A bicycle odometer (which measures distance traveled) is attac5 p,(n uurx,cs,t5 yjjgz34x dhed near the wheel hub az4g3cjp5yu,(ut ,nsxj 5,dr xnd is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant an7t ex5nc1v/pu gular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s anxetv/ pc51 u7ngular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular vex; xv5tn:u1v2b*2z4vos+;7n whkg h;hc fk o clocity $\omega$ Explain.

Can a small force ever exert a greater torque than pe*npq5r16kgy six) 1a 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 /lxri h;y2:mjyour hands behind your head than when your arms are stretched out in front hr /2iymlj ;:xof you? A diagram may help you to answer this.

A 21-speed bicycle has sevscf jyxfyxjx.,- :/d4me3b;n en sprockets at the rear wheel and three at the pedal fsdxyfy-/,;x 3j4jxb.nme c: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?

Mammals that depend on being able to run fast have slender lk8j.* w5g *cjzq lhk8v-+aeidower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis iqea+-v 885dk*h jk cl.jgwz* of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow b:/:68y btg( n)* jarnr mjstyweam?

If the net force on a system is zero, is the net torque a(gjx.b5t(jl45a k7ea2gr xn slso zero? If the net torque o x.jk2 lrg55ba(j4s( n x7aetgn a system is zero, is the net force zero?

Two inclines have the same height but make different angles with the 59+z m2alplcqbfuoa:0 xm z1/ horizontal. The same steel ball is rolled down each incline. On which inclinez 0xfamu lbzm91lcap2:q+/5o will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously sta vtpxwb*z11 )1pm(8tddkv 5kprt 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 incline first? Whikt) 118mkvxtdpb51*p(w dp v zch has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius ,5,uif n2p x0ulai0cfy88 aa+e u7 dumand the same mass. They start from rest at the top of an ia0y5+if 70c2xefm auu,d,uia 8pnl u8ncline. 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 KE?

We claim that momentum and angular momentum are conservedbkwi o9xr.3 t,o q5*dk. Yet most moving or rotating objects eventually slorqod oxkk3 9,b wi*t5.w down and stop. Explain.

If there were a great migration of people towardq,*wl3x( x fjq the Earth’s equator, how would w * (lx,f3qqjxthis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatinewxjati d(n /k.;1 c2on when she leaves the boardekin .tj2/a(; xcnwd1?

The moment of inertia of a rotating solid disk abou.vji 0gfym;-e t an axis through its center of mass i0 vim;j-e.fgys $\frac{1}{2}WR^2$ (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-kg4y g5h8s56zp ;jc ftfd7eh o*b mass in each outstretched hand. If you suddenly drop the masses, will your angular velocpyef s*545 oc g;68dh7jhfbtzity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, o ftu1n2-4 wz6am:9b u uvx,yhrne is hollow and the other is solid. Describe an exphau f6rt 4z uby-xv1w9u :2,nmeriment to determine which is which.

The angular velocity of a whena j.-(fpp3,y+ybx 1jrqy)k jel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the anguyj-kf(j1,.anb x)yq jpp y+3 rlar acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntabf ikymhy g 30dubvvz) t6,,v81le. What happens if you walk tk0,tv3m1 ,gv ubvhyd8yz6i) foward the center?

A shortstop may leap into the air to catch a ball and throw it dgp1 npcp 54el;6to02)loxpv quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite didlpnpp )o1tv 5x4;6g2leo0 pcrection (Fig. 8–36). Explain.

On the basis of the law of con:16okr :.gp omn4l2ou 7bsyerservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Dik 72um lo1reyn::op.rsb6g4o scuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a)sy6gyh dg9k gididb(ep02 8 1) 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 amavhy ncpe-+3 .7q xw9wazing coincidence. Calculate, using the information inside the Front Coverx3+9neyqpw aw hv-c 7., 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 h ua95ns*1. rcayxkrf bnr3) (the Moon, 380,000 km from Earth. The beam diverges at an rcu*9k 3(fa5rr .sby1xnh)anangle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 65a,6ywua 1k ( vcyl6rcxw9u t/j5rws4)hj2z1j00 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What 6uwwy 1c24h x vr9 /sy( ja6uk1cwl,5 r)tjjazis the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level -- mytj:kt r8m9ehf.h v+g z,ffloor 3.5 m to another child. If the ball makes 15g-jh+v9r mfz-yh,ktt:m8 e f..0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travv. tqq55g;rpmels 8.0 km. How many revolutions do the wheelt qq5v5;g .mprs make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate ia/gh9sq1h3 gr ts angular velocit a3rh1gg9s /qhy 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 , iul,snb vv8a+wdo-m(n,ho6 4.0 s (Fig. 8–38). (a) What is the linear speed of6ao(,o, v,sn+l ndh8vmiuw-b 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 ite lgp+b/25 j mgv6h0 ;vlo5wkks orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spx/ej7 y tp xu7;k+c)zymyb6+ leed 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 7.0 co31u+wgnv w3 8xs +pomm from the axis of rotation is to experience an acceleration o u3pgw3vwn1s8++mox of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 1o-fumxjx g :9/30 rpm to 280 rpm in 4.0 s9:fjmgu -x/ox . 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 radiuf tvf-vqsj6rt7 )etwep a, 8)bq,3t(k s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronmt+tx r fd4iwub2+e,9 auts aboard the Apollo spacecraft 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 with a diameter of 8.b4u + 2 frtwd,exmi+9t5 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 15t 8x l;, y/ sk*autd3rbrt2u:d,000 rpm in 220 s. Through how many revolutions did it turn in this time 2luk/ *:tsd8r;dt bxt3ar,yu?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcula bzo6+3 pbx6w *+ha2xk q::+xgiweo ohte
(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 flyobd -atoz8,fc a-v59 king highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its fac dbkvf,9 -5toz 8-oainal speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rp)kun 1z4wq+gk(pv v5+ztfme :m in 6.5 s. How far willn 4mk) 51zkvv :(qwu ze+tg+pf a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500mjb (l :u.xq/m revolutions before m q/u( lmbj:.xit 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 the carm hdq, llqi 2dp.7frj0/a5m/w 9. bxht reduces its speed uniformly from 95km/h to 45km/h Thejr x.m75qafqldtw. pb0dml/,2 h/i h9 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 a5 de8g big jp8vtrv/1+s the car reduces its speed uniformly from 95k+ dtpgb/e8ir 8v 5jvg1m/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person ridinn1s nlm vg384og a bike puts all her weight on each pedal when climbing a hill. The pedao1nlvg 4sn m38ls rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 75fgxn-c j( a6c4 cm wide. What is the magnitude of the torque if a5cnfjgx( 6c-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 th-gisv:v8dgyw 0o+w+fy eog d v2/d)n7e wheel shown in Fig. 8–39. Assume that a frgg)80 vsn /i f:-dow2yogy+dvdw+e7 viction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attacw0k8 j2h)y bpshed 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 releasp8w yk)2sb0jh ed. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head5o p/cw2 z *lts-4dpzm of an engine require tightening to a torque of 38 *psplz 2tcwzd5 o-/4 m$m \cdot N$ 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-hfihk7n *8nn 3w-la.yqds5 /lkg sphere of radius 0.648 m when the axis ofn *h da83f lwhq/nsiy.5 7nkl- rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The ripyblfi*)i7o a7 g hgrb ,;:d+jm and tire have a combined mass of 1.25 kg. The mass of the hub cf:,bp)7jih a+l i y;ro7g*dbgan be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated i)tp .yrjsd9ei6x5bm9( g e7h bn a horizontal circjix h)mb6ep 5d9g r79 (yt.ebsle 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 *y is;89 -cpuiroo(wza bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1. * p9io8c(zy;ori-usw5 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 ine4e2 sr-wb p6kertia of the array of point objects shown in Fig. 8–43 arebe- 64p2wksbout
(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 yo,di 2 sa yaqwq+f)sa6+zn0 8of 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 sk.y1ufl/rj )g pinning at the correct rate, engineers fire four tangential .jyguflr) 1/ krockets as shown in Fig. 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 cyliso/ hpq9 ;hk6mk( f2rqnder with a radius of 8.50 cm and a mass of 0.580 kg. Calculat/fphq r mk 9(o2ksh;q6e
(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 qu.p.a -)hs3l qngt* qit 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 zy-v6zqx)c + ihand-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 un zi-)vc6x+ qzyiform 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 andhzf3kyd*c p x;v1e.nl5ig* 5el1 wxb* is eventually brought uniformly to rest1k fix p wdezbl **e1g5clnh5*yv .x;3 by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates ag52; fdtxoqf00pz+zupd9w9d 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 solxd8twy27m4n b ely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. t8bw4 x7nyd2m 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+c cry)pjb*rvf. ,uxbu d4- b6 be considered a long thin rod, as shown in Fig. 8–46.rpfdy +bcbrcub v).*4j,ux-6
(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 jzfb 3udp,loc)qqr8+xt: xn-.n rh 380wzj 6ctwo 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 rest within four(x m: z 4*irx/7coze*ewvz0lk full turns (revolutions) and releases it at azv(*e o c/wr 0xx:mki*zel47z 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 i *;2vnxye h 8.+hylgvznertia 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 5 sq15+pxlxyr torque 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 sr1l+sx - n 4dq*chye+qlipping down a lane a1n + x-d*q y+hqsercl4t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Eartvjh:uzu*6 v25 i u1t.q 4bgemihln))kh with respect to the Sun as the sum of two terms ji1.)4tmui5k2lubv: v6e )quhh*gnz ,
(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. Hq v h vgl:m1fz9z/8w1eow much net work is required to acce18mzl: g/ezq vf w1vh9lerate 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 resreiw(-s+g ,8p.f xcxst and rolls without slippi .wr-igxe8p+ s csf,(xng 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 it.d/b3p gprzk a)4r b1fs tip. It starts to fall and its lower end does not slip. What will be the speed of the upbp4rk .rb 1g/) p3dfzaper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.2 +x9 (81yo)yvuvvt0t/vkw k h/v (;jqonh e6gp10-kg ball rotating on the end of a thin string in a circle of radius 1.10 mg1//k y uhvox v8 phtv vy0oqkjt+()w9 n;ve6( at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum nfb 1n) hjz/:g9vm9 w nv(8isoof a 2.8-kg uniform cylindrical grinding wheel of rmjbovn8)9s/zf9(gv 1 wi n:n hadius 18 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 th7x icvk8l * :pi+zr99gt uraaoj,q)e (at is rotating at a rate of 1.3rev/s If he raises hlqizo98+ag ei7 9)jtr(p*:c vk,a urxis 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 Fig. 8–29) can reduce her moment of inertia b u/7kfppm5ss 10j+9 ekaixet0y a xupm+ ea/7 kisekj50fp09ts 1factor of about 3.5 when 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 hpm9c je7y1+ jter spin rotation rate from an initial rate of 1.0 7c pjmjy+9t1erev 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 throu;cfjq q b+8*fcgh its center at a frequency of qq8*jbf c+f; c1.5rev/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?    $rev/s$

  (a) What is the angular momen4 7d xyptqs9 m+p ww65:g4mhhumeey6:tum 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 na hhk+1ck/b*e40d)-z+ fyd b xri5bpmomentum 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 7w 2n*km wlj2iis dropped onto an identical disk rotating at angu mjilk n7*2w2wlar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2. .f lbd..yina,3xtv3 ws+v;fp4 $rev/s$ 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-rounb;egs- eou r zm4l,qz0:0jtn3 d platform of radius 3.0 m and moment of iner, mtb:u s4grn0- ;l0 zoeze3qjtia 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 merhv8 b3 f(;0s kocbuvk:0qlz/f ry-go-round is rotating freely with an angular velocity of 0.8 kc kqbv(0hf:u ;b/ol0fs83v z $rad/s$ 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 wj p )pmql-dfu p35a,*-kpp6 sphite dwarf, losing about half its mass in the process, and winding up with -5 l)pujfpapk p d-q spm,*p36a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 1dbc ,s2k ;hs u1tt)h6tfwz b.720 $km/h$ 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 dejcq- zn7;ol,,1i:2f/v4bjft yh sa$ 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,.q9md) m bhz/oesg,c+ :qt(n u initially at rest, that can rotate freely without friction. The mom qud9)bhnmoez,g./cq(:s +tment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge o.honc6vrcp)3 f a 6.5-m diameter merry-go-round turntable that is mounted on frictionlessv .pc 6n)ocrh3 bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the grox gix bh/yxa 2ya 73h;oa03o d,gsbo11und with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rol;,/oyoogb 7yb xa ahshx0i a313d1g2 xls 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 1al czp/ifhfx jyt s*g6-3(w3that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (aboutlztyw/agspf6( ji fx h-33 1*c its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates hfxbc8. ag-duxu+d/;,qex c4from 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 unifo5 ,u ;*ogjeqaf xy *)2mgm;tpkrm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate thf;m ,gxoa* jqe 2)kmu5* tygp;e torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, sv0zvp*1sj jq4 2b2 be*fs-zn each of mass 0.050 kg and diameter 0.075 m, joined b*2jbqjeszzsp 2* bnvfs-v 401 y a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 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 angular speww 2 r0x + nazs3-t*fiu:mkaq7ed 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, b.q, x+b spg3 hmc-tb(*y1kofput 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 -+ gf,3psypt box q.*1cm(khboff no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution autt.fnq c.oo)0giv2or 5zo :ar; omobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without neeo) rqf0izot5vc ro:a.2no;. gding a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates anm8/iv ;n /lxbi $H_2O$ 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 at speed v=3.3 ;l:2 opx /uigj$m/s$ 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 free )k g,9ogxmwe u(ju-8tly (i.e., we ignore friction) about a hinge attached to a wall, as ingk 8o(gw9umut jx)e,- Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of 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 snl juig9b+ql36 h: u:ktanding vertically on the floor, and we want to exert a horizontal force F at 9ul l :gb6khi:n3 qju+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 a flat r5 p v/mxq9th5g3cjdnm .rl/,f oad is making a turn with a radius The forces acting on the cyclist and cycle are the normal f mr,5lg pjm9tn/ dxc3q.5h/ fvorce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 1.5 m and begins whinzbja;+)ow j) rling 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 required to achieve thw+j ;)jbz )anois feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and ijev8got a:z 5 t((b-cher 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 tightlye(ac 5jbi-8 (zg:vtot against her body.   

  You are designing a clutch asseafw468p ic0 jkmbly which consists of two cylindrical plates, of mass wj 0684 fapick$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fi qi +e.vk:7qfgr.e4xl g. 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 witho.4. qk 7lgv:ef+ iqexrut leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do (t,lsi1ijjh oo(uleoy)a r9 s4+2y2jnot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniforsve2xf i) fcr0*b8o5 smly from 90km/h to 60km/h The t0srbfsi25ev) oc *f8 xires 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