A bicycle odometer (which measure3itsmm0l 2 ofohaw,.5s distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycliflm3.tamso2 5o0 ,hwe with 24-inch wheels?

Suppose a disk rotats20g/ u r qc1tmukb)n2es at constant 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 of either comp/)ngrucbm q1skt20u 2onent of linear acceleration change?

Could a nonrigid body be des9g dzt;t(hcdfg; a:0rcribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a gdz m+:dpyrb 2(reater 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 do a sit-up with your hands behind your head than2 4k8j-q u s.u5tqux1upt 35g* drgtysdip8,m when your arms are stretched out in front of you? A diagram may help you to a*y-5.msu qtusu p5itrd dg,3kpg j412xq88utnswer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thsz o. f5bjrc9,ree at the pedal crankczbfjo 5s,r 9.s. 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?

Mammals that depend on being able to run fast have slender lower legs wiuwks;a6g:t f;th flesh and muscle concentrated high, close to the body (Fig. 8–34). Ou gsw6a:f; ;tkn the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fi03p b,m4ihcajg. 8–35) carry a long, narrow beam?

If the net force on a system is zeut4t+ln2;6g jmo ii2s zfep40 ro, is the net torque also zero? If the net torque ou2e 6+ 0mpo ifs;lt42nt4jgizn a system is zero, is the net force zero?

Two inclines have the same heig-8dbhzz 3+ sqi.1gdzl ht but make different angles with the horizontal. The same steel ball is rolled down eachb1zgzh q. zl+8dsdi-3 incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an 5 qbcg1b,26hcs 2tijdincline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at tbcb hqg1i 2d26tj5c,she bottom?

A sphere and a cylinder have the same radius and the samec efoo/r92 jbv28m9ni q27eu z 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 jf28n q zbo/c2e iuo7929emrvgreater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are consex2rn3 e. g eoaxs9xv*a7h s*n3rved. Yet most moving or rotatae a7 h2xrgovx*9s s*nnxe.33 ing objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, howqpbem(g i4*be6+i ap) ydm-0q k 6kd9h would this affect the lengt)khe aiim(d+-0bd9 b*p46myeqp gqk 6h of the day?

Can the diver of Fig. 8–29 do a somersault withokmlxx8 l1jn no-6dz(1ut having any initial rotation when she lea nm(j18ol l6xz k1xd-nves the board?

The moment of inertia of a rotating solid disk about an axis through itbu ;h3sa 0yq7+7gnapbiw(y* vs center of mass isaigyq 3v+bh(pba;7y* u0 ws7n $\frac{1}{2}WR^2$ (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 each o/z k+d9p) mr 9 va,68kxwyo uctvyy3d.utstretched hand. If you suddenly drmzy8y ao)kd+dpy6 uk/ .xr993 tv, cwvop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have fxc(9hv6c7 pwthe same mass. However, one is hollow and the ot7cph v9 xwf(c6her is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotjh62g-. lt5v esw ifpa.:- kbeating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed i-5:kljvsh2.bwf -e g6pt .i aencreasing or decreasing?

Suppose you are standing on the edge 9b7l i n5v: jt)iw*qlgof a large freely rotating turntable. What happens if you walk towarb 7iq:)i5n* ljtwlgv 9d the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he thqo:9flcjbizg 62 015*aieyu orows the ball, the upper part of his bod5 col fib i:oz26*1eay0quj 9gy 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 angqc(uft0pfjf emhj7ri 307nz63 c.e 4z ular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss onef n7p304f0i6jqeh zc erc(u3 tz7m.f j or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radianzm52il +h5rahs.tfa 7s: (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 a 0olxcycg14obxx v-43s rj39 tmazing coincidence. Calculate, using the information insi xx- ccl 0t134x3sb9oygo4vjrde 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 ki- dmmif/x2 1w k3j2rym from Earth. The beam diverges at an a/ dikj -23rmm2x1wfyingle $\theta$ (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 qkyr l)6d:zddcv y rra;. 1o5sc/bb,4at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s.;d4c vydc l):51rba6dz bkrys ro,/. q 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 another child. If the ball 5ub)vfzci4m.+u(h s h makes 15.0.muvsfu( i)4z 5cb hh+ revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. H y(spi/s1vbs l c0j8z,ow many revolutions do the wheels make?z,p8s(v j/cbyls 0 1is    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculatiz p7-mnt4: kme its angular veloc4mi:- pztn7 mkity 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(2,e nj+y4ve);w- htaj aief f makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from 4hyfaf2 ,j e;aj)(vene+ -iwt the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity ofy 2:l -0tc nc7mslp9lq the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poin 4rq: z;c4zlkmt
(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 2brjd wf;*w5k,,ykfzy76a dd centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 6 w*wy,ajz 2ykdkd;5rb7ffd,100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates un7u)l nmp1fv n)iformly about its center from 130 rpm to 280 rp1)pn)mvf unl7 m in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiusvby-ts,mipt.p9b qzan-z 2du :q9 ) z* $R_1$ 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 themselxv( :q;qz msfpoo7g4-ves into a slowoom-:pxq f;q4v (g sz7 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 with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 2z-dmi eeh -03x20 s. Through how many revolution3-e edz 0-xmhis did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rp(ae 1u)6xi7xhf7vrv1 v;xxz bm 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 highspeed jetss r k2,3,sz2zhmhb -ll in a whirling “human centrifuge,” which takes 1.0 min to turn through 20h2z- m s lbk2zr,s,3hl complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 5gnfi+s t;m: 12c an4u alwf3okr ny7;360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in thsw lk24;t1fnnum3 7a yo nacg;+5i:r fis time?    m

  A cooling fan is turapg yy/fs/ vb li)6siina7fax07s6:5ned off when it is running at 850rev/min It turns 1500 revolutions before it comes to a sti7l0fga si/sa/ :spaf ny 7yb56v x6i)op.
(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 the car reduces its speed)uq:mz /ahgrylj/.sz 1 lw88 m uniformly from 95km/h to 4wu lj.q8zmg y/z)ls:a hm r/815km/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

  The tires of a car make 65 revolutions as the car reduces its speed uniformlqq3 o z5n)eby-d/uf0j y from 95km/h to 45km/h The tires havef/3quynq) jd0 5 eboz- 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 pz7l;/j9 qo h(3kdu*fwt7goit uts all her weight on each pedal when climbing a hill. The pedals rotate i9;wqkz(fidj o77ol u th/g3 t*n 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 wiwef bugn 3y2)z) +r,xtno+;z do.2v wcde. What is the magnir yo23bgf))x2oc+ + zwnv w;.,tzen dutude of the torque if 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 th96 g wi(i0xbj5qup-zh e axle of the wheel shown in Fig. 8–39. Assume that a fp5g-0jxqi bh69(uw z iriction 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 mav6w bu4fzvh*w+0xi2zssless rod which pivots as shown in Fig. 8–40. Initialb4i0v2+uzw z*xf vw 6hly 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 cylinvf:rx djtnz ta eu(s x6nv+rdp6 gh*,l.-*6n4der head of an engine require tightening to a torque of 3866 t(ux-xpj,d6rg lnzv*.e*nfsthv:+ad n r4 $m \cdot N$ 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 0.6 ./jfk:7ash9o .gl x bgc0s1lafw h8+y48 m when the axis of rotationh/s1+g o.akc hyfsfa:l wj g97.0bl8x is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. T,9j 5lbyyy)chkb+ n h dhs69y,he rim and tire have a combined mass of 1.y9ydybnhbh,5)6j s l9hky,+ c 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, lmwh 9p7j f7r -q1c:vsdight rod is rotated in a horizontal circle of radius 1.2 m. Calcs7qmw7 -jrfph1v9c d:ulate
(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 s,ynd hk)jz a3r*1dhn. peed (Fig. yh a*3)nh1,kd.jr dnz8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the a2s/lxx tpag -o53q 9flrray of point objects shown in Fig. 8–43s2 5fg39 x/ l-qaloxpt 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 atof:8ltjw08 8gcz ui o0vms 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 correcq hh8.1vvevwk 30wnh +gu1toz*f (m5 lt rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass ofg*nwhv u3mhq5(e o+w1 t lv0.v1kh8zf 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 and a4z *87mb hurzf mass of 0.580 kg. Calcb 7uzr8* 4zfhmulate
(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 awih7bdp*zux +4 x6mqd8 km;m t9t(xt6 bat, 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 merry-go-round and iug 2lsy:csj;t qpo 0:y93otc9y;mb4l s able to3tb4y 2l: ;gsoso09;p tqc9lmujyyc: 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. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and 9 .c6gfc,gdy 4s3tdcnis eventually brought uniformly to rest by a 4cn,6c d gdf9 3ycgst.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 accelerate 20+sj(puhobn- zlxok .0g;+jqf 6axos a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by))cz ewzj..- dwmfua-vek;x9 the action of the forearm, which rotates about the elbow joint under the actioncu) kvwj)dza.m fex;zwe-. -9 of the triceps 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 consideref q1ayn t6n*cr-u0m ;cd a long thin rod, as shown in Fig. 8–46. fn-0c mnqury ca*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 t :3 mz*spbeneq0yaze( ik+-loj61 8duwo 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 hamfudy r/wxkn :9:(p8e gmer from rest within four full turns (revolutions) and re:wru yp d/f(k9n8: xgeleases 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 ha+zeb)am m 93mc) ur9ygs a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque6wmw 0c2bn eh6g/kh4z of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipppk)y 2v/n .fgvn,etxm4ha7us j*p55 iing down a lane a xtgu7*,f)4pkevih/n. 5v sjn m p5y2at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with res ffb :0)gs7gok.wnq+ zpect to the Sun as the sum of two b .ggf7:zk 0qnsw+fo) 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 a4z7, lc is4tsl radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution z4ls74csl ,ti 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 qxrmuj6dy 3 .5bzjm(z ;e zn-1rest and rolls without slipping dodz6 (5embzn3quz-1jy .rx;mjwn 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 y3 uocgd:rpk xcy69h3:z3(ci starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it u3i9h( 6rp d3cz y:3yoxkg:cc hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of 0teu r*ex) gb1hnr:(* ylc l)fa 0.210-kg ball rotating on the end of a thin string in a circle of radius l rh*yec:ut)le*)r( n1 xbfg01.10 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 cylindrical grinding 4gdx3v0q3rk iy,+oe b wheel of radius 1v30y+4bixdr q3,gk eo 8 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a ratea x83yox4si f j2j/hd8 of 1.3rev/s If he raises his arms to a horizontal position,xsa3/ o8jjyid8f h x24 Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inertia by64s-a.;op,brx6erwl )1 /c nud jecwmow og4; a factor of about 3.5 when changing from the straight position to the tuck position. If she becws ; 64 .ejugormlno4); -16rpdxcw/, woamakes 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 initia)i+ xi,+m*u)eizvj bf3mca :r2ayh 1q l rate of 1.0 rev every 2.0 s to a fin2:a*3i uy1z+xmh) e bqjr)cv aii+mf,al 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 allbp( l7)t ot: 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 frequency of the wheel after the clay sticks to it? b7t:l)l(lop t   $rev/s$

  (a) What is the angular mome*oaanrdur 5,j; p7r s18/dierntum 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 b. tare2ugyye41a+p6 of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is droppdcu;r3zc1m08hln:yii7 way.v 2f 0ived onto an identical disk rotati0iu v70 v:381cnr.2 ;wf zyhmaicydli ng at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at yik4h+ /slkb1q.j8 yc x.i*a p2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round u1b/q m 6,diee)tvr*oplatform of radius 3.0 m and moment of inertie ve1)*6u d oqbrt/i,ma 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 6 g2xdybt -.ct3yxf:97f e hu,9d ouqbangular velocity of 0.8 fd.3qybt 7, bxf-d99 ugto2ex:hu6cy$rad/s$ 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, losingaz 9eq eh2x6/f about 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 wo2h f6/9ae zexquld 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 involvek2 0x gz 3op4uto-;jv5dfx4yi winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass g9 w/nca/n9dh$ 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, ix-7k8fk9zw a g.vtv k w08vj*qnitially at rest, that can rotate freely without friction. The moment of inertia of the person0qtwgwkj*8- .v78x9 vkzkv f a 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 diametemd..s-zdcyrf i7 x( 8 6cjdns4r merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia cfm (s87njd cr.6y4z-xs i dd.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 zxuqais;0;t5++x kia+* gv wfof 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 person without slipping. What length of rope unwinds fromwu; +05g;*xavizk fs a++qtix 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. Deteraa86 (9bot5jv hp/ mbgmine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particlev9 /tmjgpahao65(b 8 b orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from w-ew6t2tzl,r uoo,n.rest 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 uniform cylinder of radius 0.2ecow 8xof5l0d4x(zrc s8 f(c30 m acquires a rotational rate of from rest0o cwfo5 x(xe(dlr3cfs c 8z48 over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindricamdgmtam,6p6k1 a5tk 0b+5 rh pl 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 dikp5k+5ha 6rtmb m,gam1pd60tameter 0.010 m. Use conservation 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 y4fdp0f1u q 6oangular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but wit7 kmts )2mfqcmw xjh-t3lj; --h 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 thatswm3)7-txmmh cjk 2; l tjq-f- 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 .w6i;9hmk3oex. hhwyit 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 be able to travel 350 km without6kw9e;.3h.mwi yhxho 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 illustrakmk:e f8;.ed mtes 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 j8a04, kqcchvon a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ig62 .qkb4hxcn2-x dh aonore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held h4d.2-c bxho knh2a6qx orizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of 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 2sk(* pm0tzwelly on the floor, and we want to exert as (wzte p20k*m 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 traveling with speed v=4.2m/s on a4 vspp1si z :s3m0f(94 kdmtbkd3aq)r flat road is making a turn with a radius The forces acting on the cyclist msp) f(zti40 9q:m14dkkvs 3r dasp3band 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 lengthhj xs3mc: g2t8 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, acceler8m2tc3 shg:x jating 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 bo9drm s9o)crg(9 ogdw(kmaq)2*y )ti kdy as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tight9)gcywr 2) (osgd* )kk9tdr m9io mqa(ly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plate z4:h .7+ai7m;ue i nr4afgsjhs, of mhmeas+i a fz:7 r .iu4ghn4j7;ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 trackrjep tf32t0d2 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 pointj3 0efrpt22d t of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do4gn z(q:elqt - p9pd0o not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reducesi rnicq;4d// k -ixlra3-7d zr its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was th;l-kr/3air7c nirq4zdi/d -x e 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