A bicycle odometer (which measures distance traveled) is attached nearb/(szq cb5 hp :jberu1x1,y0i the wheel hub and is designed for /rp1q0yx b cs 1h(5b:ibejz,u27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant .2hy-pue 0i9u b8upajangular velocity. Does a point on the rim have radial and/or tangential acceleration? ahje0pu9b -2 .upu8yi 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 component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocizhh;h)l g)bq0c m-f9 cty $\omega$ Explain.

Can a small force ever exert a greater torque than a lavh7hok:e bs.5 1 hd*iwrger 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 you,r6i .dn-wxrw1b(jl n r head than when your arms are stretched out in front of you? (6,iwdnb j-.l1 rxrwnA diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel a+xgvwiwzq(ljh .o/)8 nd 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 largz8x)( w jh/ql.vwg+oie front sprocket? Why?

Mammals that depend on being able to run fast have slender lower lewztt 9m3m/,m7p +fg,w9 p tleags with flesh and muscle concentrated high, close,m/t, tpawm m9gz t3+p7lf e9w to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (a q, 1q9i iqx4tr-wl,oFig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero?9p*c1jw5cox )waxl0;( a6fgje q:mgn If the net torque ocjn p aag): w*(xwf9l1c56m;e 0goxjq n a system is zero, is the net force zero?

Two inclines have the same height but make different gd2. ;xxr ks zyfbe.q+l)1gy :angles with the horizontal. The same steel ball is rolled down each incline. On which in.de y:k2bf q1 ;xylg)+.xzrg scline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) dow pz(8uo*dmog9r f v69it+hw0v n an incline. One sphere has twice the radius wm9fpo v8to+*v6id ur 9z(h0gand twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mewuw)tp17yk :1xs f 0k q/v+r2dazmk8ass. They start from rest at the top of an incline. Which reaches the 07s :k2 de1+8qw1a/)tfx kumpzywvkr 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 5f gje3r(ee(a and angular momentum are conserved. Yet most moving or rotating objects5r af(ej(ge 3e eventually slow down and stop. Explain.

If there were a great migration of people toward thy,smsjn53s g *e Earth’s equator, how would this affect the length of *5mjn s3y g,ssthe day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatio+ 0x lrb4s7mh9ug4hq3gwa f x*n whe+l90 4m hr ax4fwbu *7g3xshqgn she leaves the board?

The moment of inertia of a rotati -hyaecuu0,1e ng solid disk about an axis through its center ofuyu,e 1c-ae h0 mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outs4.:qhkc kfg4z tretched hand. If you s4hzfkq. 4ck:guddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one i 1olvsvdrj1la;- g (2ts hollow and the other is solid. Describe an experiment to determine (l2v1 r;sv g tolj1d-awhich is which.

The angular velocity of a wheel rotating on a horizontal axle pointj( hjhirduc nra-+ -5:u)tf3e s 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 po)cuudt: rnjahf (-+rj3i-e5 hint 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 turntable. What dgraa*;xqpo m + /my+hp*mj47happens if ojy4; mp+7pm/h qg*dma +ax *ryou walk toward the center?

A shortstop may leap into the air to catch a bshk05aql y62eidtg42 all and throw it 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 opposil65kga 4ys iqd 2e20htte direction (Fig. 8–36). Explain.

On the basis of the law of conserfcqpsr8 /x 9 *jzl u:h( a2x(t;ifay5wvation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more waycx 5zysiwh2lp :/((*a8xf qfj;r tau9s the second propeller can operate to keep the helicopter stable.

  Express the following angles in radianp,: 0vwdn .hyttlo0wmt3 c+-z s: (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 becau(g fmrs rrt)(s wc+*f0se of an amazing coincidence. Calculate, using the information inside the Front Cover, the an(rrf+(*sw t gs0f )rmcgular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Eart/73fwlbhq yrmlm24 k/h. The beam diverges at anh wb/mmlflqyrk 73 /24 angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned o4i;ugdjp / g2a*3px clff during operation, the blades slow4g*x3/2 pigd;ajcpl u to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chi4gbf0 / w-vlfbld. If the ball makes 15.0 revolutions, what is its diameter?4 f/lwgbf -bv0    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions di9(d .s ck6rpj;ld2 tvo the whpd(.vlid;2k 9j rs6tceels make?    $rev$

  (a) A grinding wheel 0.35 m im:*,q x i-xlrc2e9dben diameter rotates at 2500 rpm. Calculate its angular vme e,9:lix* q-d2crxbelocity 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 revolutio o7+8kdf1fanwn in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated owaf k8f 1n7+d1.2 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 itsjf ,, all7l+,; d6oa2 bs:fr3dhm6z-a eytgic orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a phsneu(y3+rs/ td 2 lwa 1x2mw9oint
(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 cmz7g;e (qj6y/gu ry b,*hgv;sw from the axis of rotation is to experience an accelerar6 7yqgs u yj ;*z;evw(g/bhg,tion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniforml+/q - mu+k5pkhss.fwyy about its center from 130 rpm to 280 rpm in 4.0 s.+wsk/yh s5 um.kqf-+p 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 radiufxzgpe)au;.c ,i9qa c9 whv, .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, astronautsgm qox;5w2nehdf-s 0 2 aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip,ndem; h52wfq-g x0o2s 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 rp6coyzp9 i uakr5 kot:vq-nt99fu q,1 2m in 220 s. Through how m , -6k 2cyo at9nq9fu:zoti9 5kvurqp1any revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 12iqhd:z:m7t,k 00 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flyino8a bkfb z0;1hg highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn throug1obk8;b fzh0 ah 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 unifoglbh;46 ni0sc*so f 8mrmly from 240 rpm to 360 rpm in 6.5 s. How far will as8*goc4fm 0;6 shnlib point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is rj0tizw 3i3hcu4l j g6(unning at 850rev/min It turns 1500 revolutions beforgj3t luwz6i0 4 ic3hj(e it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed un3 zm ,bnc v*)pt41ufi7o5r aeniformly from 95km/h to 45km/h The tires have a diameterrz *fo4n v5a3mnpec bt)1iu 7, 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 revolutionsejjk)7xjoj6wqo2 n tsb8(p y8 7q;i*q as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 086jnjx q7j; tqs) kq e*jpoyb8wi2o7(.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pe qnyl xxc7cr:o v+l5d3/7c5ll dal when climbing a hill. The ped y 5x7qnr c5o/3lc:+d llc7vxlals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 zb. +5yavp:l((+al(.g)eptxd gol dw cm wide. What is the magnitude of tw.(zgp aa.(lb v etdp5 (o)l+ld gxy:+he 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 the axle of cz 6hg5 :pbvz/the wheel shown in Fig. 8–39. Assume that a friction torque of c5g/6bz z vhp:0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod which pivf z. e yi;5o4cc)s-1sycwuk4vots as shown in Fig. 8–40. Initialcc.euswks 4c-;v)fy4 i1y 5 ozly the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightz snzwm,5v :d7pyfp)c)) mkd :ening to a torque of 38c)m)zz ds,:k5p mwdn )7vy:pf $m \cdot N$ 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 x -qvu62yo e8(v upf/)r j-jm ;qyijd:radius 0.648 m when the axis of rotation is yuf2 :8qjr mop/6i;xq(j--ujy de vv) through its center.    $kg \cdot m^2$

  Calculate the moment of inert ti:7 0u8)hqa7(c x5hapnoa:veh9kcfia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of 8 af)e::h 7vucn0q oxtpakh ia975ch( the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light)cc z sck9*jz/osbz *5 rod is rotated in a horizontal circle of radius 1.2 m.c )zsz /o zc*bc9s*5kj 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 potlq rfvsed + 6;m/.-bhqter’s wheel rotating at constant angular speed (Fig. 8–42). The/l.fsb- 6q+mvdreh q ; 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 array of point objects shi-tma zft 74483j -jy yc*hbrfekp9f. own in Fig. 8–434jm ae 8yfftif -ht9.7pzck3 b4r*-jy 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 oxygen10dgiet v 3kg;lvr*0 n atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correcsqg06fm3let .k+a0 j-wy qro7 t rate, engineers fire four tangential rockets 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 minfq3-7mkjl gs6oaw r. qe+ y0t0? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with amk:b8s q+,i( jruidz9 radius of 8.50 cm and a mass of 0.580 kg+ q, szi(i bm89:udjkr. Calculate
(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, acceleratin-vch1:e.j e7s ugj u-c 3u1kczg 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 smobiyz0n i8 v;(b.gc ta wxlu)4d6- nd1all hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, anwz84t; .lg bb nia 6cdo01 (ix)y-vudnd 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 is evo20 q1g)sem q 53lnuiaentually brought unqi13s n0ao2 gem)qlu5 iformly to rest 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 a 3.6-kg w1xq/5te aop6cf9/mj 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 sorpzpm:a9c 9jd/ x(.p(it . drvlely by the action of the forearm, which rotates about the elbow joint undx:acprp/zm.9 ( .9 vptird(jder the action 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 considered ,h ipgr*e63naa long thin rod, as shown in Fig. 8–46h,3 6 *gnriepa.
(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 consistsdve b0o5n fy;6t*6zp l of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the c t*p7ru.w k)g+o,kj zhammer from rest within four full turns (revolutions) and releases ittrw.co7 zgkpk + ,*)ju 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 inertiag+p fy e9ubm+. oi50vn 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 aw j;c5b2u3kw n 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 radc uv0smtekh /bjj(h+1++ n2nxius 9.0 cm rolls without slipping down a lane auxj(+b1n 2k+0 vjmhhcn+ se t/t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of edyd:zh1oy r(,c)e ; etwo terms,zoc)ye :dd;e(h1,e yr
(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. Ho ys3x71 hvg2rpw much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is apyrsh 217xv3g solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withous0g v ywps,j+3t slipping+,swp g3j0v ys 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 stanck ui/pv26 4y4q d3dsrts to fall and its lower end does/yd dqv6us pk4n4c3i2 not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the e 3bhgh a-vr+db0/cm3g5g lff ;nd of a thin string in a circle of radius 1.10 mf5f/cabrv3d h-bg; h+3gml0g 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 bim9;,e/ck fn grinding wheel of radius 18 cm when rotating at 15f;/n k9ebimc,00 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 pbb:(3gj 2ll 9afi.pia rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the spb9(jip f .gp2bi a3:lleed 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 on gnryo; y,y )j*k/m+vve shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes nmvvy+;ry,* oyg /jk) 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate ofxo/; iy7rva +a 1.0 rev every 2.0 s to ar;7av/ aoxyi + 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 veqysu d-yz0 y.)rtical 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 of0yy zsyd-) q.u 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 angul,3.j+hxylt i dar momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of loafog1.c r28 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 inertigu6k zfr8u2)rpp 96i:bq0 jhw a I is dropped onto an identical disk rotating at angula2f6rp gk) :0u6ub8wqr 9hpzjir speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ag:h2 xlvm;rl, 98sis94wkyb gt 2.4 $rev/s$ 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 xty/b g +t2do ,r9c2lvilfn92of a rotating merry-go-round platform t,x 9l/boyic2 2dr f+9tlgn2vof radius 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 rotating freely with an angular velocity593kcmg -wb ghg59xf s0iz u3q of 0s u zm55w30q gcxfkgg3hi -b99.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapc:lo(+dtdo9 jf7 j/hgj)*h7 il:jm c (pvqj0bses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass ca jbj0l vto9jh(+ cf/dpc*j)7gm d :l(:qhji7o rries 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 4, h 8ooag-hf*)g kfsdin 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 bz1sn *q91;plw4 w0ljnuquc ;$ 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 aocp0kup8+ws-:d6sf j,0sg y, initially at rest, that can rotate freely without friction. The moment of inertgk0y u 0,s6oppjswas+dfc-8:ia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameteej: e;ztqco(/p7f:x 9cyk7 ltr merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 17 tc/7tf 9c:q:ej(e7yko pxz;l00 $kg \cdot m^2$ 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 l(-k0 nlhbjhr 7ying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding o(l -0 7jhrbknhnto 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 always faces the4mx80sz e; ka)ay6lqe Earth. Determine 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 pamye;l8 e4a6a)qxzs 0krticle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates l .wxsl6wo.1kvj- 0bgfrom 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 unifoike r 6yuidt)94y ;ez2rm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calcul;td9rei) k 6ziy4y u2eate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two s0ru r qqm60b-rztw qf8w-v sdd,l)4 0xolid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solidmb wq 60-ru)04xsv,qr 8 wdlzt- dfq0r 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 angp,dn)by6z.r r ular 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 of7 f8y/m*fve8frt juait0-9jl our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, antu vj/f * el9jyrf 7f8i80am-td that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution ac-u4).xb8xh3t fidi h utomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg,xt) iixc3h .8d fh-u4b uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrasm/pk-ge*ul: (;fc nctes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface ap h;g4xp0tn cp0wg,j 3c7o i1it 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., y)buta5z 6xcua.: )gf we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of tzb6) .f5a:gy)xtcauu he 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 verticallysvt lc7z9f l2/ on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The stet9llv7s/ c2z fp has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat roa qx-60re3ym +tsg ,jk7s mbf6td is making a turn with a radius The for0xrk-7,6smt feb3 qjmtyg+s6 ces acting 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 slicq6d:b h904 0 qmrwmvk(6qhj ung of length 1.5 m and begins whirlinwhj0kr9 d :m6mq(vbh4u60 qqcg 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 this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s -mp duxeq9 :r vekyy)(ir-c *)body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular sp p-yxmer:*)(9 ik rdvqy cue-)eeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two ol :d s+xct60avuix4 p44;l iwj5ij/ jcylindrical plates, of oj l+06 t vj5/4cxsd;jaw4ii:xl4upi 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 rollse.o agvtu 21m/ along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving th temvu1 a/.o2ge track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assuu( 2u5.l)zjsn1zw8v1 q mm /r l)chlf; xl/wutme $r\ll R$ h =    (R-r)

  The tires of a car make u og7qaw0 *gk885 revolutions as the car reduces its speed uniformly from 90km/h to 60k oag 8qu*kg07wm/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