A bicycle odometer (which measures distan 51wlkcx *x/g5k mfqq8ce traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on lx85q c15m*qkxgk w/f a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Do 9x9*gne(y v:bbi6lmzy de/ 2res a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleratione9:gm *n(yr6b2v 9izb/exl dy? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid bodye)wmg uc a84o0 be described by a single value of the angular velocity $\omega$ Explain.

Can a small force evsq* 3fz32;c-nbzps 1 mdayr+wer exert a greater 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 dx tx)qw ;9u ,kdehztyo4k:1b t)3siybc 96lh7o a sit-up with your hands behind your head than whenw1d oih sk3k 9eth9tx) t7byxy:c 6;uqbl ,)z4 your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets a43mtnaw2wk dr0y-4g9 s7kdo st the rear wheel and three at the pedal cranks. In wmok w 4920str4waks 3n-y dd7ghich gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being abvzi v,v7w sp;5le to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, iw5 ps, v7vv;zexplain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35)(ei41 59v j+3ggoju ;pand xe9weyn6a carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the ne:x32icpi 9ti04fsz lsgf 91wyxq9ou jo:e)n9t torque on a system is zero, is the n9os4fnji 9pyw9)gcqt1 2 : e osii xu0x3z9:lfet force zero?

Two inclines have the same height but make different angles with the horizf l. 2f/um6kqwontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom b f .q/u6wfk2mle greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down ant 7dv o (i*xczj9:k1af incline. One sphere has twice the radiu7 iodaf tvj9z(ck:*x1s 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 have the same radius anh-;htose0e 8sd 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 botoe;hh80e sst -tom? Which has the greater rotational KE?

We claim that momentum au6)pzuubvz8h4inq2h, -- odynd angular momentum are conserved. Yet most moving or rotating objects eventualluu4vn8 hydz p ,6z-ho)uq2 bi-y slow down and stop. Explain.

If there were a great migo 2;qg0dvfg g+ fji8-q w yv)7ggk*ko2ration of people toward the Earth’s equator, how would this affect the length of the dky0gov2gjw ;gk +ovqq2 -gi*ff gd78)ay?

Can the diver of Fig. 8–29 do a somersault without having 0/ljr3(swqez9hl 6h y any initial rotation when she(3 rjh9 qsly wl/ez06h leaves the board?

The moment of inertia of a rotating solid disk about an axis through its centen;zz pe xj+dxi0wg1/pq+t ,j /r of mas/gjx/pit j1z0 +e;dpw +,nxqz s 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 roclsod0.u aeb; n22+be xv*-qttating stool holding a 2-kg mass in each outstretched hons ;2v ec2*d xlbe.utba-0 +qand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow 2ys( 3hdlxb u8and the other is solid. Describe anhx (ld s283uyb experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle poi)w b(o;sfsoy,nts west. In what direction is the linear velocity of a point on the top of the wheel;)s,bowf so( y? 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 bgfnnlhs 2ww* *+v* j z6gs/v-edge of a large freely rotating turntable. What h-svb*f g ngv2h+ ns/**w6 zjlwappens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. Asqx 8:i90dp5gs b) hldn he throws the ball, thg sd0:5np98xh)ldbiq e 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 conservation of angular momentum, discu6bxa o*ghu:-st +ooau6u+ /niss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the o6a-ox:ts *nuiho guu ba6++ /helicopter stable.

  Express the followingiqy63pwfuer j ( ,j;2o angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing cd 22hzt abv/-g eyg.my-8x0 lloincidence. Calculate, using the inform.y he8y x-2dtz2v-mgl0l/ bga ation inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverg1x usm/6wca ,gcn/4fmes at an a gw fccx6n/mu,/m a14sngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned 5-s1tn3,pqowq i8rre(b h 0zm off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as 8wtiob-(0m rqenhz5pq3s1 r ,the blades slow down?    $rad/s^2$

  A child rolls a ball on t; oteme rh8 q/n-kzvevzia p78+2k93a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is itspre3tonq/ iv28eatmk9+7 zehk8 zv;- diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many rey h12e l,pibs(volutions do the wheels mh1e,pi(b y2 lsake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 258fbd.2 / 1knfwu5 scpe00 rpm. Calculate its angular velocityp5/d82kfne. usbcf w1 in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig 9f8ikf+ uz9cs. 8–38). (a) What is the linear speed of a child seated 1.2 m from the centefis 9kc+f9z8 ur?    $m/s$
(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 orbetxgzc ;cc l3-3d 4h7o,( dxmyit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spexx6:1atzter: wmd4o .ed 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 centrifr b1p rliyc6.dllpys(8)* v5o uge rotate if a particle 7.0 cm from the axis of rotation is to experience an accelerationb651yi)s*l dv8pr oprlc .(ly of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheeaqs :5)b9+ jyhrg)jxml accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s.sx)9g hbjyar): qm+j5 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 radiuse y;wefuhcz;w 8c ry76 yv+,i4 $R_1$ 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 spacecra bo+cxl0 ildxf*5. vi-ft 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 spa*ocx i0-d.xb f5 ll+ivcecraft 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 15,0/r+-7b iuzwt kw8hw;g00 rpm in 220 s. Through how many revolutions;wh /rzi+tbguw7wk-8 did it turn in this time?    $rev$

  An automobile engine srd tjm .)b9.(o3hst(c x6 zzqdlows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying high ;pt8y:y q a0bc2-i/rg sxh8r ;rbssn8speed jets in a whirling “human centrifuc8 ;-aby 0y:gxspqrrt28sn; 8rh/ sibge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 03x r7u 8fhp*ivf3 qn3,hlwodrpm to 360 rpm in 6.5 s. How far will a point, uxn3 hfo8ipfv w30q3drl7h * 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 revolm*0bwtdqt 0 9qks :lp2utilm0tb tksq*pq:dw20 9ons 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 65 revoltm-:7hx bcn foa a552 i)rwj.lutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.xf552t a.oihc ab7 mlwr-):jn80 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 maztl2lw8:xvr rl qql6 u*/x;s,ke 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/, :t2lsv ulw ql6l xrqrx/8;z*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 herb58- -dpw nkwc weight on each pedal when climbing a hilc -ww8 nb-5dkpl. The pedals rotate 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 cm wide. What is the mapz:/,z nyywt m 4kzd(*vz 4)hjgnitude of the torque if the force is ex*z k,h:(jmzv zt4yd) w n/y4pzerted
(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 torajlfgcy)2aq *b u93s3+ ijf.mque about the axle of the wheel shown in Fig. 8–39. Assume that a frict)c.a+ iujaj gq b3f*msfly293 ion 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 to the endstxx0 g :wb:zn;f i/4i dkhy1*m of a massless rod which pivots as shown in Fig. 8–40. Inixmfb y :0xw/1k4gd*izith:;n tially 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 evy/jnc5vs9) zngine require tightening to a torque of 38 szv/c5j)9ynv$m \cdot N$ 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 spher*q:dx +u6vm g3 vgjmag5v fuz7 o,k/b+e of radius 0.648 m when the axis of rotation is through its ce5z/qg3o vm+gvm 6j7ug, ax kv*+df:bu nter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm i anfcn 1vxpo/- ,nq:t3n diameter. The rim and tire have a combined mass of 1.25 kg. The mass on x faqo3cn-:v1,nt/pf the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizoe0;jd ij aamhizer+7x, 6.-odntal circle of radius 10oe e;,ha.a ii j+-dzmrx6d7j .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 ahnx etwwzmrx,lo(0+r 3+v*p42:jaopotter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N toxmjv(rwo,a*t :zr ha32o40 w l+xenp +tal.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig. 8n.8xrzgka-g8 skg ts 1,rc p0.–43 01.tsgrk ska.g x 8c8p,rzng-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 of two oxygen atoms whose a.4y0(lwnazuuios)l*3k +jilm ug.,g*qnx ,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 aty+xg*ugh,v.l.ph; mh aoiem +ny- 5.w the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the *h n.eygli ;..hu5mhvmag+x+o,y - pw satellite has a mass of 3600 kg and 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 y ;lbpi+949eyso mtt *- zgof2with a radius of 8.50 cm and a mass of 0.580 kg. Calculay99l gbtisom-o+ypf e;4 z2* tte
(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 batefxgp53 wq,5 n, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven i8y0la5;akit rdh0 k js 7g+a.merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate g . 8+ki50aaltydkri07s;jh a 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 10,30bxv6)(n-- 0 8wjnu 7jnhd dplt0 rpm is shut off and is eventually brought uniformly to rest by a frictional wpdb0dv-x( -jnn6)87nlhtu j 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 accele5zwkfe-+ dw ypc f4z54rates 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 tt;sy x9. tqfv7hrown solely by the action of the forearm, which rotates about the elbow joint under the action of the trice;xfsty97 qtv. ps 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 considereqhp . nd .r p85gp,6gv4dc,4jpft*ijt d a long thin rod, as shown in Fig. 8–46. tgr,tc 8j .p4*d,d. gpfnv45h 6ijpqp
(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 co0vg78amourl+7ra1ewh 6c4 f b nsists 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 accel6f vyk35qgkx(5r0 cbferates the hammer from rest within four full turns (revolutions) and b65q 3( fvfrk0kg x5cyreleases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has aq h4ardf2g4c4 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 c,l/zzi 2g2 bqbohb7( 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. rs;uflx3m gq1v)/wmw 5yw:cc;a 7nz,0 cm rolls without slipping down a lane at 3.3wwcwq sg5x7um ;:3v c r)m;1lyzfn/,a $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of thew(9 jim yotwug(+ ,b1i:o5m- gwgndk 4 Earth with respect to the Sun as the sum of two (jw-51 y du(oi+ tik nmg:9g, ogww4bmterms,
(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 massxfr 7+ jwx6w s7ohd9/fwk9a jbre5d( . of 1640 kg and a radius of 7.50 m. How much net work is required to acceleratb57foww(e+kj h969ax/r d 7dswrj .f xe it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from ,rphq vxx ab3kd, ,:3jrest and rolls without slipping down a 3 xqh bap3:v k,3d,rxj,0.0 $^{\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 t0gfk ox9uz(y(qo0 o5lip. It starts to fall and its lower end does not slip. What will be the spe k gqx90f(0u( olzo5oyed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotatinlocya7 ogn ,kxfh47;m08b d9jke:gb1g on the end of a thin strjc o 4m lx78geoh90n:ka d kbg7yb1,f;ing 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 cylindricu k+)r7v vmh/aal grinding wheel of radius 18 )av v+u mr7hk/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 p0nho dn6;c.nk i7mgb4latform 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 rotation decreases tomngn ;oh46i0c7 . dnbk 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one9xd 1.e()4pykr sbyyul73hkd ebh; p8 shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck h8r)9 p3(pd 1 4sle.7ehudbbykk y;yx 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 sp5r9d3:c agrex7lv fw3in rotation rate from an initial rate of 1.0 rev every 2.0 s to aw3 c dax9rlf53gvr:e7 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 rofeg mv.vi;(5vz3ua 5s tating 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 k a;(iefsv3vz u55m g.vg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular mo3jzdo 5 yw/rw*mentum 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 thjpmwpe) j, b1i/c,ry)e 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 divc;x c82; 0pjbfifvs* yra;mt/tn x6),h; julsk of moment of inertia I is dropped onto an identical disk rotating atc ;6bffm;vrti 8jx;* u)ntlcj0h p ,v2;ay/xs angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 k s18jf w6idn rf6d,9k$rev/s$ 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 krfs3ov: 3a6wd g stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment6s odawf:3 rv3 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 an angular cm)9:*2/n5 wq hceg-hskq tcbvelocity of 0ekth ) hnqm -sb2*w/cqc:5cg 9.8 $rad/s$ 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 event)u9z -a mvvzxzsde1vat82( k 0ually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost maase9)v(uxv z0t- 1vk z2a8dmzss 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 e,3 o6e2 g khj2sv-a3 gfpoh0c66yjt qaxcess 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 2y3ira3 fq0 ), whupl./ peavo$ 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 d7,u w7nt s7zc6ri z7mat rest, that can rotate freely without friction. The moment of ine67 s i,7d7cm rnzu7twzrtia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of ml a- i,qkc)o;yi:,m ra 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings ariim;-k:l,a ) qmo c,ynd 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 roped+7;gcpts ,4ku l8bs s rolls 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. 8b+s s,7s c p4;kugld8t–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 that the same side always faces the Earth. De0raygs.97imxn 4s zs- termine the ratio of ty4 z x0-smg7si.9rns ahe Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1w op4+6jdt1 tm 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 cylind 9w )uo0jj /07ssfafmqer of radius 0.20 m acquires a rotationals97ow m0jjs/)uf0afq rate of from 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 sozmpc k;*sn -5 zw:1zillid cylindrical disks, each 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 conservatio5:p;z*c1ziml -skznwn 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 a p0b1or:8sbeg ncga s;a )n:1fngular 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 rv53tl)-eo-f5.h.m 0y zejxzt t r/cpof our 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 gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off nf-t toez zye)l.-r3pjr.ctx05 m/v h5o angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is fye /dqg imfpk687mt /9or it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should kypid89e/qm7/6g mtf 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 illustrateu-bvo r+ m4p -u)ddarx 2kyel7:a5nr4s an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface .ywuwzzrw ca7t8s65- l8jvq7at 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 Myk:u- ibcal0. and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of relyc.:- aulikb0 ease, 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 standing vertica)w80c r csezg+lly 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 (Fig. 8–55). The step r8s) ccz0+weghas height h, where h

  A bicyclist traveling with cjx; +w89(r cep fndh;speed v=4.2m/s on a flat road is making a turn with a radius The forces a ;h jc+p(xfre98; ncdwcting 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 se - x4r(cqxs22vo:cta5 4ziwvling 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 i 5xecw4o: 4s-az c2vxrqv (2tafter 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid9zxw4kfvw +f; cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speedsvw4fzf;x kw 9+ 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 cylindrica7 zf1m3bmkq- v/ khm7rx:gu4 sl plates, of massm xfrk77 hm/4u1s :km-v3gbzq $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rqz, hc(l; i9ym+lg s/oough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without 9hi ,gs;l+qzl ym/c(oleaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do n8zc8 xrl*wnymlu j067f l)*znot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions -l.m6f l*vzgp:h l+)hf .bhon as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. g o)fn. *vl.lzflbhhp+ :-hm6(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