A bicycle odometer (which measures dim1 -e1cwyrwe.stance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24.m1y- e cww1er-inch wheels?

Suppose a disk rotates at cons8b -j )wadse7otant angular velocity. Does 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 acceleration? For which cases would the magnitude os - 87)abdwojef either component of linear acceleration change?

Could a nonrigid body be described by a single value of the ah9rhy rl,h zrye*7hcnbc p;c, s4x:-0ngular velocity $\omega$ Explain.

Can a small force ever exert a greater torqr.nby ).fqte6 ed;/aoue than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your handsckit 5iy8b 2l)l f-ynyv 4 9)9oc/hpdo behind your head than wheivcl )yc-) 2b/5yl8p iy 9oo9nk4 tfdhn your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets a:oe(dh;u;hmat 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 to pedal, a small front sprocket or a large front sprocket? Why?mhe hdo:(u ;a;

Mammals that depend on being able to run fast have slender lower legs us du) o/6o: aigg *s,lzdy6i5with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advant sg:ay lsoid)z66iuo/ g5*,udageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow/dd5n.* y qud h5paxk5 beam?

If the net force on a system is zero, is the net torque also zero? I 2k+f vy9a3f06blfm gdf the net torque on a system is zero, a 2 +d9b0fkf6mfglyv3is the net force zero?

Two inclines have the same height but make different angles 6r(k twqlma1.jei j2l6 )9djef.u 3f qwith the horizontal. The same steel ball is rolled down each incline. On which incline will the speedmjda 6f)2i e(6ktllu jqf1we.j93q. r of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously star6 .mwm dm;h: oxlsms 6v+4 /gxzu8gl9ht rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the inmgoummm;.g xhx+4lw: vhdzl9 6s s6/8cline 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 and the same mass. 7zk+adu g-mzf 6-m*wm They start from rest at the top of an incline. Which 6*w 7fu-k madzmzm- g+reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are v9iwar-n/fq+0 +f pd iconserved. Yet most moving or rotating objects eventuap/d r af+qwii9+nv0- flly slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how woud:81m wr1 pqhgld this affect the lengr1m1hdg8: pqwth of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial ro 0qti8c(08d jp/9:4do qmgwmqi wbz)ytation when she leaves the boardj80q4z0 wmmqgbt:)/cdiywq i9p8 o( d?

The moment of inertia of a rotating solid deqo8n xfov :x2)62kel 1es 6ukisk about an axis through its center ofs vxe6x)e:1f qln k ou62k8o2e 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 et2 rttf+r x0o6 a3j2uon a rotating stool holding a 2-kg mass in each outstretched hand.aot3 t22ert0 jfu xr+6 If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and has:sid6h2rs bihi8 :2r6 vck0jve the same mass. However, one is hollow and the other ii8 rvs2bs2dr hs0ick i:6:jh 6s solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizonss ,o6.pevsyy 1 fzvu9u/g0 i:tal 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 the wheel. Is the ayvu.v6: os,fszi/ 9 1ep0gysungular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotatingxu1 ixdwwkd d-r175.j turntable. What happens if you wwj57.1d uddixk1x r-walk toward the center?

A shortstop may leap intiy(wbdn iy2ys-44j 4 qo the air to catch a ball and throw it quickly. As he throws t(jbd iny4 sw q-y42y4ihe 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 conservation of angula+9ml( mbbotb) r momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can oper)mt+(9mlb ob bate to keep the helicopter stable.

  Express the following anglesgw(7pym.wl u4qr 4zy.op g0b2 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 coincidence. Calculate, using th,vjj:1e6hsfs e information inside the Froj6sf s 1:jehv,nt 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, 380,0o kpg 6lsm5 .8ezkn.;xm-l.sr +exd3f00 km from Earth. The beam diverges at ankse +.m6x sor3 fl-;x5nm 8zk.edp. lg angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blenderwm8p4g.x d jt9 rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow t9w8 xjdtg4p.m o rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anolj0 8uy - 7x4z i.t.jtduro8kpther child. If the ball makes 15.0 revoludy-.jkuu txt8. rp 8jloi z470tions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revol2yw:6z. ahmp uutions do the wheels mak6zu. 2amh :pwye?    $rev$

  (a) A grinding wheel 0.35 m in diameter ro 5u804fok++kxjr l m//ikyifotates at 2500 rpm. Calculate its angular velocity ifk++ xji8/ ko4m/l0 ikry5oufn $rad/s$ $\omega$ =    $rad/sec$
(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-roo *exksrt*q*sp;8ec8 q qz::cund makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is qeqsko*px8 ct8:eqz c:sr* * ;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 of sjcu8eb d , b(. x(o0gobxp1hq6j8,jkthe Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe gxp* 5h3i2+4 wzmmr9rmpcm a8ed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7. mv q-8cvry0q;t1m*rw2 ibhr70 cm from the axis of rotation is to experience an acceleration of t ;7yqmirvb *q81-v2r wh0cmr100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly a8xm-d. qiw8rfo je ;x,a4;lp3eok ft2bout its center from 130 rpm to 280 rpm in 4.p3 8 qmw-;eaf4k x,2lo e;o8xifdt.jr0 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 radius vp ah(q5hax+ u/03wba $R_1$ 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 equ-)3 463 irpfdcv+ cwsz7huspacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder wit) 37urfz+c p4 cdi3u e6svq-whh 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 li:oz*w7 ai5u ab dvf: (nf37lrpm in 220 s. Through how many revolutions did it turn in th fvl77 alibn5uz*di :wfo a(:3is time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm ino x.ph-if:n) c 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 highspeed x,5riw 4j+eqh*-jpw in6z te3jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revz*+ e-n4p5h iwjr i3w6,ejqtxolutions 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 36.7ct6pfn 7 rnd0 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled indtpf.c7nn67 r this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 revn:g oz6zs+fj3 olutions beffj 3:s+nogzz6 ore it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as thwob3 ie, vu9nk:+uvb aan,kj6:smc:0e car reduces its speed uniformly from 95km/h to 45km/h The tires have aivsounwack :bj39: u6 kemn:,v+ ab,0 diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformsu q- a q)6xzl.igyg35v zz:*kly from 95km/h tigq -a) g:uv3xqkzyl*5 z6s. zo 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 bu f8 7dz0p:mexike puts all her weight on each pedal when climbing a hill. The pedals rotat78u: xmfezp0 de 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 ofv 7.istch, ,nmdiw,- k 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force hn, wmi,d ivs,t-k .c7is 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 to xqvfwy;7v b l4; ts f5ctm-k-u(c+(ujrque about the axle of the wheel shown in Fig. 8–39. Assume fuqv7w;ul-bfxvt(k 5tm ;jsc +c(y4- that a friction 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 ends of a massless roel6g l+g6i )+-adqgand which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. ga-q +n6 d)+e6lgi aglCalculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder.j d/3ry+qipfd, 7r 4at6ik f yh,daq2 head of an engine require tightening to a torque of 38 y d/hq dft7d6rajy,ai+qp, r 43kif.2$m \cdot N$ 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 1d4w, be4jw th*zg 6w1f0.8-kg sphere of radius 0.648 m when the axis of rotation is4 zfw w6g1b4*,twhjd e through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle z/p*tz) ;fhekwheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub canz;phf/)zt *ke be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is r q8;2+nymbl;smnk()0e nxb rt otated in a horizontal circle of radius 1.2 m. rb(;etmqn k2bm0n8+; lx )synCalculate
(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 rotatn,s690-.n sp pjm az a/ym6wyting at constant angular speed (Fig. 8–42). The friction f ss.ztp6-mj0,6aanp9 y w myn/orce 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 ofit1x-14oij et point objects shown in Fig. 8–43 about jxi io -141tet
(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 consi 3pvxl 7xtv7x3sts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, eng8n mdd9m-t 2f oi vtz++et(vu9ineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a m+ vdmditov t-z2em (tu+899fnass 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 with a radius of 8.50 cm anw o)i;wr5r* s xze,rg)d a mass of 0.580 kg. Calrwrz)e5)ix g w, r*o;sculate
(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, ack *cfk1 x k;qv2diw0-kcelerating 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-w:.5k4i;bpkg lvm mu8 driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to pkk:i;m4g5u p .bv8wm lroduce 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 fco)fac. 90wwn r p)ixpj 0)7yis shut off and is eventually brought uniformly to rest by wycfnc xa7 w0jfo9)0ir )pp).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 accelerat;l8fs *ho1 lsxes 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 s(903t yux4b2bf qlgg is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to l9 b04xb2sg 3q(y futg10 $m/s$ 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 rodb st(n4z st71l, as shown in Fig. 8–44( bs7sltnz t16.
(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 twoy yp.5rb(f*/ 9xkprapzlh g90 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 ,:wueuo .tpd5rest within four full turns (revolutions) and re.t deu,u5op :wleases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia oocm;u5bu0/k *entxi ;sr 9v 1.m yct2gf $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 28gd g*:2uf.8 do )i(ima y(arve0 $m \cdot N$ 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 q7c3eh8.zzlbwnn-4qhjl .o 1 rolls without slipping down a lahzon3e1 7. .b n4l-z cqljwq8hne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to th*p,* ft gq,come Sun as the sum of two terms, *gocqp,mt ,* f
(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 7.50 m. How (jtk)u 1 iq:g a0qj6bymuch net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Aiay0(k)jtuq: jq 1b6g ssume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg stardzzlmzcl + 53f*l-/rlhj2ku (ts from rest and rolls without s3(- ldm cjlz luzzl/hf2*k5+rlipping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and itp10ehy q 2xfa-s lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hinty peq1h a2fx0-: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin12o75s ho2bo zh .qcdq string in a circle o.oohqq21o2 bc d 7hz5sf radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of: vffvh t7ut k3mm+yh+;q9g/s a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when r kv9 qm :vhh7 mf/tg+ts+uf;y3otating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at ),g;hm yitc4( k/o +ys7( o2dwocvurza rate of 1.3revhw t;ckv,zo( y27ur/co) gm4 i +(ysod/s If he raises his arms 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 Fi-zmf7cl54pnx g. 8–29) can reduce her moment of inertia by a factor of about 3.5 whmcl zn7 4pfx5-en changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.0 30g.wg xn-o7ts2tdrj revt7 3txo0n-2.gwj grs d every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through 1 c4c+okv-s *iuiny1k2(jqxs its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after*oc 4v2 i(i-y1qsnjs +kkuc1x the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3.5wmu)s:m g lnbua(j-.id2bb,:/wj7e i $rev/s$ 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 the Earthc5m66p2w h tlu
(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 g scdi+9 +yb0*q ikl)fof moment of inertia I is dropped onto an identical disk rotating at anguyik)9 bd+lq + f0cisg*lar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2rr)i 6avvzv3991h8w ojn3kps.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 rota.npcs6w2po doc4 .da zuy85 *fting merry-go-round platform of radius 3.0 m * yapzod46wou8nc. p.sd5cf2 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-rouiqb -jna65 f*ytdyp gk9p076pnd is rotating freely with an angular velocity of 0.8 pk5fiynb- j gqy 0pd66 *tp9a7$rad/s$ 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 collapsb0(bsifz* 6uv es 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 mass carries awfb0s( *zv 6uibay 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 12,gbw.0/t nd2d lvx c;o0 $km/h$ 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 f7cyi u 6 tum8kmg0f *(bup/,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 platfor, ug9*g v4dfhfmw/s2om, initially at rest, that can rotate freely without friction. The moment ouo dwgg49v ,h/ms*f f2f inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diametvf v oi6938aufxnbg r )vps*0ri) ;x5rer merry-go-round turntable that is mounted on frictionless bearings and has a moment of ineva 3uii)rrsov xxf9vr;0 b*6f 5 np)g8rtia 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 wit 7;wspfv5bjewhj0f, p4kt- z+h 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 pers,0zh w-ps47+;fbjf vtk ewpj5on 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 alnl824:wg*q3h3cy 3yb lqkur nways faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbibqrn cy2n33yq: u4*3lgl kw8 htal angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from restwf7v 1y h8t 5:t)ugqaf at 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 unifu8(pxpljk 9: vorm cylinder of radius 0.20 m acquires a rotational rate of from rest oup :9jxl p(8vkver 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 rbvm l.9pozo2916so 1pn7qp( h-4mllax lli0 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 linear s(lnm9p o2l b.pl x6m7 as i4r9h0z-lvo pol11qpeed 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 of the rea kn0exsvx:squ i,0.l .r 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, butngo:f uu (ju.x a*x if8090*i xexio3t 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 o0f. * 38xaei:inut g*xu u(of 0xj9xioff no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobu-cvp;.6afu z+z kk3 mile is for it to use energy stored in a heavy rotating flywheel. Suppose s;6vazu3mkk+ ucpz.f-uch a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to 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 illustrat y6yzglr7a7pf) bmx ;a azm:l3pz)+k9es 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 a 5*swq 2auc+i5v 6qawhk sy5ur)zy01jt 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 leng6d f +:jsu2l0kp9zy rsth 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 release, determine (a) the angular acceleration of +29:frjlusyks p0z d 6the 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 vertically on the floor, and we wpwcqwuk*8 vqvf,r m8:0;lz1:yf mx 0q ant to exert aq:18qu;vq wf,f: 0r 08 xkpc*lymzm wv horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist travelinglc6k nvo 40 v29;ztshwt.x2i n with speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the normal forcl i n x9to.cv k2s;06vzwtn2h4e $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 slinr* 4 pnwhzr/ln*j;4zkg of length 1.5 m and begins whirling the r4rn/pjh *lz *k;n4 wzrock 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 torque come from?    $m \cdot N$

  Model a figure skater’s body as avl7pj / a6trgpw+1- ua 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 wit-wp lj7 ag1tav/6urp+ h outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindricawxvif4o0b4 g)z av.y 0l plates, of mas.zx40yvv4f w) aibgo0s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track lm*3jb: wo k2r8mk.koof 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 leaving the track? As m:32lkkkro8w b*mjo.sume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, buthtrn 7lzo1 1)x do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the , jvhg:rbp m1j3u *r5acar reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0h1,pr: j vbu35 jma*rg.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