A bicycle odometer (which measures distance traveled) is a/s11o sio lu 7k/uq(asttached near the wheel hub and is designed for 27-inch wheels. What happens if you use uaq/ s l1i k/7uos1o(sit on a bicycle with 24-inch wheels?

Suppose a disk rotates at co) mqrhd t tj3zd m:;+1u0(rmhjnstant 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 accelerat)(h mj;r3+dt 0u:hmmdt1rzjq ion? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a singlqe6-oq8hn qr- u-dg 4xe value of the angular velocity $\omega$ Explain.

Can a small force ever exert a ,(ej0f v 1 l;eym9lrktgreater 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 difficqnle ;zd :fq;q;1 usg8ult to do a sit-up with your hands behind your head than when your arms are stretched out n8qu1ez;;qqdl; f : gsin front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets fs3 2oecni781er p 9gvat the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear s1 eo8fegirc72pn s93vprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to moov1 b)zd m28run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain wbmod8v)1zm 2o hy this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lxd hn** t/g lmv+u5al7rwe/g*ong, narrow beam?

If the net force on a system is zero, is the j0 sl 6o5vt9 sf-zsj:+ja3nd rmv)9xq net torque also zero? If the net torque zjl s-tvovs rm5s an99fq 03x6):+jjdon a system is zero, is the net force zero?

Two inclines have th ngg/3y)6ai zte same height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of tizy ngta)3g/6 he ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. On6 *u8 2luvudufe sphere has twice the radius and twice the mass of the 82u*v f6uuludother. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start fre vq7v 3moi;z,om rest at the top of an incline. Which reaches the bottoviem oqz,v3;7m 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 momentum9 gsjyo 1m.9gwt *qp-g are conserved. Yet most moving or rotating objects eventual1j.ostmw gq9pgg*9 y -ly slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equay 9vutltlf3gk83.li6gwi* k4 tor, how would this affect the length of tk gt.8 u3t3*gfyv4l il li6kw9he day?

Can the diver of Fig. 8–29 do a somewqh 8 4;g1w;jvs.kv mqrsault without having any initial rotation when she leaves the boa;. jv gkwwqvqm;148shrd?

The moment of inertia of a rotating nik: eu 7h2o2ysolid disk about an axis through its center of 2uieny7k2o : hmass 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 ou xxq/28q;60ap- a+jnrh ynsf ftstretched hand. If you suddenly drop the maqy2 6na rfa qn;/+fp-xj08h sxsses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. h4mi- psg, r7hkr,j8aHowever, one is hollow and the other is solid. Describe an experiment to determiira47-,,hprk j8h sg mne which is which.

The angular velocity of a wheel rotating on a horizontal axle points wtbdq vb x779 pd,2ptjvs*b40cest. 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 pointb7* 7t4d jxp,ctp0 bbd2vqv9 s at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turnts4h82d4;zyv) 5zb 9xrw iir2 -hjdwntable. What happens if 9iyrr44- ns2z25 8hi t bx);wvzdwjdh you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickl.vloy;j 0uann712wrv,vo dl 1y. As he throws the ball, the upper part of his body rotates. If you look quickly you.va 2l7ru,j1v;l 1noo vwyn0d will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conserv vfp i+ydy41gie- 7g/tation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the s yd-1piy gfv i+/47etgecond propeller can operate to keep the helicopter stable.

  Express the following angleybn/tc3q: 0;.k bshpss 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 -h9w6 3wjnx fa7qc.bg coincidence. Calculate, using the information inside the Front Cover,qw.f-ch9n7g jw b6ax3 the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 3k;e dlxpi 3jmbc-jar ff2-ub3 49e a,380,000 km from Earth. The beam diverges at an amk;3abf ce,e-rb dl-94 f j3p xuij23angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender ro smv2v 4 (+)z6rpypbpotate at a rate of 6500 rpm. When the motor is turned off during operation, the p+4m zobpvvyr) 62sp( 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.it. -duyo-yywb)4*7t+ uqsdn4p 2erq If the ball makes 15.0 revolutions, what is its yypo t4-un4 2.uy+ter*q bwqi7dd)-s diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do 19t kb,hxh5e/bihxz; tz ehb hxt5h;9 xbk/,1ihe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 ri cx1 dfa5 iy ib;5*+5jfi(gqwpm. Calculate its angular velocity ini ibfgcjw +5;(5faq5*yixi1d $rad/s$ $\omega$ =    $rad/sec$
(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 4.0 s (Figq7k; y:1rp3v alq56iq,olkph . 8–38). (a) What is i5 ,:hv;lq6q7q3ol1 kppyrkathe linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit arounk-ioe9x 8rh1 opthq3: d the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofz- ( ,lrkqs4okvt(to0 a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 c(m(o-oq3)bg pg y)qenm from the axis of rotation is to experience an acceleratioe ()q(- o3pybgmq)gonn of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to q78+g(b q8)6ystz( cqxbwe em280 rpm in 4.0 qte bmcz6ye8xq)(g+s (w8b7q 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 244v nhnugnls7bj( 43ou co,w$R_1$ 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 aboh3)l12,esg5 d rwslr nl1 sn9ss h;,cvard 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 thc)s51segwr,l lh9shnv r2d; 1s3 ls,nought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. kpe 4woca i:bjo;ovshg1()- ,Through how many revolutions did:j1(o4w;b sa e, ocopvk -hgi) it turn in this time?    $rev$

  An automobile engine slows down from 4ydp** kaj k*qv n.f1h1500 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 highspeed jetmib-yz w3ng -sb;63wns in a whirling “human centrifuge,” which takes 1.0 min to turn tby wzsigmb3n nw6-;- 3hrough 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 frold3 ce l7l 5-*t,xwhkfm 240 rpm to 360 rpm in 6.5 s. How far will a point on,hklfxwc- le*dl t735 the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is run0yrbj;ih s3w53-hv3a :5;lu dqoqfqsning at 850rev/min It turns 1500 revolutions before it comes to a stop 3rdyfqo-qu3aq;s:lh; vw5503hjib s.
(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 ity8fb: 2abd9 zps speed uniformly from 95km/h to 45a9zbp 8dfy:2b 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 65 revolutions as the car reduces its spen9scbjbp4i0 7q4; pu bed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80bc4j;p9npus 0bb7i 4 q 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 a 9,k7fmj /ne.remk,j 8lk, 8rb gviuz-hill. The pedals ro8 k k9fibegr,eum,j ,j.mlv8- kr zn/7tate 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 ew5a9-tfxb 7v bvwb,* mnd of a door 74 cm wide. What is the magnitude of the torque if the forct7*wf ba 5xvbbw9m-, ve 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 axla*lsd53 cub;c exo rzu- 6dc)/e of the wheel shown in Fig. 8–39. Assume that a frictiono/czc* dcbu )3udaxr;l5s-e 6 torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m,) xq;4oni)q nb+xxu,a are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then relex q b4n,xqa+xo)i;)u nased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of -j wiluio1i;6 an engine require tightening to a torque of 38 iowu6i;i1- lj $m \cdot N$ 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 when :3aetleu7u 6v v 5f1bythe axis t1y6a5ub 3feeu7 :l vvof rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Thei kv++t3i;6:wfc q1f czwn7 kc rim and tire have a combined mass of 1.25 zqki c k3wfvc1c:+7;ti nw+6f 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 rod is rotated in a 3 +(- gjqyms*9a xrdptb 6wywmt(xq9o q5(7 smhorizontal circl-5 9r (yj6mgmqqps (yx(tx3mq+ w*adwt s79boe of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular s3wsb9qczpg ug; cm 8)1peed (Fig. 8–42). The frictio1m pgb; cw9)qgczu3s8n 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 poio:du idy- +klh440*kfrck5urnt objects shown in Fig. 8–43 aboukuhr0y+fu4l* dcr: k - k45doit
(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 wtm1 mm(s3 nzx4hb vl(0hose 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 cylindr*w ng46p3ey 5 mdn +pyvor-yeu,w :tx3ical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what y 5-ppotyvx4e6*,3 mr3 uw g+ewndny: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 radze 6 1fnor 5anm--gqj48,rcc pius of 8.50 cm and a mass of 0.580 kg. Ca- 8f r5m,p-oq6c4zrnn jage1clculate
(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, accelerat/ ;xzt wj*qa9sing it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round.ukgbkul8(;bk dp h(4 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 onb8u(k 4uk; . d(hkglpb the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventudm5trd7) lzzg( nuu /0ally brought u7l) g5nmuzrd 0z du(t/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 ball h:v-.b c z7+7 9d uqqgmic2akhau6e:eat 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 th du7q0kj.rx,qldo37 yenk sr d8n, *,hrown solely by the action of the forearm, which rotates about the elbownu r .n7 hyqjl7,x,reds d 0d83*kko,q joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a lo3m gz ;p)ldy,ing thin rod, as shown in Fig. 8–46.;gymzp3 il),d
(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 consis5ypqt4fq0 ;rmb bzt): ts of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four. 7uukl6 ga2yv full turns (revolutions) and ryuu7g akv6.2l eleases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor hi w78ad8 v4l 2mbk8i (fpmq y5l:ekip6as 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 -87 il)wec-,y+p8qrf a3d pzmaviu:ftorque 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 wwweoo: )saig0es( xl;-l yh 4b 1l+kz,ithout slipping down a lane at 3.1shglsw: ei), -yl 0xel+zwo 4b(ko a;3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as:z5h9 22o np8kio lnjf the sum of two terzki5o2o9l 8p:n2njf hms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a rad(.; 32tnu r cie6duiojius of 7.50 m. How much net work is e6t n;r d2iu. uic(j3orequired to accelerate 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 masvve;or)w .qv *s 1.80 kg starts from rest and rolls without slippin.*;r veq)owvvg 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. I9(b.wpezbq2.0 y xfmzt starts to fall and its lower end does not sl(. mpwq.x0z2b feyzb9 ip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-,*tqll.--namzs .f zokg ball rotating on the end of a thin string -m ,*.satqn lol.zfz- in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentulvdb:9 n.kqgu0 kz4+lb ) lc2mm of a 2.8-kg uniform cylindrical grinding wheel of radius cdm9 lkq02ln +) lvu4kz .gbb:18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rqfz0*70 lkpd votating at a rate of 1.3rev/s If he rai* 0q0kfpvdz7l ses his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the onjt .,5:wjfrkly( xeyip 4 l,;qe shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in(wxjk f5t.qi;pylrl,j e4,y: 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 spins3 alvexm1of61(42ts c547 llo hb clr rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rathclal4 641 s7(fo2co 3extl5 m lb1rvse 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 it5dhydp1 (qqg9lf+nsk* ):.rh j:si rotating around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius sn d9kqy.gi5q (t: shl)f+p* i1hjr:d 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 skathxk u6zz d0y-me8v 9n22 b3jhver 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 moggp.dg f(vf ;9-n3duzzxzn 3: mentum 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 an identii++4;ao uj9 ek( ac5b7llxd sqcal disk rotating at angul(lcbl+45u;o sda9k je 7qai+xar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at .;rbkj* y)yqz2.4 $rev/s$ 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 of a rotatusw:jtdrl(w 9 2j1)fmh vt t74ing merry-go-round platform of radius 3.0 m and moment of inr1:4 (dhwjm2u t)9wtj7lvstfertia 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 oh k *)1gknnk :x.e8expg 6smm1f 0x1es1kmgmkhx 8k 6neg n p).:*.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 dw vvu wm9s2;he7arf, losing about half its mass in the process, and winding up with a radius 1.29e7 h mv;vwsu0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in ;yngo azx9bsmr) 74:y 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 p()e6)hnwpqvls m/n ) $ 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, tr+j z8 4v-jci16hbs oghat can rotate freely without friction. The moment of inertia of the person plus tji68r +o-v4sbgz c1jhhe 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 erradv pap- 4.wkkw:6/dge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionlessk-:6 vra 4kwwppar./d 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 rope v1hx n:(op8 m1qoqlq 8lying 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 persmq l xqn1v:881opqh(oon without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces t .;c ;f1ts/0aa 1p2wjpmzusvzhe Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbimvws2u;1as;tz1jc a .f0 p/pztal angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of rm:cdx0exj6 .en a27d1 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 ac6l v/8cew 6q-0upxg z1m (*bgcyn-d c uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the dql0 pc (xc6-1wny e *gz8c6vumbg /-ctorque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks,,pwf4qfp ty h/ i4b5 j+ppmb64:lqle. each of mass 0.050 kg and diameter 0.075 m, joined by aej + ll fbqyp4f 6ih45mpwtqp4 :/,.pb (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed o3av1c,+u)p rs)vmie 1z dk*35 r czucxf the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sd99(/sbi6hpzz5ys aiun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform s9izb 9sp(/ia hsydz65phere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobt5g8l5y 6 3ufwqtey8l ile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has l5 fgy5u38l6 w yt8eqta 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 illustrates xpvy1b m 19 s8ov8si wmr+n;*man $H_2O$ 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 (hoo:6hzuv:r130t: gqkdt eku3x ip) is rolling on 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 lengtnn y4zzawbh1;g*d )7 zh L can pivot freely (i.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. Assume that the force;7 g*n4adz)w1 znzy bh 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 vertic i693mtt qk7 kfx.)zxpally on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against xqfk) timk3pz9 .67xt 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 road is making a vy96 h vj2o r,izq,;oj5rxpo2turn with a radius The forces acting on the cyclist xo5vo;9pjvo6y ,rh22 r qzij,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 int + jsdp-g gdrtd8xco4)6z3,cuo a sling of length 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 after 5.0 s. What is thgd38jo 6gt)dur cp4 czd-s, +xe 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 2f *nypr.w ,w e-tfsd(cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the f-e(f* yrwts nd2wp.,ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assemblsr7) / az8kewny which consists of two cylindrical plates, of m7az8esrw/ ) knass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loopfvgl*/ag9i: b+ 3x ergl-k e 7r9krl9med rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is a3 g+gv9 kl9-9ke/l7*r ferglbxm:ri 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 pv-w.h*4)u xws;; *pkxkw lzcdo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 iouzwxhpdy hnh e- ,u,(/9n.,revolutions as the car reduces its speed uniformly from ,dz(n h, po-9ux.yneuh,hw /i 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s