A bicycle odometer (which t.b2ixl/tm3q measures distance traveled) is attached near the wheel hub and is designed tbl/m qti2x 3.for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on twyjj6 f30hi *k,sy-p the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does 0,6swt i-hp* y3jyjfk the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described ywrwk*l thz(g 5u6slj /n)( f6by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a l3x3avat0( * c9hs dxaad 8z5vwarger 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 behinszk4qzl36 m-o7ktv t ,d your head than when your arms are stretched out in front of you? A diagram may help yo7zo tl4-,k6 kt q3smzvu to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pcios- c2- bix2vzp/s 8 l/r0ocedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder i0vo o8c-b-pc/xlrsz 2 /2sicto pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower jvxx1uxkldxw- )53+ vlegs with flesh and musx)u+ jl1 xwv5vx3dkx-cle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long,r p.9(atf65 wmhywjvmsm-+ g, narrow beam?

If the net force on a system is zero, is the65 k:i: ylinpe5 nd8y8zjrg3e net torque also zero? If the net torque on a sysdje pni ryyei :nlg:8z86 35k5tem is zero, is the net force zero?

Two inclines have the same height but make different a ai+lh/ci7 uz7,oul1hp3bjd 6 ngles with the horizontal. The same steel ball is rolled down each incline. On whichi7zu ,+ u7/olhi 3abl1 6jpcdh incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously star x02zatj:dr w8t rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the02rj8 dzwx:ta 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 samer:pk1;w p *.( idwwwm5yvsb)c radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greap.s:y 1cwdm5) ; wip(rw*vkbwter speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum+ dgrs98u)lre -c:iu t and angular momentum are conserved. Yet most moving or rotating r s er-:9l8u)ic+ut dgobjects eventually slow down and stop. Explain.

If there were a great migry,cq.58 ph z8x+hnkl6apj uoecx1 19tation of people toward the Earth’s equator, how would thc8. ctyap,9ehu5+kopq181 xxjzh l 6nis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault withojr 0el m5pt6ct y3 i6q/c+o1jok.bsg mr 2t-5kut having any initial rotation when she leaves the board5j6l mperoc+02y51cism-6g3 q/ktkbjt r .o t?

The moment of inertia of a rotathl-f* k6dvl pg)/p: q;bas vh,ing solid disk about an axis through its center of mass gl/;hvlbfq-p p* dk )a,:sv6his $\frac{1}{2}WR^2$ (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 stolzsy 80j))irsd . rqb2ol holding a 2-kg mass in each outstretched hand. If you suddenly drozrsy q)02dbs8.j ri)lp the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and9*pyv6 soea;1. mhhut have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine woa6m.* sth9euvph y1;hich is which.

The angular velocity of a whereg nao//pb;t nzhs3 9 iha/j8gajq160e ti0 3el rotating on a horizontal axle points west. In what direction is the linear velo6n;a8rb t3g0n/p es jaiz0tiogh1he/3qja / 9city 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 freelsu h -6.jl:ux wpvc3x1y rotating turntable. What happens if you 3cup s 1jxvux6.w-lh:walk toward the center?

A shortstop may leap into the air to catch a ball andgw)lbxe.*yyfa n t0 r5j8;1i g throw it quickly. As he throwsyle;gbj8ay5 t0)x1fr*n.g w i the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the u tb r:a445/h hyip3vz4jbdm( law of conservation of angular momentum, discuss why a helicopter3 ta( 4i ph4z5/ryubvbh:jmd 4 must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (g h+tt(/e a uvd14j7lua) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calbnf0s/ owh+y+ culate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on f+/hw0bs n+yoEarth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 3ubc54zyb82v b80,000 km from Earth. The beam diverges aucb2v bb8 4yz5t 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 3o08rhw vsq:h6lm-o g rotate at a rate of 6500 rpm. When the motor is turned off during operationml083-sgv:ho6 oqrwh , the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child.aqrar)o6keo)z t jit)/gb3m6 :c( uk1 If the ball makes 15.0 revolutions(or t 6r:uzco)taj 1gqak b3)6imek/), what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolw . sqv0:5u7uauqy4 ceutions do the wheels m0qs:5 veuu ca7.q 4uywake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotaa(.jjyyv)5akq0q j+p tes at 2500 rpm. Calculate its angular velpvj. y+aqa)q (y50jkjocity 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 o1v.2bgdxd4 l nne complete revolution in 4.0 s (Fig. 8–38). (a) What is2.b 4vlgnd1xd 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 itssk e0ryb(*zr- orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ov klh8g;je0 -tj6+u w1cplmk 3f 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 a0qd ,bq5riv ag0b:*fa centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,00dag,fia0 qbq b*r0v 5:0 $g’s$?    $rpm$

  A 70-cm-diameter wheel a bcj)hmx)r 5 1uxmck-e3m/ddcsx03a5 ccelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 xmrm)3cbdc)/kd -55 j013s xahumec xs. 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 radiuj :zvtk/ty/*h 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, astronauts aboard the Apollo spacecraft putqv. ilg)jw:n 0 themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotagji.n)vw 0ql: tion 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. 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 unifori rijaq)k3) g 0f4jbi)mly from rest to 15,000 rpm in 220 s. Through how ma3)q ijribg))k4f a 0jiny revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculrqrj6idu.cu /g/)vr ,ate
(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 hs25j 1rpze +dv0t08hpxexgrg g*u 0)stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 shgd * 0e x)rggp1hjzpt 0e2+x5v 08rucomplete 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 360 rpmt a3vx/yb q x/5;9bgdq in 6.5 s. How far will a point on the edge of the wheel have traveled in this timgb9vyxt;5 qa 3bqd// xe?    m

  A cooling fan is turned off when it is runltx; g+rfz)1pbl( d m59 h3 ui6 ckpsw7bk+;urning at 850rev/min It turns 1500 revolutions betdru7sbb+lz9 6 r ;xwc )pp5i+(1;l mghf3kkufore 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 makp)e0;oi; 7g nkhe rm)+wtnb8de 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The t)b7onn;i0) h km w;p8r+edget ires 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 spee*;ft9nqv e-qnbg* myi/u2 o+jd uniformly from 95km/h to 45km/h The tires hatuyn-9qim*ojfe 2vqgb+;n */ve a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each 2bhi. c ;emuwj e837uwpedal when climbing a hill. The pedals rotate in a circle of radim23ebce8uw whj 7. u;ius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door qyg p0rr73z (r3m0cek 74 cm wide. What is the magnitude of the torque if the r 3m eryp kz0qr3(g07cforce is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. A2n ly u+k-x.k)nj:-ai 5upav fssume that a frictionf- :aj+p) vn5.ukiu2 ayk x-ln torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod which pibq7s8bt,7rc4y)7y dp gezk*d vots as shown pbd4 y,7 )sbgk8 z q*etycr7d7in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine req i0w3-1wep0jh bu(ua cuire tightening to a torque of 38 cw pw0au-1je0ih3(u b $m \cdot N$ 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 the x b760md rfvqu-c9n 7taxis of rotation is through its cec db7n0t-uxm f67v r9qnter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Th3eq ;0e-9odydr 0rrxme rim and tire have a combined mass of 1.25 kg. x drr90;mey0r qd-oe3The 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 horizoutb;1x9.qr i ontal circle of radiuso .btuiqx9 r;1 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 ccr; h :w2xef5n+jt5kv onstant angular speed (Fig. 8–42). The friction force we;+xk tf 2r5cn5h:vjbetween 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:fbv 2 um(g;fh3bk3bo inertia of the array of point objects shown in Fig. 8–43 about2 kbgb;vhu:fmob f(3 3
(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 jig*6k 2tf cm(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 uaqygjd+ z3-( satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. j+g(y uq az-3d8–44. 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 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of yj ,arbdeuoi ld( w .801hvc7r3yz.v8 8.50 cm and a mass of 0.580 y(i,e 80w. j38vb.oadly1v7 hcdruzr 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 swingsl q)4w(3vmo5j wvm9 hm a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-drivena6,5 ae lc1df;tbinsd;xlg; - 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- ;-;elsc d,1fgtal5;dbn x6ai 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 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 :d34v kmj9l, l.w7 6safokmjz is shut off and is eventually brought uniformly to rest by a frictionk96d47m fl.v,mal:w3zkjojs al 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 boemki9 wm(f2 h*if59 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 isnh81,jsjqc (iu,,a) cgkico 4 thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is ai(co4h1qj,ickagns) j ,,u8 c ccelerated 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 c cmp +wq(b2t6cke/bm,onsidered a long thin rod, as shown in Fig. 8–4ewm q tpbbk(cc m,2+6/6.
(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 czs (v+vbl 5(le5 qa ;n;okgcq8re8/qi onsists 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 withrh* n k8kz0s4kin four full turns (revolutions) and k4rnz0*k k8sh releases 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 has a momqb+ k,t 0 m8g1 (tgxo8;erwtys)i t;wxent 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 r(k6 m7 2a2(yc zgs oruiqfinc3ui ,f (k63yw(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 e + bfm np4a0yvla;6w6kg and radius 9.0 cm rolls without slipping down a lane atv ma 0lpa nwfyb4+66e; 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun asjf3m)a s. 7.gqtho (hvn qr51f the sum of twtq gh53(v smfqo n .r1a7)j.hfo 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 7.50 m. How much net wor8 t ryln4y-zk1k is requiredzn81r yl t4ky- 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 mass 1.80 kg starts +reu 7 rmef(+qfrom rest and rolls without slipping down +m7qufe+e( rr 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. It starts to fall a q27. z;h7w zjbm;t/ j6 -f1xketfewsend its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation ofe jhwe/;w;q1zt76 7mzx.f sb2tj -fke energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thi/a/eue sq61xc n strixe/q 1asu6ce /ng 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 momentum of a 2.8-kg uniform cyl2as7h:*0kie w2ns qg s5lq3hqindrical grinding wheel oqq s2h2s 5in 3kqgh*:sa7ew0 lf radius 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 i3jn(vn mzp-fk-uzzr w99)p1 js rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation-np(f93zzzj)unkmp1v rj-w9 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) camnqb: a ider870*b3htui:t*/kyi. az n 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 1.5 s when in the tuck position, what is her anguliue7mr/:akad0b 3**y t 8 ii: hntqzb.ar speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her jrrzf2 z ajtx 7g7m).4spin rotation rate from an initial rate of 1.0 rev every z.trzx 27)fr7jgm aj 42.0 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 through its center atq61llyu9g .-gv a*kxr 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 thr*g ay16qu-r.vx 9kllg ows 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 the clay sticks to it?    $rev/s$

  (a) What is the angular / **tox0z)hvblf; s9 7vkzxxc momentum 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 momentuh7dz u/8jd f2ym 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 is3cqz*sn7(z ppac7 i v2 dropped onto an identical disk rotating at angup *as v7 7qic(c3pz2nzlar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at .ilbw2v.yy5 5lli5r,v uupx3 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 o xdbg 7 59q*xnto56av d*y7tkgf a rotating merry-go-round platform of radius 3.0 m and kgx** n 5d gbvt9y57doxa76tq moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating fre tx0 8oqrrew.kd7dds)5.si5 v:nnmmjt/ :b4 oely with an angular velocity of 0.4.r70ow 5 5mn t:.qs)idoxmdk /des njtr:bv8 8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losbwlqo3./t6unh*x o z9,non8h ing 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 momentumonu./n o 9 8h,*xqn6zl o3hbwt, 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 c m /kdfgf7rf b9o:u6dp(0zt9 winds 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 s5k5 vpniq ry)h8j-e 2 a:26evu:rocu;x mi x5$ 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 platforukmofl9,h9j 5 b ps 7xwcez 93+z*ny+zm, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platformz9*9y+ w b,35koh s +mf u7xceplz9njz 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 standksw2.e gphq9 ei3(mo1if3 vt(s at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of h. 3wvi (1 tk3(qimeg9ps o2ef1700 $kg \cdot m^2$ 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 cn i66mvh+jol8os+:v the rope lying on the top edge of the spool. A person grabs the end of t+:s 8jon6 l c6miov+vhhe rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always facesp go-h 3krsdzs92- n8i the Earth. Determinz --dhoi2 k3pn g8r9sse the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerate6 xpyt.egf ,hs 3e,ug.s from 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 xy 4q2j qkmbdy+ e+2/mc0lxy(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 th lx +0q+ymy42/2cmydkxe(bjqe torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylind*fo9sy t9d*j7q aka/5 *g ;sxtsy*sx crical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use cooky;q**9fg/c7xss9 ayst* t s jdax 5*nservation 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,gzoe tupon;9 */ggh+ 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 Sun, but with mass 8.0 times as great,g s3d3+r gi1bu were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of isd3g ugr1 3b+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 r5qvzl*2s9 smlow-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses2z*5s 9 qvlmrs 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 a) y6v- sx8noxi3ux+j ln $H_2O$ 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 xppfdk5m27ie l5ppz8/ 5mv -rrolling 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 9ga:) n +df ie ixset7opm0u0jy5e6p2L 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 them)yas fo2 g+5up 0t0e9n6xii7jpe:ed 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 radius R. It is standing vertically on thebla:kb*2 )sye floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has heig bykb2ale:s) *ht h, where h

  A bicyclist traveling with spe.jqh-du qd(nim8; 8pued v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are u.i8 (dudn8;- jqqhmpthe 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 me4q *bk( /h-ipy/gyo 0.50-kg rock into 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 rp *yo-4 ph//mgqe(bk iym 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:r 1i(fvabh3f fj)ud + body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms hr 3dfui+(bj)f a:f1 vheld tightly against her body.   

  You are designing a clutch assembly which consists of two cylindric,qv aws2diu p).q ah0(al plates, of)2q,s 0a(uqp w.dhv ia mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 uo*,mxic+5rf track of Fig. 8–58. What is the minimum value of the vertical height h that the mirfmu5cx* +o,arble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not 5f h gop4jl1 5w(loelb))w c-bassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as tmxmhe4n2.z h r1i n6f elw;,pg*w i3)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 tef iwmn3*hgi6zlp;xwnr 4em,)2h .1tire? $\approx$    $rad/s^2$(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