A bicycle odometer (which measures distance traveled) is attached near th(o7pj-zlbl gt+j /ootf2 z;pu0qzu to77 ;8 zae wheel hub and is designed for 27-inch wheels. What happens if youfo/t; gu zop2t pa z(+lq778oj0lz;toj u7zb- use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular mj9c+ vz,ee :k ysn;)uo4z e:avelocity. 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 accelers;zzvje en4:cmuoky9e+) a,: ation? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid bodyllh8/m :)ni, kp*s ude be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert v)7 8fnoyisu8n p0ad: a greater 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 sthbjsfzo g9.0c0/ euc :r-uc9 it-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer t.jtb0zcec:-ours fc 90u9 g/ hhis.

A 21-speed bicycle has seven sprockets at the rear wheel and tklq8pi.mn2tdses0/ g)vz fb763 s q5z ;fjz (phree at the pedal cranks; zs sn tpq /2zdf.bz jp)8le3vq0 fmi7k(65sg. 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 tamit2p-wd,mnl 43s * o ahd8w.o run fast have slender lower legs with flesh and muscle c2wd .*ts,min3p 8 ahwmdlo4 -aoncentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkeb97h8 juyz 0 .feo *+*conoqvsy(vn1ars (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net tpd1kk+p 5yy+km2 7vca71v hgjorque on a system is zero, is thv1 p 7p+cj7g5v1ymd kha+yk k2e net force zero?

Two inclines have the same height but make different angles with the horiu+duvla ):/cs zontal. The same steel ball is rodu)/c u+svl a:lled down each 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 inx k/-p3l8hbfuc 6xy2ncline. One sphere has twice the yxbh3 fup6/l82knx c- 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 the bottom?

A sphere and a cylinder have t *sp oxepx 41 1 0wn02lab83of9bzgpuehe same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the 24*o8 xf unepl1opa0x b gpe3w s1z9b0greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved.l-u/fbf76-lxs- h y na Yet most moving or rota f /6n-ulhxy-fsa l-7bting objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, hosfj8 zo9i y:hpps6+/kw would this affect the length of the day /+sk6h8z j:pisfo9p y?

Can the diver of Fig. 8–29 do a somersault withmey xa1;w3f 07e gyx;d.,jeenout having any initial rotation whee ax0.ee;wyn1e y;m d7jfx g3,n she leaves the board?

The moment of inertia of a rotating solid disk about an axis tuqy-) ji ijo1a2 -fkg02 ndlwdt)(6aghrough its center ou()y i6 2dnk) 1ljg- jdta i2goa0-wfqf 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 on5ivsm*4n9bmaq1 h 9w y a rotating stool holding a 2-kg mass in each outstretched hs54vb 9aqm 1h*mywn i9and. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and hap-q7q 1ac bc)ove the same mass. However, one is hollow and the other is solid. Describe an experiment to determpq cc-17o)q abine which is which.

The angular velocity of a wheel rotating on a horizoni w-*m ctd/2rpr2* /psc(ympttal axle points west. In what direction is the linear velocity of a point on the top of t( srct*d *i/pp2ctmy2-p rwm/he 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 increasing or decreasing?

Suppose you are standing on the edge of a large freely t. dhb+isd qfj3* )es3rotating turntable. What happens if you walk toward the cente t s.i3sb+dh3f)ed*jqr?

A shortstop may leap into the air to catch a ball and throw it q.gs k8 r(/plhz75kyisuickly. 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 th.hg kir 7yls85pks(z /e opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angulwuum orgqw/p :i.8w9eb c .3-tar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the secondrge w.-mt./o bwi8qu9:wpcu 3 propeller can operate to keep the helicopter stable.

  Express the followinza 9dnwlvln(h0*8 )iz g angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, using(jhpu 59ma ;je: y)jxc the information inside the Front Cover, the angular diameters (in radians) of the Sun aujxpy): a;e(jh5 c9m jnd the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,0-qw 8d +6zjd 4romlh3e00 km from Earth. The beam diverges at an anglo-+ q4 erd8h6w3 lzjdme $\theta$ (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 ratei;hspnzk6l/ 8 of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as iks;lp6 nz8/hthe blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makevr.bk.e;0b,.c ggu r xs 15.0 revb..b 0cg,egv;ru. kr xolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions qxgjivavl;j27 y h,96 do the wheels 6;jq hy,xg7v2jvil9amake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 bh9tr +ef0u3i rpm. Calculate its angular velocity re3 +hu9 ti0fbin $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4/3w r.wn*.rq qexvy g*uj 5t;1up wyt8.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m /.u wrvtn3p*x8uqjq rt. yyeg5w1*w; from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the hezb 7 n: dn7p5.4byoaSun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pozpui*e3a tup e5gf0s+,i. 0pfint
(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) mustea3g2p 4bw.ag f lg3riw: (0xf+j xp3,,xpbwd a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 1 fw0rawpe 4(2,p+33gx:,b a 3lgwbf jpgdx.ix00,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unifoq/8ks0gg(v fhy he5z4rmly about its center from 130 rpm to 280 rpm 48 y0/k(e fhgvgqz5sh 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 radiummf n w jtrich,x/pg* ,l :,2.o/wnp-;pscw s5s $R_1$ 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, astronv :zt-p :r)durauts 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 8.5 m. Deter-t r:purdz)v: mine
(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 i 8u3fnvfs(vr:a9xm2 pn 220 s. Through how many rer8 f(s 39xvfv2mpun:avolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 c 9,q3kjly mh7 u;iwb*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 ( nobo0a(x5kuflying highspeed jets in a whirling “human centrifu(uk bxo5n(o0 age,” which takes 1.0 min to turn through 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 unif97.uxs , ygtuxun8t , k.n5t jsdyx1r0ormly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveledss,x.u k9uryxjdt.nn,71 tyu 05g8x t in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turnsc 1g 23*t-)6*jlb ouhprttf skncb28t 1500 revolutions before it cb2tb3-pu6t)j c82 gskf* r 1lntoh*t 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 reducesx)zyf xc h5g -v,b8p-f its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.gh v y-zfx8 -f,pcx)5b
(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 un +339f 3mtcjx wxe/kjiiformly from 95km/h to 4+3cwxe3x jf tm9ij3/ k5km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all h e f8vuy;a6;zfsb4gzgn7c +- eer weight on each pedal when climbing a hill. The gny 4gs8;;auezf c6 be-f7 +vzpedals 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 cm widef9lv. y0xbihgs.w +w(. What is the magnitude of the torque if the force is exerted +by.w0wxfgs( .h9ilv
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel showny5llkimf3rdb* 28 1y hy.9pgzi-w g)q in Fig. 8–39. Assume that a friction torque9zby l. 2fg*3p5kwhg)q1- y iy mr8lid 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 +fyfkgj vm9 .ms s.1g-the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this+g s -y1 .msfg9kv.jmf system.

  The bolts on the cylinde n -rxq9:n:zk);gyeqjx1 t sz0r head of an engine require tightening to a torque of 38 :zyrex gtq9x0: ;n1s-jkqz )n $m \cdot N$ 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 radiul4f2 kjh y4up)s 0.648 m when the axis of rotation is through itky p2u4)lfjh4 s center.    $kg \cdot m^2$

  Calculate the moment of inertia of a b+czzqzz 5( io pvcy0hx* z615r0kt( zricycle wheel 66.7 cm in diameter. The rim rr z+ip5zq1zc 60 voz5z*k0x(hyt(czand tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rodx95(ra5)ezx7 u2wl xmp)hol+ q3jyez is rotated in a horizontal circle of radius 1.2 ze3 7hxp)rz+eo) m5l9j x 5q2ua lw(xym. 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 potterx 91yhto,g2ylp5 asu n3+of+ y’s wheel rotating at constant angular speed (Fig. oso ,x5 +t nyyhug39p+l1fya 28–42). 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 ou 2z5 (rn0 ,tcxqcadfue5i3p;f the array of point objects shown in Fig. 8–43,ndp;xeiz(f5 t2 q cca5ur3u0 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 a byk3upo24a)btoms 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 spinningxq nhn46zx;-0nb2va h at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the sath4nnxxbq 02nahz6-;v ellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm a5snzh* j45dax nd a mass of 0.580 kg.j sh5ad 5*zxn4 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 swin7t*egu 2,fpq cgs a 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 ou,l v5;gbi1 ra small 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, and two childrenlbrv ;,51ugio (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 giolr,-cknilr1s0 0 yn- u:xlq)bj-1 is shut off and is eventually brought uior0 bci qnljn)1 ,-g-sr0u-ly klx1: niformly 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 b*p5tahr )ilz7all 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 balacdtfvk3; :yg r2/ /jgl is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig.:g2/t; v akfjd3c/r gy 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 a long ;t0l hw ofq7 .h-n*ujd;-jl9qtp;tvr thin rod, as shown in Fig. 8–46. 0 wpqtn htl;h;tq-jovul; rj.79*d-f
(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 t1o.gg7zcd j k5wo 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 f:j xb 4nccm)e.v)dvhtp7t p7)ull turns (revolutions) and releases it at a sp7x4.hv :cdb )nctmpp7v)e)jteed 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 -rwxho8*1d3me/*swsk p,e3dm g1ll rof 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 torqur:u;n lw-3cppvw td u:*bo y)0e 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 r 2pkclloy4y3ua,84pa olls without slipping down a lane at 3co4lkup y, a438pya2l.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of v-y hndg-,/gmq.zv2 qthe Earth with respect to the Sun as the sum of twonv2vyg hdq-z,q.m- g / 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 andabtg;+blr*n bsn3 ,7 w a radius of 7.50 m. How much net work is required to accelerate it from wlnba*bsnrg3bt7; ,+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 staz ujz7m d/k*h8g2d, ejrts from rest and rolls without slipping dowh2zued/d 7mgj*j8 , zkn 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. Is(,msa / e:f( w0thmhwt starts to fall and its lower end does not slip. What will bwt,(e /0ash m:(wf smhe 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 o yzd(, *hatfv4l3wmi/rc h./sf a 0.210-kg ball rotating on the end of a thin string in a circle of rad * amr wdiyf/z,h/. (h3vs4cltius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mom6*4qwo t, mumimww(7i entum of a 2.8-kg uniform cylindrical grinding wheel of radius 1 uiot*4iw q6mm( w,w7m8 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 rotatingen8 ) 41nw*)1nl c,jmz7aopj-s nnlbd at a rate of 1.3rev/s If he raises his arms to a horizontal 1-ba8n e nmpj4) coj1l*d znsnn 7l,w)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 hecl35s)r g q.hnl( xzk-r moment of inertia by a factor of about 3.5 when changing from the straight position tkg5hn)s l x -r(z3.clqo 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 he7slg*g5 gidju 7(p;9gwk)bp**sucg8: poomc r spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of *ko7bu:g*pwc cmid* oj58);gg9uplg p(ssg7 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 througz:f. n.;z-;zhe t l rtuxsuy/;h 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. zlt y;ts-;nx/:heuzur;f z . 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 mofeu rzg j2zs+5,8) ecomentum 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 the Earth * unc s2zvk*u:
(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 drork)*j)7ctig ul j/ 3swpped onto an identical disk rotat)lswur3c g / 7i)kjj*ting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4y5xh *pntr28f $rev/s$ 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 plaje p(o y0kok7xp ,89smv upd+5tform of radikd0voy,(pp s pmj5 9 8u7+kexous 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 velocity of .ku+z+csb c x20.8xu2.kczc +sb + $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half its mo-ui6pth pi66tu20 xu ass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost masihtu-op 6u6u x0i62tps 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 win;ey)zh4 .kxmt*/jsp 7ecu 3vj ds 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 qs4 6*vi8s4sz (2t5mcrlhybz$ 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, initially at rest, that can ro*09ejblqyudya.g,y w pz8 c *.tate freely without friction. The moment of inertia of the persol.,yac* bywqp*duge8 jy 9 .0zn 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 xvoig0r;x:ym 3 s)r2g diameter merry-go-round turnvx;m)s30gg:i2 x oyrr table that is mounted on frictionless 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 ground with the end of the rope3czq/j,j 4wxu 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 from the spool? How far does the spool’s center of mawc3xjz u/jq ,4ss move?

  The Moon orbits the Earth sc33 ,zat+yfw+ w2k9byy o8w* hnmlv5such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case,2c3w,aw y w ny9 m5fy+*vl38sb+tzhko treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at-ue 0nu2ialpy00lef yy4tn(:09 hk hg 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 0.20 m acquitl8a19t /sukkglpg3u9mm2 b i5oc/c ) res a rotational rate of from rest over a 6.0-s interval at constant anl8b s9)cpugg uimc/5mt o3/91a 2tkklgular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid(v(1zc4dlg gs cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use consdsvg 4lz1(g c(ervation 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 spe g,cd-daq )l;9way); kltky7hed 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 oxe0u 8rhcviw les2d/zt5,k;,f 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 tr8les, c5vik h2eu0 ;x/,dzw 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 off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is formbcp,0f 9 ;y8hs/rvz0d uli k; 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 t98ycfm0 /,d k pu ;rzihlvs0b;o 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 illustrates w8p6x gbwp3 6lan $H_2O$ 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 surfacx249h ; t99wixuvi8uw5qowha p:jn8e e 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 (l*(dnec ) cn7pi.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod.p)e*(dcnn c7l Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and wexq5k8 2 dvw7jf;kr.km want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–5d7rq;mk5x kv.k j2fw 85). The step has height h, where h

  A bicyclist traveling with speed v=a y.sg *x(zke:p*de9a7g9j w*xapt :0p2in ow 4.2m/s on a flat road is making a turn with a radius The forces acting on the cycagi w eknt9wa7* s.pga:(exjd*9 0o2 zxp :*yplist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of legou. qs eu.+s50.g axength 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm afus0u.oq5.. gse+a xg eter 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder fbc0i sb;m;fjl 4+3p p,jyi7nqrbc57 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 tightly agains yc,77nj4 3bp ;b0pms5c+f lf;jiirbq t her body.   

  You are designing a clutch assembly which co)b; b ab p*pg pg6auy7 wjb;,j/vpm0:wnsists of two cylindrical plates, of ma: g0bwp,ampbpu6j*v /w) jabpy7g ;;bss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ala92zb6g2lr uv ong the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop i rb6v9l22azu gf it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, b) mwmzfk,crdeq zpr7lij)jx- -51rspk 6q 985ut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions age3suiw kpn(tg j9,)4s the car reduces its speed uniformly from 90km/h to 60km/p, (g)9ug4jn3wik s teh 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