38 .t*l n2ndwarf, losing about half its mass i. *en,1iby2mby 8fqti34tul.vz4s ynn 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 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 ina l k;gv2-01i3l i/ms(ztcua( upu4pn 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 y ul1,a+ zetn5o* ot,m5r5enc $ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely wi1d3vq3) z2*pyz,ntju ikp; ac thoutnt u;vp3d qai,pc23zy )1kj *z friction. The moment 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 theh+mmt t+5eyq jiud8b-eds*rl j9-jgv /o9 h8 4 edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bh mg +i 9r9oevl td*heum d8b--/s54jt+q8j jyearings 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 rollm 80h(qo6w s al4578polvt6xhv m(wms(e1h yrs on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the perssmoslvtwx mrq 1(h(o485m8 elah7y6vh6p0w (on without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Eartvnnmx2ba ;g 7zi1vhe,ow 2m*4 h such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital wmia bo41 vevm*h,zx;2 gn 2n7angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at0ce :rn2qpcqw , ext/g5y72vn a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquires 4 oa*yzhr 8rtr5d9l,o a rotational rate of from rest over a 6.0-s interval at constyl4dzo,9 rr5 8tohr*aant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eacp4b,xu3un .xd4)rp p qh of mass 0.050 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 of energy to calculate the lidr xp 3uxubpq4)4n.p,near speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed ofimu cjt81q p0tn-kf/) 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 mass 8.0 times a;5k;lmxs1ot0 f ttcc -s great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform spcftxt0lk-5o ;smt1c ; here at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobiled/in4o)mh:,9/aox +ih ywch r is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has :ych4 +/a)o md io /rhhwxn,9ia total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates - ju3e6 vx;em48fdqn:idgxs 8an $H_2O$ 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)ltw tu p3o6u-sb fwm )9xy0:n- is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (iq kg98,s )wf5uc/qa uc.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of 9w5ca/,ufkgq8 cqus)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 standgbv5 sbwh4y+6(j.km sing 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 (Fss(w46bgjvykm5+ h . big. 8–55). The step has height h, where h

  A bicyclist traveling wd u-e,zw5vu b:ith speed v=4.2m/s on a flat road is making a turn with a radius The forces au5 vue- wdb,z:cting 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 lengt2ql:8ripq 3fl x1r9i lh 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 thpi2i lq1l3xr8 9qf: lre torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body ases2m 4pyryn8al 0ca5 2 a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her bo pml a45a2cyn8er2 ys0dy.   

  You are designing a clutch assembly which consists of twozirm24 y2zy4n cylindrical plates, of m4 nmy2i2 4zzryass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 r99lv depxkg4j)8jtwxs3w0/yolls along the looped rough track of Fig. 8–58. What is the m/xy 9)w 0s4 kvlj8j gedxw39tpinimum value of the vertical 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 nulvq4ckw8ts dmhjb1pn66g 1q,g ./*yot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces i, bfcrr(e p (xs:-csb+ts speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angula((+cb ssxr- ec,:bpfr r 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