A bicycle odometer (whic qq vb4+qup6w+r6wb r3,s4igch measures distance traveled) is attached near the wheel hub and is deww4b rgipsqrucb6,4 ++v 3q6q signed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates atacm*+p6*e ip x constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of eithe 6*c expm+ap*ir component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular 52hvl7y+mwf fvelocity $\omega$ Explain.

Can a small force ever exert a greater torque than 1gz;uz*oi tj: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 he4brzmnf5j. l *ad than when your ar5.mr*ljbf 4 znms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel h,vk+m6dp n;l and three at the pedal cranks. In which gear is it harder to pedhm;6dvl ,nk+ pal, 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 j/z a8*pow/ xwbeing able to run fast have slender lower legs with flesh and muscle concentrated high, close t p*zaow //j8xwo the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig.qd mc jwo8jq2 c;t;5-m 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the /445k rc0 jxq ae7xdf0jauqa*net torque on a system is zero, is the net force zedeaqx 47 4k5j fx*ja0 0arq/curo?

Two inclines have the same height bi3 b/exnj*b,0t7wqv/ vm2qsvut make different angles with the horizontal. Tnx /3qtvmwj/s *,7i0 e2 bqvvbhe same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolli e8ed2luu) myjx4 i6,png (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the g )y2 i64u8leu, jempxdreater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start fro0 , f3ud3rnhufaqiagx8ra :/ 7m rest at the top of an incliner3 urau3n7qfgi/0 fx :a,had8. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most onk+domxtrt j/+l 5f ,n17nl79i/lbs moving or rotating objects eventually slow down an7t+sj//nom bfk9d t5x7l,rl nloin1+ d stop. Explain.

If there were a great migration of people towazcepb+3.g,h7wh qu*p9,vdsk 1e ya 8ird the Earth’s equator, how would this affect the length of thea w,phzc98,isee *ghd13y kbpqv. +7 u day?

Can the diver of Fig. 8–29 do a somersault without having any initia*dd-hd ufb d4spw+wux* 3+x2ul rotation when she leaves the bo4d2- sx w*xubu++h3dpfdwu* dard?

The moment of inertia ofeukhxd- qr.-v/uho ;. a rotating solid disk about an axis through its center of mass is.uqv /o-kxuhhe d;.-r $\frac{1}{2}WR^2$ (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 so /ayreupg0,6 tool holding a 2-kg mass in each outstretched hanayproe/u, 60 gd. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identica-bx,a6 qlwe pbf1.pl5z:9rk g l and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is which.,plwfb: p.blkr156 zeg xa9q-

The angular velocity of a wheel rotating on a horizondwi xw1gjs04(tal axle points west. In what direction is the linear velocity of a point on the top of the wheel? I(gw4di0jx 1w sf 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 odwh-fviiq+c 0c 5:h4 e-fk/mlf a large freely rotating turntable. What happfefc:qm k/wh4v-0ih- l+ci5d ens if you walk toward the center?

A shortstop may leap into the aic3q(-a :cmnj or to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotatesqc(oma- j:c n3. 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 consers2 -;(rp de+chjj wskuvbp;e(cw99 ;h vation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helsw;jhcj2ebs(d- (;r epuk h9 cw9; +pvicopter stable.

  Express the following s/6s ek y-ackb)qp3r5xdfl::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 coit3um1 o 0*p13ohinzbkncidence. Calculate, using the information inside the Front Co1ko*o3 1ubiz0 hnpm3tver, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed:*vf +koi3aq y at the Moon, 380,000 km from Earth. The beam diverges akqio+y*av: 3 ft an 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 6o6w ano n+zw3:-fqi m(500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration nfq w-o36aw+om(izn :as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to 7 6 y ofuvx8t7f( p*nasoqz-.lanother child. If the ball makes o-8.*(fa v yfq7p 6xzst7uno l15.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many reve9* v,ioewu*uali 3 ,yolutions do the wheels makea9wu o* *iv,,y ui3eel?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates atqr yn. p2x l0p* t+v;8n-x bdc3.grskv 2500 rpm. Calculate its angular velo+ dynspx3n.2r b tq-*; vl p08.rkxvgccity 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 rev,,qo 6qr1, )pm4 ei0jldn7rqvap xs- k ojuz/7olution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the center? p m0ja4)p1sq- ji7, enqxqor7 ,k ,zvodr/u6l   $m/s$
(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 bv+f/9 4bosm u;b** h9ochj.wzxdt u95yoz0t    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a p rt4w2:* d vovtzll74moint
(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 ld 2v nyy/-rqd7*6gvwparticle 7.0 cm from the axis of r yrddqlv/y7nv-6g2* wotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly r03b)u5 mh b6qi+fnaz about its center from 130 rpm to 280 rpm i3zn 65f0ibuhm) +qrban 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 radius g cyl97; guqnl;wv2u2$R_1$ 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, asto. iwfk lc .us: ,kc4b-wr-.bjronauts 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. Determinew,cu ..4- kkbbcojsw :lfr.i-
(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 7w5e xpstq*z9s. Through how mwxe*7 sz9t p5qany revolutions did it turn in this time?    $rev$

  An automobile engine sym0zrqu -q:: klows down 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 strejeg m2yr970hqrz *i-w sses of flying highspeed jets in a whirling “huma rm2i0*h-wze7gr9qj y n 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 from 240 rpm to 3sn1;5 ;c nasx.)esvai60 rpm in 6.5 s. How far wv nasa5 1s)i ;.sn;xecill a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is rfuc1dfli 0h ud 3rf(0j6(wsirr (j1i) unning at 850rev/min It turns 1500 revolutions before it comes t( ijrdfu )ci(is6rff10 d0 rhu3lw(1j o 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 itst6y( p.-xrqfd speed uniformly from 95km/h toqdft 6(p.xyr- 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniforcpksjlks(43r f/. 1vqo*b0 qd v(5lpjmly from 95km/h to 45km/h The tires have a vc3l(f.jbs l4pqvd/q 0s(1 5* projkkdiameter 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 when climbing .s rtmk ;w+,ina hill. The pedals rotate in a circle of radw,. r+ inm;ktsius 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 )g5 o at3kvh:b6k*gpa) 7 wxecof 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the forat c:p*a)56)vgh7kg3 oebx wkce 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 thn66 o+g/ ai0xqs(by u1fy.wzv e axle of the wheel shown in Fig. 8–39. Assume that a frictiiy/ 0x(v61 wyuzba sf6.gq o+non 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 en 9sixc*n cd x92nikl*7.sb;nwds of a massless rod which pivots as shown in Fig. 8–40. Initxk;b d9sni * xn7.ns9cwc2*ilially 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 he78 ie-gjr ff ude38c-dad of an engine require tightening to a torque of 38 id fu edf73c-8grj8 e-$m \cdot N$ 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 10pjs1n)4thc e*zs)s5 /o-phd d.8-kg sphere of radius 0.648 m when the axis of rotation is ts /hh-es*)tzj 5 1)poc4ndspdhrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim,32j vu-s l1kczoe 8tq and tire have a combined mass of 1.25 kg. The mass of the hub can be ign81,q 2scvojtuek3zl- ored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod i8l t zm-9) kcqr)j 4bn)k,0esh;xttzhs rotated in a horizontal circle of )k rt8,ltezjb ;mzxqk- tc n4h9)0)hsradius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s whn)b tkv g *b1*ni9h/pbeel rotating at constant angular speed (Fig. 8–42). The fr9/kn*h*tv) nbbpgi 1 biction 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 shown in Fig. 8–43yig6wb,f)9 7hh uh,t k aboutb fi t7gu)h y9k,6whh,
(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 atoms whose tota;,6sm ; ntmrj8heu)ufl 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 5b 59g:4c +d sz wnrgwzl2.gfesatellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If4gdgs:flw9nze5r+ bw.g z 25c the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylind, 8cfy4xby 1s3s3diuner with a radius of 8.50 cm and a mass of 0.580 kgx 4fy3cdusny 83b1is,. 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 rest tk4soar+y)et )f;ei :6f/ir ; f w9tdqpz4pjz1o 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 m-dnin1vgn 4 e*1 2sgrsmall 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 other on the edge. Calculate the torque required to 41e*s2ningvr dn1 -mgproduce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rkt(31)ofce oxc5k dcj,q:id 5pm is shut off and is eventually brought uni oxc5( 1) d35 kie:jqd,cfkcotformly 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 a( eob l59sp 3g7:finuuccelerates 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 byj1jjv 9n bqac ,40mbpfk(s74w the action of the forearm, which rotates about the elbow joint under the ac a w 7jbvbqs(f1ckjn4jm4p0,9tion 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 consi olk,0 hk3qs(ldered a long thin rod, as shown in Fig. 8–46khl s0ol(qk 3,.
(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 two masses, euwkh e;qc-2(mbg)5qgd( gv . $m_1$ 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 fultx9( c1i j:lm xjme87 mbsa*/xl turns (revolutions) and releases it at a sji mmc(mb 8tex jla:1 7xs9/x*peed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor ha)2 ;)yyji2p5zca jitss a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque o bqz7th+ : +yh1ztzt4.u) 66rolqngsh f 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 rox1jm f-ucr/ /blls without slipping down a lanx-b1f/ ujm/ cre at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respg8f0yhceq- ckxkw-1sj6:( uapr 3t)tect to the Sun as the sum of twoyf u(t-gqkcecj6-w0h )xr 1:83stp k a 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 ofde c)1c6wepkl:pp )k 3 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assuk1 c)c)ee6w3k : dppplme it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rr piz73vd2z*qxy -2vbolls without slip2b2zxdvyq -r*zv3i p7 ping 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. I65b*xrlm dwq5 t starts to fall and its lower end does not slip. What will be the speed of th6x5l rw dbqm*5e 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 end ofm2l:a4 hva moghw18dv c l 0ny/.1d0*oebnwg / a thin string in a cv0on11ycv:2 gelww b8ahhmdlmo 0 n./a*4 gd/ircle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angul )0( cmmy:hzhz 34(wbegs*x mkar momentum of a 2.8-kg uniform cylindrical grinding wheel of radib0(z(*cx )msez:g4ym 3h hwkmus 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side5 rfmd42w(r ay, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8(4rda5 yf2mw r–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 sh kypg-dk78;amz+h ojb qhu1 ,;own 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. Iyb1upz;m gajh7o ,hdk-kq;8+f 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 initial r hnq(i 1m3yt6late of 1.0 re1ylt 3q6nhm i(v every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis throug qv km 4fj:c7dyn6:x c4v3 fljm,;noe6h its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after :k4mxjf6jo nmcl ;:7c ,4d6nvq fe 3vythe clay sticks to it?    $rev/s$

  (a) What is the angular mome8g c,kr-r41+xd g,4 eecyr r az+mjdw,ntum 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 mk2g2d9;- urdwxvye4yy+ k/0f d(o rqwomentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto a:e:x z*yh: b78uzv dakn identical disk rotating at anguz*by v:7 :axkzed8h:ular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atobnn+** ac)r/ ldp-fhwjnd2:5wb.m q 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,uqmk7 m+3 myr of a rotating merry-go-round platform of radius 3.0 m and moment of inertr +q3my uk7mm,ia 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 ritb byk1w0uez *98i6x-du n v3otating freely with an angular velocity of 0.8vebiu9kzynd 01* t8 - xb3wi6u $rad/s$ 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 .s h8p nrfe t8v0i.tugt831mhinto 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 carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest intsntvi0 1 h hrm3. p8efu8t.g8teger)$\approx$    $rad/s$ (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 excekx+4y 1 joof w6mjnw2(ss 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 rnc 0(-1d lj9cf dvxh i(h.w3p$ 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 s 0*6e1qhxrvv 1rf o6yplatform, initially at rest, that can rotate freely without friction 1r y1r6 6osv*fxqh0ev. 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 6.5-m diameter merryl ct- ez;lp;e.5chlaw ::hpq 5ud.*ql -go-round turntable that is mounted on frictionazdcct -p5lq: p5.; huel;le*ql:wh . less 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 ro2 n4/+ rhpco+i;pkpp cpe 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’inp/rp+ pko;h24 +c cps center of mass move?

  The Moon orbits the Earth such that the same side always fav00 vihcgmuqc7(:wta()t1t cces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis)wa tq :t(c tcu(0)vvmhig c071 to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates frh:ju;b7r- o jvom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquiresmmvs-r 7 f4:l8bookg; a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the mor fokvl;: s7g 4-bmom8tor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diskgfp k34ya j lodz41.-gs, 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 cons4 1gd3gk4ajy- o fp.lzervation 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 speed of the rea1c0 ey4p am0vfr wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as yncwn m0 )+q3ygreat, 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 th)mwqyy cn+n 03at the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use ener stsj18/x **+dp urdx sbzi)/0ln6rffgy stored in a heavy rotating flywheel. Suppose such a car has p *su6zi f nd /d0+/flxjst *)r8s1bxra total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without 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 illustratmydq:nrn(+g;pbl2:i,b o j9res 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 oqwjy 5 -3lepw2ks*,kpn a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we igtv+ fq4 ljdkj3codb*// ,2ycbnore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then re+*j yldjk4b,cc23tvo/bqd/ f leased. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has rad.;h - cy0y3gemxy:rfhius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which itgmh fr0yxe3:h-y.c;y rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn with ak )toi7 a4mnz n)*uf2t radius The forces acting on the cyclist and cycle are the normal f2*mfz7 kou)t4 nt i)naorce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 oxf, k, 4ojqyh5f length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a xoh5,4 ,jkqfy 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 cylinder and her arms as thin rods, maka7ezrlj6* x9mp*xnf ,ing reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, 9fr jzp*xamn l,x6*7 eand with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrica;7)vg6ozmj ri dsknm,lz.*hb023 s fvl plates, of l zrkh.)0*2vimd sf;msg zvnbj7,o 36mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough (a yp) q j0h1:o)jiyh3xk9ppbtrack of Fig. 8–58. Whxp(p 0)1yok 3)p jiqa:j9 byhhat 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 the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but d4:)bofjv9i)xo a,x drk5f4 xzo not assume $r\ll R$ h =    (R-r)

  The tires of a car mannzxu 632.t hb,jn;soke 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tirs3ho2, n nbtux;z6.nje? $\approx$    $rad/s^2$(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