A bicycle odometer (which measures dk; )+y zm(k/kdi0jtysp r7f*z istance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens ifk) j rk z*fdp y07s/im+k;(ytz you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates;9l uk3cm6exr0fpo 6g at 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? F3fxp6; lego 9 kc0r6muor which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value ov0m*kx 6j4 e rj8l8pkdf the angular velocity $\omega$ Explain.

Can a small force ever exert a great qj)ktdq+n 4xxz.1fh8er 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 head tha4y43mj:n:izx(zks 9ks c k7 cln when your arms are stretched ojznkyc x7ls z kics:49 3km:4(ut in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and t y+yp6wsp.: 5w.;f gb;vrywuzhree at the pedal cranks. In which gear is it harder to pedal, a smalp ;wsf yv+5wu.. :wzgbpyyr ;6l rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to :t-1kc30 r2rz0sq*jqwxfnqv;-i/tmrq p+ip run 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 why this distribution of mass is advantq:vtm irzc 32qpf;r1q0+kp0w-jsr i / -* qtxnageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow belwtf*04.2 ubn*k. pccceh 6ocam?

If the net force on a system is zero, h6y q20ze,h3 c llnbshey(q 2-is the net torque also zero? If the net torque on a system is zero, is the net for(el b0e y2cl nyq6hh-hz3sq2 ,ce zero?

Two inclines have the same height but make different angles with the ian ec 6b5v vocho:7r(v/n6;x bn5gi0 horizontal. The same steel ball is rolled down each inclinvn6/6ao0obix;:grnv 5vne(5h bc7i c e. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) der- 9bfpd dr ,q8w;t:.rl.7*yegrnb+ 2 gdjz 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 ener9ry8z g7d-;lrf.jdr, pb+wd. e*bd 2gqert: n gy at the bottom?

A sphere and a cylinder have the same radius)/j,w3 ihqffq5 lj 70)o vjxsh and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic en h)l5307vqhxfjsj /w ,fi)qojergy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserv-jf7o )/lynel tv e:)eed. Yet most moving or rotating objects event :ey)7ljnlt/ vee of-)ually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how woul0.w +zjd hny3pd this affect the le+wpjd hn3.0 yzngth of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotageaet ud6 w/t)a2b-,ttion when she leavut/g-b 2aaed ),t 6wetes the board?

The moment of inertia of a rotating solid disk about an axis;kt+sbe q; g+y through its center of mas; ty e+b+skgq;s 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 o +8 o t by6o+.wu tqczkt- .1-.xs/ryzolfce9bn a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, -yf z ts9yt z/.wu1 q6+ rk.o8+c- xoblbecot.will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identica*knkc-,0izptqk: +qy4 vvxq;gtvl54aqgd 0; l and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine whgaxcy-:dvkktgl; 4 ,p0nq;0*+q 5v qt qk zi4vich is which.

The angular velocity of a w. 5l zs1ooe.mqheel rotating on a horizontal axle points west. In what direction is the linea 5o1e o.lzsmq.r 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 angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating tur:wf )y5w6 bv4( ep.xa wk-oxtpntable. What hap4oxa)p kwvt 5(:xy w pbf6w.-epens if you walk toward the center?

A shortstop may leap into the air to catchq g-c;c 24o:8iju pzxa a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you wi8xoq-4pic: u;gzja2c ll notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a :s ;hxud61nzfhelicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter s1 zfhu6xs;nd: table.

  Express the following angles in radia(u gxpxzu5ev9 a w1 8/v9bcup:ns: (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 ama2b1flay:* v/;gz v9fw- qycc gr.dop 4to)1blzing coincidence. Calculate, using the information inside the Frovc yfv12yod -o/.*4 ; 9zl)tclp1abgqrg f: wbnt Cover, 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 Moonk*u, xwrnf d.-eu8;a q, 380,000 km from Earth. The beam diverges at an adn;uf-x.a u8 kwq,er*ngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rothg n(.4umy q;mate at a rate of 6500 rpm. When the motor is turned off during opemgm.yq; n 4uh(ration, 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 twxmmcsk0: 7yg(tr4 o (o another child. If the ball makes 15.0 revoluomtcywr74 k0x( ( smg:tions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do t2 l2f fpnx6 *xzob+u-5g nj8vbhexopjzg 5nblvu+262n- fx f 8*b wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2zn bm*wc dt/36500 rpm. Calculate its angular velo63dcnbtm / z*wcity 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 com;54vh 1ba tz ksp/rr2tplete revolution in 4.0 s (Fig. 8–38). (a) Whtz ra2/;r t1psb 5kvh4at is the 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 arou; ah;r pq tyhll0-2k (:ps-iaand the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of ari cldn95;)4wb1 rikj 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 rsr; xfm e9b89 /-aobvgotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,00 brae9vbo-;s m9/x8fg0 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fro2zpab gt6 z;.ax 4kieorh3. z6m 130 rpm to 280 rpm in e2 kx;i.btrp63 a4oz. 6gazzh 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 radiuepkucjyo *oee 6/x; ;( bgqq*g,* 9dsnvf+rv(s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon,*) vfxr//gmcy astronauts 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./cfmr *g)y v/x The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Tji f:32jju/4 h nzk o.a+:bvb n2nn0djhrough how many revolutions 2:junjo3kfhdb+bna 2i4v.jz nnj 0/ :did it turn in this time?    $rev$

  An automobile engine 4d+;os 5aarws slows 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 stresses of flying highspeed jets is g qv3u.wc1d5qv9;pl n a whirling “human centrifuge,” which takes 1.0 min to turn throuqqvvlw3d5;scu.1 p 9ggh 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 t:cax.7)jm yb 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in thim.y:taj7x c)b s time?    m

  A cooling fan is turned off when it is running at 850rev/min c d o5dxh/.34idcs/*sm4nt uzIt turns 1500 revolutions befodcdz. c hd45msxutn/3*o i/s4re 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 revolutionbf5lm3ktz3 +o:qp,l;,yns yw s as the car reduces its speed uniformly from 95km/h to : 3,ptl3kfo;y q,smlzwy b5n+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 makemgp .95xq 4..9v mznmnyjet6b1g lqv( 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a dvv. .n5z 69eybjm4mqpnxt .1(q gmg9liameter 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 htelzoz)ugx+a/7bni 8cq59 a a +f c+hp4nur15er weight on each pedal when climbing a hill. The pedals rotate in a circle of /i5tbnarcaxp+uha gl+c7 o z4ef z591qu+8n)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 ofx6gld6kt -pk fr96xb* a door 74 cm wide. What is the magni6k-9xl gtr*k fpbx66 dtude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the )exn p us675lqaxle of the wheel shown in Fig. 8–39. Assume that a friction torquqpl7 u6nex)s5 e of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attac9 (y(wudcf27k2crq lohed 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 rel q(ko cy7cf 9rw22(uldeased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening td/a 2v p9gul0suvb4/o o a torque of 38p/g l0do/ 4 vb92vuusa $m \cdot N$ 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 1s/yf v f) yvt(1vzppyx12y,l7 a)l *fgc .bsn00.8-kg sphere of radius 0.648 m when the axis of rotation is th, )pv1*.ysl 0 )(f sy7xzvv1 fplb2g /ctyyafnrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. T,8dbj0z tom f9he rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignorzjb,om8t09f d ed (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod j n;tot)h*0 6zx3qkjwis rotated in a horizontal t3xhko6z n); qjwj *t0circle 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’-pkr k0upq 13q)q;yr.jhfo0ws wheel rotating at constant angular speed (Fig. 8–42). Tkqo)-1r y;0k0hqurjf.p qw 3phe friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects ,ai0pze j.y- k0sn wd1/0m6kj t7dtfyshown in Fig. 8–43 abo0,d zt1 k0yj -0dne7jw sk6mifat.p/yut
(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 con4qxk1 g9/mlskd) 4*bt0z2+ vc ypwl bmsists of two oxygen 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 32gz bclu 9h tzl(t9z+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 is the required steady force of each rocket if the satellite is to reach 3239b zc ll+hg(tzuzt92 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a unifo0 2ehz1ibmqu 2rm cylinder with a radius of 8.50 cm and a mass of 0.58ibue0 z2qm2h1 0 kg. 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 to 3 4 kryuzr 8-t3:i0hcr.) 3fscnxa m)7/cgu lhc$rev/s$ 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 /u1p 9qk uau(mgy *5xxand 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 othepa51( kx u*m9gx/uyuqr 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 k9 5eg29cz wabat 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 25kawc9z g9eb 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 bqvkn7bc:myt /z:(:5ybyqm6 v all 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 (c*ccq z c77yxsolely by the action of the forearm, which rotates about the elbow joint under the act xqcz 7cc*(y7cion 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 long thin rod, as shown in Fig. p: +qv9 pq+ogs8–46.:go sq++ qv9pp
(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 ue-)ib49w c.cq1x u lxconsists 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 evu66u3zj9b (kqw- 8r)mnwzdi 5,y fwwithin four full turns (revolutions) and releases it at a spwvd6uun q9m-f),8k 5 zw6 yjbwzeir3(eed 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 mm g4)q.. 7fehtnlgk5j oment 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 a5su1 c.tgc7 bc4k(m ux 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 7c1iw8 mz sjq9+.3 kg and radius 9.0 cm rolls without slipping down a lane m8 +zqj1wc9siat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the :/myz u6,w*ra0s wv p)7(9wmrfift hlEarth with respect to the Sun as the sum of two(7w ht9i s*pmwyfwr06:alurm / )v,fz terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.w:/9ma8ih itmc16 s d:*bryvo50 m. How much net work is required to accelerate it fthm9* :m6riai s v:c/y8odb 1wrom 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 ku 9;h(idu(bsvaco3teo,o. 4d g starts from rest and rolls without slipping d osu a(c9v4e oio;h(d.,d b3tuown 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 i:npdls0lo jv;*gf p(, ts 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 before it hits;df :ll(onj0p*,g spv the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular m.4 liv6ru r *kck(lsl0omentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular spel(6*irc krull04k.s ved of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular 42wt)rs ( nqab +v7(0yiivk vhmomentum of a 2.8-kg uniform cylindrical grinding wheel of radius (t2v 4ya vn 7 b)r0hsqiw+(ivk18 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,raivww i+frmz 3*z2+ 3 on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Figz+r3 iwa*zf 2 ri3vmw+. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inerrwc j 64nl/0octia by a factor of about 3.5 when46cc owrl/j 0n 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 initial rate of 1.0 weijoa6-ie1o36p 1sy( h j)surev every 2.0 s t3jo) -s6ew 1e(1h6po juisyi ao 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 through its cent)anno6 gxtdof15 5f w3er 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 thtn3odawg )ofx5f n 615e frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum jk8n*84vr91e in yps hof a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum o ks4u4*7*mk a/jfzag+h /a wsef 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 idenahsv)7pyb.k 7-/ izzatical disk ra7 h.pva/y)7b ikz-zsotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at wc ::dhyp2b)j2.4 $rev/s$ 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 s qt9h (m/elg l.vj+vb:tands at the center of a rotating merry-go-round platform of radius 3.0 m and momenblvl tjqgv(e+/: 9.mht of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of 0t q sap;mr00a8 c;)wfp.8mwa crtq 8psf; 0;ap)0 $rad/s$ 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 collapsev u x7ey3fja(* 0ii8-h. ppabhm1obk :s into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rub1ah p3x7y0f:o kh-jmae .vb*(pii 8otation 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 wnabqqt8gww kdy:wy ;3 *a 5+5ninds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass -te h xew1d)+ pu1m3px$ 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, init1ksfw. fsm6- mially at rest, that can rotate freely without friction. The moment of inertia of .sk-f1f 6m wsmthe 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 aig,ff- s2f4tz5s)uwps h)3 ddiameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1 sa4iu53tf-ss ,2h zw )d)fgpf700 $kg \cdot m^2$ 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 groulpuger33 fs) 0nd with the end of the rope 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 per )le3p0sr 3gufson 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 src arm3zsf n)i0d7qd 1-xn -;the same side always faces the Earth. Determine the ratio of the Moon’s spin ang1a sfsqd7 mx-0i);n-dc3n rzrular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate7qh80fi d dz2wcm ml79 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 tf 9; 6uzsu6htgvehq2j//uz u/ he 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. Calculate the torque delivered by the motu s/fzzu6h;u2/tv qj9 g/6ueh or.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and y 5)iplw2lem2 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 speed of the yo-yo when it reaches the end of its 1.0-mpw mliy5e2)2l-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 5 -r flgxvs fjas(,x70rear 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 timesj0 movou k :y8y;j2nb6 as great, were rotating at a speed of 1.0 revolution evu o6jb;0yk8v y mjo:2nery 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 mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a lfk4-7svi x n2dow-pollution automobile is for it to use energy stored in a heavy rotats x-24 7ifdknving flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able 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 illustra */gbu6m -r6zwg jnvn79u (ssntes 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 rza +tj fbs* ybgm)ww9x/q *txf5o+.j/olling 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 length L can pivot freely (i.9t3 :pyflq ck u.uq9x2e., 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 of gravity acts at t2u9yql :k uf3qpxtc 9.he center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standingk -(a ex87ktqv vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step agaaev8(k7kxqt -inst 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 turn with a; 2pd.tc+ 6je p5xzigq radius The forces acting on the cyclist and cycle xd5t .cp q2i g+zpj;e6are 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 intqy3(u;h/m zzzs 9dj 3jo a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerat 3zm(9qjz ;yuj/s zh3ding 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 cylinder and her arms as thin rod0drxf2mv-vc ; s, making reasonable estimates for the div 0fx;cdr 2-vmmensions. Then calculate the 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 assembly which consists o/os v(-djp/ tlf two cylindrical plates, of mast /l/(vsj-d ops $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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,cidoxb d.p5je *cot( 0:e9if looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assu:jc5ox, d0*9 ibf.odept cie(me $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not as v7v-u4svpp0iq+ wdbfda 2b6 5hmg1w/sume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speedwia5vbng *nr sf) k9kk 4u0.8p uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular an5 sa vbku8*nkw.) 4p krf0i9gcceleration 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