A bicycle odometer (which measures distance traveled) is attached near thvrjd i2jfh ;u 4yr(-20) zfd .renej7x:dx4kue wheel hub and is designed for 27-inch wheels. What haiu rrz d-:njfe2x 7d 2u.e j( fvrxjh;yd440k)ppens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim 9.5a hrtwauix1i p4 ,fg560ttldh se;have radial and/or tangential acceleration? If the disk’s angular velocity increaseea5hrg,5itx;h fwtt 914 .la60p id sus 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 mu4lyh0 .573snms d m,e pkcn.value of the angular velocity $\omega$ Explain.

Can a small force evew ym+r:yvwh34 r exert 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 sit-up with your hands behind your hemj9 a;g6,)-czpqcjn*k r r/ tqad than when your arms are stretched out in front of you? A *)cmj p69k,jq/; ctz-rgq nradiagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at theh o8t o,zx:owx a;2ch* pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?wtcxxo*o82 :h,;zoha

Mammals that depend on being able to run fast have slender lower legs with fleshwc.rrfjr w cw5/ )xh28;tnl9r and muscle concentrated high, close to the body (Fig. 8–34). On the basis of roc wth5wr f/rwx; 8rl.n)9 cr2jtational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow bef33h v 9 ecw+dpmovz2:am?

If the net force on a system is zero, is the net torque also zero? If the net tgz 0e 9a5gn5 j,:d/m7c;bqmyo0u kzltorque on a system is zero, is th a 7; 0l et5mz:5c9gnuj/,bgo zdq0ymke net force zero?

Two inclines have the same height but make different angles wi i;epf/e md+)nth the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the beid+/mn; )e fpottom be greater? Explain.

Two solid spheres simultaneously start rolling (j c cj5m2a(*9epnmii8from 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? Which has the greater speed there? Which has the greater total kinetic energy at the ba8ipmmj cnci9j2 5 e*(ottom?

A sphere and a cylinder have the same radius and the same mass. They start fro nt3 e-g;3fhog c(o4iym rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater r 4ch(3if g; yg-ooten3otational KE?

We claim that momentum and angular momen2m+ xm 6ffnylx kcl)88tum are conserved. Yet most moving or rotating objects eventually slo+x8 m6 cy8xl)2n fkmflw down and stop. Explain.

If there were a great migration of people toward the Earth’s eq)h ha v+g:sryi0 ,huj+uator, how would this affect the length ouvgh a0+h sir,y):h j+f the day?

Can the diver of Fig. 8–29 do a somersault without ha/kg2i*l .zts ;gmf0obz+ cy;k ving any initial rotation when ;0 o fty; ./z2ickg +*gmzlbskshe leaves the board?

The moment of inertiat;uoais+ir7 / of a rotating solid disk about an axis through its center of oii u7;t ra/s+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 o5xed m88ab h4uutstretched hand. If you suddenly drop the mas58b ame4duh8 xses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the samell8r 6z+gs x7tb17sd d:/0qgqc o5bhc mass. However, one is hollow and the other is solid. Describe an experiment t1+ 0q:glghc5 8zc7rxo 7/dbd6tslbqs o determine which is which.

The angular velocity of a wheel elqmp0k)9gp 8mw ps*5 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 angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the ang*ep0qklwpm9m8g s5)p ular speed increasing or decreasing?

Suppose you are standing on the o,cs c1*f gg(lq ):ydw5wex t2edge of a large freely rotating turntable. What happens if you walk toward gfd2ocext, l:*g(qw)5ywc s 1 the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he thpj-joi5e1t* fr+benz6 d ke0; rows the ball, the upperetjb-k+ z;6 0j1*eoip5nefd r part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of o; ydp62lzgjasp:n46zt-jw5conservation of angular momentum, discuss why a helicopter must have more than one rotor (ozjs lw o z4pj2;p n66-ydag5:tr propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a uqph;vl;g ljep 88 r(-tcp,9r) 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 h)f3ef ;)po nkamazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radian);efhfk n3)op s) 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 Eajy:k)vawlrnb:o /gd09 zo:s*jh38b xrth. The beam diverges at an )z/b alorv*:bd0:83g9 nho jyswxk j: 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 the4948kb )nf- l8gbg +nhtt uelb motor is turned off dul hne8f+k)-bl t9utngg4b8 4bring operation, the blades slow 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 child. If the ball matyx0 a2,p+c,td3,ajcht70 yc ztwl1 ckes 15tp ly, 7,zttdacc 320 txya0w,ch1j+c .0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter/ccvd ss1k: oix 00fxd* tds4f9u)a1d travels 8.0 km. How many revolutions do the wheels x0dsfk9svdtc1asu0 xf41)i dc:od*/make?    $rev$

  (a) A grinding wheel 0.35 m in diameter 4 r-5e2r9;r-tpp ykh7w:affg dja+dr rotates at 2500 rpm. Calculate its angular veloc; e rp +: dgkf45-a-2fayj9rhd7w rrtpity 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 . / gy-(l csmw(du9hoerevolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the centersle-./ (9h u(ymwcdgo ?    $m/s$
(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 Sun .u 9sszmx.ce ,l(dk3h   $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sp q*42 aaplzo- xonhv-8dd/1sgeed 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 parti l- m;vgi.myy)cle 7.0 cm from the axis of rotation is to experience a;-m)yym . ilvgn acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheelx0*i s*,1tfd k;c vtn bir3qnr/)d0ur accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 ut0sx )ri3r/bt**cvf0,d n ;qin r1kd 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 radius 5qoy hq)oq:f3 $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themsk 63dha s*wr2ukys6hfu-6y 6. cssha(elves 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. Du ks wuy h6-sc(akyfh6r h.3ssd26a*6etermine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Thryn bakz ;wts4bc04v-jeqj / 13ough how many revolutions did it tuc4zbn wt 4;s-vbjq j3/kay1e0rn in this time?    $rev$

  An automobile engine slows down4uqr:,xa8ga4 +ma 9gp99oo0 zxfjxkh from 4500 rpm to 1200 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 flying highs fvod 9,dw2cmk:o;;wa peed jets in a whirling “huwf9mowd c2o;, ;av :kdman centrifuge,” 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 uniformly fr: 6ay1 v+ibluiom 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveleliuay +6ibv:1 d in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 15y1mi )loj:nv700 revolutionsjn ylovm:i1)7 before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the c s,n+mwqy *yr2ar reduces its speed uniformly from 95km/h to 45y+y q2rs wmn,*km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 69so6bnd; poma 48tg8s*f 13bptlmn-w5 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tired 419lf;b*8stppn68w n-oo3am g bmt ss 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 her weight on each pedal whebja-llo1m b( e9 3i(jqb*:n syn climbing a hill. The pedals rotate inebj*mql jbi93lsn1o:(b ya( - a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is tdxl:p,bf*rsfm7 jk( 1he magnitude of the torque if the fflfsr:p,(b d*mj k7x 1orce 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 the7u8e s5 p(fm;sqys.d o wheel shown in Fig. 8–39. Assume thesufs;pm5q s7d.o (y 8at a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod whil0 h6gmt2shv,4on6 *.ijmheyz 0brr(ch pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then releasedvr2 .,yb0 6hitlh sm6eh(njm*zg4o0 r. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightenp6qpng za, .3xing to a torque of 38zx.,pp 3 6gaqn $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m whe6d*mrb fsx1r68z pfp (n the axis of rotation 8m(1z pb6f6 xrsfr *pdis through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicya.ad+fb b,i*ymldbj06/rxs , zw, fm4cle wheel 66.7 cm in diameter. The rim and tire have a cr*dbjblfy6 4.wz,idab ,,ams fm /0x +ombined 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, immd3chg6j )z 9tv7 *xu; x6yvlight rod is rotated in a horizontal circle of radius 1.2 m. Ca xhm6d)m uy9j37;6i*zvtxcg vlculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular sp ; r0+qt2eaoh ,yyy*3p vr;ybeeed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N trbryh y*;t0;p ya2e,vqy+e3ootal.
(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 pw7q.gecwx(is a ,9;xpoint objects shown in Fig. 8–43 aboqewpi7x cx.9asg ,w( ;ut
(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 ox.0 z bw .2 5;jnioobjalvg6brd9q- 9ykygen 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 correct rate, e por.-.cdall 2)pa a8ak l+5vgngineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a massvalara.kgd8l5)-c +. p oap2l 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 rad2((l1xxerr/t m7 3 adtp;gwaq x2r/pzius of 8.50 cm and a mass of 0.580 kx e pt3 zd/x(a(xrl/1ag2qw;mp 7rtr2g. 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, accelerating it from lrt8d(sq/g;g (h i a6+wd:qcnrest 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 smug7)rs i kuzeq6 a-8pmzn4k-ui8 5k2uall 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 children (each with a mass of 25 kg) sit opposite each otmiuua 7k8ks) izqpkn6 2-e4z85urug - her 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 1mtylx- lgztzh 70;6d4x-2sn f0,300 rpm is shut off and is eventually brought uniformly t 0l xgtfny6lmz- ;7sz d-h42xto 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 accelerat. 5.ov v*vti ksh58rfwes a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the a7y pgw-.ya m3forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball7 yaa-3p. mwyg 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 thin rod, at+- wabyp+lkp rti7 7.s shown in Fig. 8–46. 7l+.yt+ -twip pbark7
(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 consi6)+wl d,vyyl xmr *3ypsts 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 accelerateu;i(gm t)jcp :s the hammer from rest within four full turns (revolutions) and releases it at a speed:j )(cgmiu;p t 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 mome03mp8,bv xxd4-yt aef nt of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develojuuwkj)gh2a9 ea2(0j ps a 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 slipping down a *qbva t su -+(r98mx*d:u.to flpo ji/lane at 3.3s9uq:b(xpt8 u/*.iv+ o f jd*ola-t mr $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic eo9kcgykq3* 4cnergy of the Earth with respect to the Sun as the sum of twoq3 ogkc9c* 4ky 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 aj: ;hsrb /q-ym mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate qysmb; /j:h r-it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest qnx 8vccv1oai 12m2v (6oelwjxz *4 8gand rolls without slipping downec8l zvn1(i1v 6 q2vc4x*o8x2 amj ogw 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 ,rp-/qysfad 3yf/x 5fbalanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just beffadfpr 35,q //yy sf-xore 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 end ofkskk imt-y ;56efm d*2 a thin string in a circle of radius 1.10 e2mk ;*ki tyfm5sd-k6 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel 534x lpx bjxbxitd,8, of radius 18 cm when rotating at 1500 rpibd,8,p xx4xtx lb35j m?    $kg \cdot m^2$
(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 ivzi8ut 0 -y q9,zq;,296tbfrq qdsomss rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Figbq- r809os ditqt fm6 2sqqzv,,9uyz; . 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) zep(o3c i:a(m can reduce her moment of inertia by a fm3(oa( ec:iz pactor 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 her spin rotation rate from an initdora2lev3 bee mpx.4+ b32q-iial rate of 1.0 rev every 2ep 234b+lxvaq .-bm de3ei2o r.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 ve uo jwaya q0qzhs2s7w5fr,f1 m,: b6h7rtical 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 offf0q uy,a h7j7 q s,6swm: aow5z1br2h 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 momentum of a figure skate+h t vkk-ugm40r 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 c lu ;8x9ecy8bEarth
(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 in /l;rfrztcs3- ertia I is dropped onto an identical disk rotating at angular z ;rr-fl3/s ctspeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4bv:l2wqu fp 6, $rev/s$ 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 thehw)3zfxc8lij9 5i .+mrxm4eo center of a rotating merry-go-round platform of radius 3.0 m l8wx9i.mzeh5x 3j mi 4rc+f)o 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 3btc(jditcwc. ;n8lfp, 54utof 03nit;fw(u,5jcplc4 ttd. 8 cb.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 collapses into a white dwarf, losing about hal; u(t g*fgs9uyh/ssy 0f its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be*( u yhg f9sysusg;t0/?(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 dj ;.ctrtk,xm /sub9 f q,nt(*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 proky; 9 s/la )*f6mmj$ 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, itwx9xco6 cwim1 r(w0-t1:x w fh w0*fknitially at rest, that can rotate freely without f- cw xwtwxw 11to0x(kh6 0fi* wc:mf9rriction. The moment 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 of a 6.5qtt .b8-iia s*-m diameter merry-go-round turntable that is mounted on frictionless bearings and has iiqb 8.-t* sata moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying o+ pm,v (e(tdyan 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 mass +y((,pdmeat vmove?

  The Moon orbits the Earth such that the same side always faces thexoe8 ,5mqol3ivcwr6 va . ,jk( 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 (5ev.r a, co oiw,x3lvqjk68m a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a xurgg;+jx,( tf34*pwg og9j wrate 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 th u(buw ;7dlkx8 szgq*4e shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. ;sdkq4 uwb 87(*uxglz Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diskso 8:f95 yk vit2kwo-mk, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speedoy v8:mkot kkf9-5iw2 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 speed of the rear wheely3f kll 0u(q9 0b3nq2o 7xxiiq ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 timh) zay.u :)yyfj6:bn le1 ybr9es 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 f r)l1z)beu69:jny. yh:yyab mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is fi9jfzv/w** y,ak3 r vtor it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniforva *twvj,*3 r9yfzk/im 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 illustratek/px,6 m gmzgy3h-s 4zs 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 at spee h,u.ql.mz(qs4mtwp/ d v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore fqjqa:)p*:i- r-v q /ewymy7ch riction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the m q ecirj*m/yya7--vq qw):ph:oment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verticallwwt))w*lyh -8zr y) gdy on the floor, and we want to exrwdth)8)g )-l y* zwwyert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat cx*nsfn(+) +r6oxyrl road is making a turn with a radius The forces acting on the cyclist and cyr y+o(x +)rxl*sn fnc6cle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins wh*4e,lyys tg3vpx s(8zuj-ph : irling the ry lyp3-(zjxu :p8e*tv sgh,4sock 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 body as a solid cy51 ;,mujk7qt5nlj js28a tkgrlinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spin,8ja5ktgtnu7 1 52 l;qmjjrks ning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consiti/if ntn n 0wxfs5363ly 3k*5 sf+mkcvbm d4*sts of two cylindrical plates, of mass 03fi3 t yf m3lndbvx+wtn5 mk64si5k/c* fn *s$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rolm3eki izfli1 : ds-92ugh track of Fig. 8–58. What is the minimum value of the sk9m2-:l 3i zilf1dievertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do pla8- p+u o vuvqf3/.wnot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutio:rtq/ jl( 8sxvv: h*kigl8zm*ns as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 08g8rmj:qv hzl :s(xvi t*/lk *.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