A bicycle odometer (which measures distance trave* 6o bc3vh;iizled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle wiz*cii;6h 3v both 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the 4 3 y:p/p3kfggy bjqqt+rr(4jv+6ur r rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential a(33pr4+/p6yrj4yfvq+ jkqtr: b gurg cceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body begb 511jb l2p3pbogw 8n described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forc moao;6 g84lxay; ev pg8t5yd3e? 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 hands behind ru3vu1qh8r l+zd +zg+your head than when your arms are stretched out in front of you? A diagram may help you to answer td1qhr+ lr3ug vz8++zu his.

A 21-speed bicycle has seven spr:uv i 80 5vkg:0.jwom66w-uwttdm xbgockets 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 to pedal, a small front sprocket or a large front 0vj: wm v0w-xtbugo8656:ik.du wg mtsprocket? Why?

Mammals that depend on being able to run fast have slender lower legs ( mfx p.pu gb5hc 1b.,.8zx 7- 4vvpooohy4wzvwith flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotationaow7 4uo.h,p xcx5 -.o41p f.yzgv8zbv(b phmvl dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walki tfi)/w6 al 8239j rt6lew zdbpbd7,rers (Fig. 8–35) carry a long, narrow beam?

If the net force on a systeaz ugu*ci54e32 stq 7qm is zero, is the net torque also zero? If the net torque on a system is zero, is the net force zu2ci7*3sz 4g aqe q5utero?

Two inclines have the same height but ma pruxut/032h nke different angles with the horizontal. The same steel ball is rolled down each incline. On which i0u 32pxurhtn /ncline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. m +3dpm2rx, luOne sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the grea+l32dmrp ,u mxter speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and thyiy(6nu z4t c8e same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE8u (it4yy6n cz?

We claim that momentum and angular l rchavx29lkk33bqfd2d7y2 , momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Explain23dc3kdkq ybxl7rhl2a9f2,v .

If there were a great migratiiy5bzb3o4jri:uh 5w4 a d /xz)on of people toward the Earth’s equator, how would this affect the length of the dayi/b ow 4:y) ab5xud43ij 5rhzz?

Can the diver of Fig. 8–29 do a somersault without having any initial dg 6 df7m0m xk-9v.vu5pdfa)l rotation whek.7 9ag0mfd-6fmxd )ulpd5 vv n she leaves the board?

The moment of inertia of a rotating fo6j : q1a6ujosolid disk about an axis through its center of mass1 u6 6oqaof:jj is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in jjjao7(h4/ bup0f9 dq6ia ayct.6p++7p mzsv each outstretched hand. If you suddenly drop the massessj0c7./66b+tppdj 7fho pz 94aqv jiy( ma+u a, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the l j ,)re)6t)q4xee df2ns1rph same mass. However, one is hollow and the othrxfqste ,))e )d24 j enlrp16her is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal ax;pm, c f rv5(i/f.gmymle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration poin p ,mrgi.m ycvf/5f;(mts 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 largv-w fp1e vsr6uipd6o,2:t / ige freely rotating turntable. What happens if you walk toward the cosgiivt/pu 6pw-fe:r 2 1d6, venter?

A shortstop may leap into the air to catch a ball and throw ig fpgxnz8- 52it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you wil2- fig5pg n8zxl 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, disbqumf1d;i.uk9 i3b)fl7 og : xcuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helico7lb:uq9 fdixfou)3b ;km1gi . pter stable.

  Express the following angles in radia1*, nt5 qi ; ligfo)kb(n*dof-lkfj(k ns: (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 cbhtndjzp51n+0 gngp-;r +g )goincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.t5+ g+g1;-hr zppnb )ggnd jn0
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Eartbp14ii ( py4u d)kjqm0h. The beam diverges at an angle bmypq p4 i0)k14(ujdi$\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rs* ornmivcqt.7 *ky8) otate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to o) r*n7tm.cqi8vk*ysrest 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 anoth9i h :0zujj3d0ljp03;hb3 yyrjfifw 3 er child. If the ball makes 15h i33j;y0:ij9du0 zl3f fhrbyp0w3jj .0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm85icr4ybbdm8 ;5o p s0eeh .bz;e xe-u in diameter travels 8.0 km. How many revolutions do the wheeibze;8 u;4m bxoheb 8 -ed.espc0y 55rls make?    $rev$

  (a) A grinding wheel 0.35 m in diae;; lsnym q-u;meter rotates at 2500 rpm. Calculate its angulmqy;- ;en usl;ar 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 s s45 pm ox05win(Fig. 8–38). (a) What is the linear speepxi 40mo55nswd of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Su67g2czo .uojrn    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of; b7 a w4 wo:zayf3lozhy2ig88z uu13x 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 partic )/:kbuubhu nye)p0jc;y b2/rle 7.0 cm from the axis of rotation is to cjb u:u)e)rn;u y/y2b0 p/k bhexperience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abouzps v 9y2+tjja1.8kpkr8,b elt its center from 130 rpm to 280 rpm in vr kkj9.j,et+s1ya 88l2p bzp4.0 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 radiuse ptv 4v0cwujnjgbfm0brnw,n v64;+1) u6bx1 $R_1$ 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 spacecraft put t nq4vm zip9pwd t6w(48hemselves 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 spacecraftm4 wp69w(dzt84piqnv 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;tx szcs3d. s,00 rpm in 220 s. Through how many revolutionsx c3,.tzs;sd s did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Cant ,e6ojyo ku:tz7 s13lculate
(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 hiuatv qi7zn0mc/nwo9q:+ xhb :k:x -n-ghspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn throug /q7uxba:0mnqvnc:h nt9-z: +-oixkw h 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 :)z0 ktcwn fn5to 360 rpm in 6.5 szkn): 5ctfw n0. How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off wkk72u+ a3+q9l hlel6j o whtp3:j.okhhen it is running at 850rev/min It turns 1500 revolutioo+7t + k jhpje2w.ao:llq3ku9 3kh6h lns 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 ma4ziiyac0*; n h8uv0bm ke 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of i ;80bya ic4hnuv0m*z 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 speedj tg*es85yv7sl ,lg0zkb .,sv uniformly from 95km/h to 45km/h The tires have a diamete jb 5,gl0l yg ts8v,se.k*zv7sr 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 a8m b 8gfwfsdr39 *on.f6(l bbe bike puts all her weight on each pedal when climbing a hill. The pedals rotat.bbng8f(9 l f6r8mfe *obd 3wse 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 mpp0 roy(:*er,(nwyb k agnitude of the torque ifwy, nk(pboyr:e0p(r* 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 showr bl:4/iw-: rxa kilx/n in Fig. 8–39. Assume that aa::x biirw/rlkl4x /- friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mak wy; v1 .ahgsgte.2;kss 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 thethys2. g.vw1k e;ag k; net torque on this system.

  The bolts on the cylind(o x)+ llsf-fker head of an engine require tightening to a torque of 38so)+( -f lfkxl $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0o knuj o4osc tg054(m(.648 m when the axis cu04gk(sj5ot4o on m(of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cj h) px- 8najf11sl-akm in diameter. The rim and tire have a co8hp11a s jk-l-na xf)jmbined mass of 1.25 kg. The mass of 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 horizonthz k.mrl62 wk)bsh :8dal circ2sz mrl dk86.b)k hw:hle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular sc3fai- 53cxh 0bxw n6ga7xx7i peed (Fig. 8–42). The friction force between hergbi f76c7xaxc-i0x 3xn3wh5 a 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 objects shown in Fg.akq j,misp vc*p,wp7cr 4*8ig. 8–43 gvic78jpa,m .r *w4c, p*kpqs 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 at; +cjorvv r jo*6qz0i5ys t-*koms 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 vzjhlot 3f/bh2 p056d correct rate, engineers fire four tangential rockets as shown in Fizht3d0 vj /5bol6h f2pg. 8–44. If the 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 with a radius ofdx 9x9m:iszra:tnt 02t( 3q pu 8.50 cm and a mass of 0.580 kg. 0ias pt( u :2xrm nzt:q9xd39tCalculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from res 8r:3 pukv uj* :pfpmq ww+qkvk)h9,(tt to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-rovo)ij2 x)gow(y*gki/s )2h ovund 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 produce the acceleration, neglecting frictional torque. 2v vohi sg2* )kogo xy/)(wi)j$\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventupgqpzqm- +,m /d5hr i/ally brought uniformly to rest by r5+hq/z/,ipdgm q-mp 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 balqkx826ydg36w)vps45uf y p f gl at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solk5 roc*o98 hgbely 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 unif9rhk g8o5o*bc ormly 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 consideredvd tg8o 9pgivpqkli+2)c8 /0y a long thin rod, as shown in Fig. 8–49kodipgitlq280 8v gv)/ c +yp6.
(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 ma.h tb3q+ygd k o0)ujj.sses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full tur7z.cl- qjmr5 lns (revolutions) an -m zjqr5c7l.ld releases 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 gmtu (p1y;i.h+nc- r9d xw;r ninertia 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 of 2smm 0 ciw:ctjw - rh-vrf7d+ig8 1ee0i8 ./jcc80 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radiu,ddw*q8 nb (owt 2-bixs 9.0 cm rolls without slipping down a lane at 3.28ot dnxbqw-*( b ,dwi3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as ther.lndcu:9v7x-6l yhz pmi3 u2 sum of tm-7l zu6n hxucvp.2dr :9i3 ylwo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radiucjsl ,8 hm.w/dbgex (+s of 7.50 m. How much net work is rhw+(c mgl.8sxb e/ j,dequired to accelerate 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 rest ansb8s njh63 5ys dog2r.d rolls without sli5sos 386dhgsj n2 rb.ypping 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 star/ ( fz5nndq nq2xa+oh2va- .efts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use c(2 aan5xd qn+v2 .oenfq f/-hzonservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball reghc ibz ) : p(y+nki5k4k.5kgotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10.(k+5kk y:.bghz ikige)5p4 cn 4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grindiq0chq lh- e4w9ac57r gd8s*ckng wheel of radius 18 cm when rotating at 179 a5slhdw0e4k*hcq 8c q -gcr500 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, 647sb arm qkutn95fi8on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreas 9 manbi sk8urqft6754es 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 reducee5ti.ruv s:b4dv w7c; her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0d4.ic t7b5s:uvre ;vw 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 an1l jmgp* ifjzz 68:8bc initial rate of 1.0 rev every 2. mjj6f*c8i1g:8z lzpb 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 rotati +/ji sf8zie+ung around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequenc isi/eujf8z+ +y of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a fic (gjj*)1+2p 7arn*z) k usoghqj2v.titbd(dgure 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 :7-dysf nrlt b(cw, *uv4pbm/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 cn 68g6p-3ljyl5 8pnt am o+osof inertia I is dropped onto an identical disk rotating at angular sp5 lmpj +68nn8lsyo6tc- g3apoeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 5d je*vudznky7. n(m+ $rev/s$ 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-gof3z6u/xs/no a;;b dp xkmv em;fan;;7-round platform of rad;p/k mxo; ; 3e nbf7d ;aafv;unm/zs6xius 3.0 m 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 rotat3rss-krw) mggxy ,( u3ing freely with an angular velocity of 0.8r )m gyr3,s3x-w ukg(s $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing abouy69ayk nbpz1n* )a v5jintv; -t half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(roun-pvb*6 t nivak;9ynn )z 1a5yjd 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 ofk d(3ud; s:n7j1xrgfy ef2w:e 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 z*2 )bafxchs7lh;ab 8 $ 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 platf5vfy, / s(qjhdorm, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform ifh dy v5(js,/qs $I_P$ 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 edg6rt4 (tv j:rzq2flt;be of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and hasj lztfvqbtr:(r2; t6 4 a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lyia3/)s 9xc4znwxvfx-1 rgpoig0 4x2w yng on the top edge of01wx)3 fy xiv-rocspgwg4ax zn 942/x the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person 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 alwa.cspfgo/ns25 t xjm: -ys faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to/s s.oc2xtfgj:n m p5- 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 rategf893yi 8y ktp 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 uni* jq pj*q83porform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the qj*8* oq3rppjmotor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindricr /lyo1.gefpie-8 sl.zw: r1mal 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 conservation of energy to ca1 z-r8r l.:/if me.ly1wogesplculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular s ku e.6cmgucv,s)*j la phl exj+54/d*peed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were roh-2l 3;:hh5qjt4drqum ;m twptating at a speed of 1.0 revolution every 12 days. If it were to undergo graw-3 h :;2d4plqr ttjum;q 5mhhvitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored in( jv/l5d s*9no :iytlt a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cy:(tv*ydnos i 5tjll9/lindrical 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 illustratej;/t sm. ls,ehp24gs(7rb tcjs 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 horizo0ik)xna2e ih/g)(rj p)+ hohy ntal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., w ;*io y;wgqk:6qv)6id beew)l e ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then ) lk* ;:ivqww6 e)b ;6iedyoqgreleased. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the flok /8fzp1(up 0os 6odnoor, and we want to exert a horizontal force F at iko dfsoo npz 80p6/1(uts axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flatc nidyw p*f2l2 i kg-wg.w6(3v road is making a turn with a radius The forces acti3d6c vl g.2k p-giwfw *nyw(2ing 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 length+xjlzl,0. s dh 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque re+ 0sxl,h.z dljquired to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder aodp 9 +o(czan oiz7y.1nd 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 arpc.17yi z (ozaodn o+9ms, and with arms held tightly against her body.   

  You are designing a clutch assembly which col: l,3ij)k g,zecvhf 1c /q8n ;qx(vljnsists of two cylindrical plates, oflc(), hinqk,8z j exf;gvlv:q1jc /l 3 mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. 8–58xf x83 dpjw+l3aclu2 -. What is the minimum value of the vertical height h th-c2p8dx+uf3lja3 wx lat the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumf-pwqc:vc3:trw j 2 e/e $r\ll R$ h =    (R-r)

  The tires of a car ma6 cexrpkrw t 2aglzp03+7q9 *qke 85 revolutions as the car reduces its speed uniformly from 90km/h to 6qrp*w9r2tqa+e30pcx 76 zlkg0km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s