A bicycle odometer (which measures distance traveled)myo oe( jo npvjc)lt.o fm0;8n l:1f1( is attached near the wheel hub and is;mf )yop08mf e1ltjj oln(:1.co onv( designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constam)-bjs4q hw5j f /uw8h8b3vj1yjxv8 vy h1( nynt 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? Foqv 81snufxm)jw( j81vj5y 4b b3v ywhh-h8y/jr which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value ofve7b9g f 47trnxtb4tsk:*f :pl;a5 bf the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain.x55dtthk;id j30 .czc:cq0 os

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 zxxz 1ps0 yunhrg1 qt2:a46p j y-ze72a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help yor26pya zj7 xez42u-1z x0: ntpqsh1 ygu to answer this.

A 21-speed bicycle has seven sprockets ater(b i (;qd+8ow7r lcg the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprockec ;( l7geqdbir8+o(w rt 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 run fast have slen.rm h9qog/ s6nder lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dyna.nmsorh96/g q mics, explain why this distribution of mass is advantageous.

Why do tightrope walkers fone, fsb9lw2* bgi/r8 cq9p- (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is 6khb o6j*y- ;4 9svudgzbxyz* zero, is the net torque also zero? If the net torque o-b b6z9 x64k;*ujygz vhyosd*n a system is zero, is the net force zero?

Two inclines have the same height but make different anglesw ux -h)u-p2n0os9gmj with the horizontal. Thu9) hu2 n0ojmp -sxg-we same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (froe 6u,*csnml:e+i y x.kq.jc 9mm rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the y*uc,xsm69.k.emie:cn+j q lincline 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 anp 2 gnr 8d55xc ;8*tzakycwdq.d 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 ena8yp . k5cwx 52tqzd8dcr* ;gnergy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserve bce)/uec *b8n gu0/ lqrp,cl)d. Yet most moving or rotating objects eventuallyer) cbl0/ uc)b/ u*clqpge8n, slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wom/( p.tzm yvb,.:p bejk72srnuld this affect the length of the day?skrevm ,y2tj( pm p:.z.n/bb7

Can the diver of Fig. 8–29 do a somersault without having .n;nujfe / /c:iq7v cf lvxb*9any initial rotation whf:qcvx l ni/eu. 79vnbjc;*f /en she leaves the board?

The moment of inertia of a rotating solid disk about an h7vxyr8y55(,pj4n hcwj+fx- rvwh k8axis through its center of mas 5xrwkx4hhr vphcwnvf7 ,yy+j8(8j-5 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 on a rotating stook *sgnbf04p o-l holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular 4bp*ksn0 o-fgvelocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, ojz7ch;6ajt-db.b e4f yjw0:hne is hollow and the other is solid. Describe an experiment to dh bc6yb0:ae4tfzd h-w. ;j j7jetermine which is which.

The angular velocity of a wheel rotatingqzqn jk pb*w 0lz3u3,- on a horizontal axle points west. In what directioz 0*jzubqlw3p-k3,qn n is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turnt:* sg u2zfrvyn uo eyfv82rj+9t,dgee5 1,wz 1able. What happens if you walk toward the enuy 1vj,er 9,yz5+fr2*f2:uegwgvs8z o1tdcenter?

A shortstop may leap into the air to catch a ball and throw it quickly. As 6gz *lgnyvu(hg3r7 rss0 r):d0ti fu;+8qv iw he throws the ball, the upper part of his body rotates. If you look quickly yogl ri6 8sqi gt r7y0:3s+zrd0fvgv wuh* un;()u will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angularsy1kj v)ucsc)34k 2h u momentum, discuss why a helicopter must have more than one rotor (or proj2sc1kv)ukcy)3 shu 4 peller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30.kg4d:qukoua d (5e jp7,m.ic $^{\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 coincidehyvae *n0t/; bqi 2ir9t(uq;knce. Calculate, using the inf qy9thki 2na( qrv*/u0ib;t;eormation inside the Front 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+qp n)x1znt 0u at the Moon, 380,000 km from Earth. The beam diverges at an anu1txz0 )n+nq pgle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned o9 eu 0lm9p g( sjzi:j;9v5szkaffm5v(s 9jup l0:esja9z9zig ;k during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ballww( b ab:/rm8c makes 15.0 revolutions, what is its r m( b8bw:wc/adiameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions dxpdd;gry.y8m v6wm oq2/dz6 9 o tqod mrg.z;x686v wpmyyd/ 92 dhe wheels make?    $rev$

  (a) A grinding wheel 0.35 m :y 9ak)gwv /1r bafif/in diameter rotates at 2500 rpm. Calculate its angular veloc1akw9f:fvi /ayr )/ bgity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38). (a)ahx7i,r l 8pk+2 vzea) What is the linear speed of a child,ikx er+pvh)l az7 28a 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 around thnbj*ox3v k otxv+,x12 t6gll0e Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point hyd x/73ez0msm9f5os
(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 e7az22ok-bri+noi)v w-0be k m from the axis of r-ize7)robb0wo+-n2 i v2ekak otation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its bt(+ xb i6 la0g+6 w4zg11k et7rpkdewcenter from 130 rpm to 280 rpm it+1e d60ai6p xrgetk 1zbw4lb(kw+7 gn 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 radiunq9 c5acefg r bamk1:/,o:;q os $R_1$ 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 space88 )z4vlzs oybcraft 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 minutey8v) b4s8zz lo 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 uniformly from rest to 15,000 gp ysv52-go:oltn q1 )rpm in 220 s. Through how many revolutions did it turn in this t)ly1 g2o:-qg5vopt snime?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 rz 2, mx+rsl2*t3gbyus. 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 ofeb 860dor9md-+47fp 5ftnq *zq f gmpo flying highspeed jets in a whirling “humano0f98f ftnqbrzm em 4gp d*7 dp-+56oq centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpma wd4u3c 6amipjw 8c(0 in 6.5 s. How far will a point on the edge of the wheel ucm4 (wi j30pad8ca6whave traveled in this time?    m

  A cooling fan is turned off when it is running at 8c db0;jdp:m ;a50rev/min It turns 1500 revolutions before it comes to a stojbd; :;0dampcp.
(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 k4sf6nf8 7etwreduces its speed uniformly from 95km/h to 45km/h The tirfn ekf64st7w8 es have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly frvya 4y1, wzlr)om 95km/h w4,y1azlvry)to 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

  A 55-kg person riding a bmr+zp1q l . e6f7ftzh *1v cwfkle69s,ike puts all her weight on each pedal when climbing a hill. The pedals rotate in a cc1 t.flmhee1q9rz * ,wp 76sf6kfvlz+ ircle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is 599sg4;ppbp xw fw -lythe magnitude of the torque if the force issy5pg f w ;b-94w9pplx 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 abonbx t8, )glrv7ut the axle of the wheel shown in Fig. 8–39. Assume that a frict ,t78grlv )nxbion 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 ostwp9;.,gk 8 hjjb z3ps2mi9 of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then relea8;p,o9s j3hg 2.piswbjk9z t msed. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an/px;p/(bhho h 5+bbhv engine require tightening to a torque of 38bv/5b +p( x/phhhho ;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 ofr;:l5 :0 xiyle 4tblkxzq/8jsim+wdf9 w( 6wj radius 0.648 m when the axis of rx0(ewll9 w :5:+f/yrt4bxmzk wisq;ld8 i j j6otation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bic61su txsvmzn g (nq:1,ycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kgqm zv(x:gtu 116s, nns. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the eck q3 vqg2xgd67iz. a ,r+2wfwnd of a thin, light rod is rotated in a horizontal circle of radiusqxraki f,v2d+gzw6 7q 2w.3g c 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 constkgszevuuv+//nn0v8 ;* n s1 aeant angular speed (Fig. 8–42). The friction force between her hands and k/8vuennunase g+ v/zs10;*v 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 ih:wdp ued)6; 4 jv.wxdnertia of the array of point objects shown in Fig. 8–43 about.)dxv 6 ;ed4wjduwh :p
(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 eq zfv rt.s7rr 3b q7a/nl(,0kof 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 spinning at t*(uw;vnj4c h 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 satew4uc* vnth(;j llite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinderm fq6auj*t0jyv :e4r16kbe/ h with a radius of 8.50 cm and a mass of 0.580 :a 0 4me6r u qkjv/*1eyjfht6bkg. 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, zh u2 1-rw,zfraccelerating 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-61fi7if c lo* ,5x wkbogz0iju)cm/r1go-round and is able to accelerate it from rest to ,owib* oicucgf/01)5kzx7i ljrm61f a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,dtt9(4psm 0pwsz m4j 7300 rpm is shut off and is eventually brought uniformly to rest by a frictio9 tdt7szpm 4msw4(pj 0nal 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 acc-.8o 172usdr hun0rq1antz si2g*dwcelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of thjn 3k m1id2.ljc4oye2t7ywth la *,e /e forearm, which rotates about the elbow joint under the action o2j7,.2*1e4ym yji3wlht lacednkt/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 304;wml ywavs whd( ;vconsidered a long thin rod, as shown in Fig. 8–46;w4l0awh(sv;y vmw 3d.
(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 consi8w prlyrr3z;4sts 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 s0e4dyjx 8 :ofz2+d wkx5v1qbfull turns (revolutions) and releases it at a speed0eyxdjbwsfz:4 q18oxk+ dv5 2 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 af khayqs x,100hckc4p b )ur48 moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280mh h+q;b4 m9pr $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of ma*j(c mx dviq; 0pgc,7xus+ti4ss 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.3v7it; 0,x*4+s(cpqid jguc x m $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as themwws++un+ep mjr; 4 e6 sum of two tjer6s+w+ nmwme; p+u4erms,
(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 ne4*a:c4b2 nt v-i)w k: 5-qlwuidjbhxat work is required to accelerate it from rest to a rotation rate of 1.00 rctd 4-i5x)b4q - *aw li::bk2jnwuva hevolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80r6dd9x8./dqd0h qlw l kqc0c; n2b 0py kg starts from rest and rolls without sl lw rp9;06dnhxq qbd2. cl0yqdcd08k/ipping 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 balanxnwr 4 si b1nrx0,.tvlh775mbced vertically on its tip. It starts to fall and its lower end does not slip. What will be the i7 71r vs.l b, tbxhwr4n0n5xmspeed 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-kg ball rotating on the end of a thin st2h wm 77vdaej;ring in a ci2wj e7m;h7dv arcle 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 uni:x9h3 ql01awwt6gt +7sjfhkiform cylindrical grinding wheel of l a167xtgwfw3q: h+0hsjk i 9tradius 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 rotating at a ratwav9mo8 n 692yzuj5qwe of 1.3rn 9vzqyu6ojw928wam5ev/s If he raises 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.rz9d1jzi9y 1 c8bkl f one shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 whebcjiz1. 198dy9fkr z ln 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 bj)k-cq-w :guspc/ q6 an initial rate of 1.0 rev every 2.0 s to a sc) kw:/-j qcq6u b-pgfinal 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 centekqy;qvs ,1 0czq; z(iar at a frequencsa;,; 0 1iqqyzqzv( kcy of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular mome6ns j-njsu ;4+f, 7bcpp)am funtum 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 momentum of the Eh3(-t v,c fl (hcz f68r/movlparth
(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 ot 9fwv2z v)757rspzkw f moment of inertia I is dropped onto an identical dpw9r k)wv2v7zs57f z tisk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 ol. p oaoq3;cgvf 0gngx5t(g9yl f+)1 $rev/s$ 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 cente759 vps4:x2d, gs rnm7a pl f9fxvz8usr of a rotating merry-go-round platform f9 87f:srpv gna7x4dslzs5vp2 ux, 9 mof radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an a5)djm vt+xc .zngular velocity of 0d)zvt m cx.j+5.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing abon nz cvf6(+w.jl:hs(put half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to thnw+( z(v j.nclsfh6:p e 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 uby8o3 sb9p:*bbx s2a fcwqaa5uh67: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 2u/;,jhyfy17pm q(rfg( ptpf $ 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, that can rotate freely withnepj5 u7 1xm8 9nn6 wesmvh99fout friction. The moment of inertia of the person plus the pjwpe9m5u19n 68 fvex 9hmsnn 7latform 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 stands5c pwye;al4ia4*zq2q x 8qmta 9o0sk / at the edge of a 6.5-m diameter merry-go-round turntable that iscewz2lka*atao s;y9m ip x4q/0q84q 5 mounted on frictionless 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 (c/nb.c0uft kr 8at,qthe 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 person ankrf. c8/tc(q,0b ut 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 the Eart 6s1.kd8fsw(;k ru k6bnn9bb nh. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital awk6b(;1.9rb6 sknsbdnu fk8n ngular 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 1 -y9l1 :yfmt nwjt9-unm/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform c:f5l ufa1vweya.av je*j:.1sylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleratio:a.v.fe aujyas*5jf11l : wev n. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrics8e1 3 njyc-ym+1gtox al 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 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 y gcysx1jm3te8+o n1-from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speetw.ct 2ia b+j3d of 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 Sun, but with mass 8.0 times as great, were rota5 ry,sojmr0ur *75,xh2pihbq ting at a speed of 1.0 re 2pb5o xuj7ir,h,mq* 0 y5hsrrvolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-polluix:su gtd-4t4ss1ukh( nvsv775 k 3thtion automobile is for it to use energy stored in tssxg 1n(h k3dts7:sv5ukiutv44 7-ha heavy rotating 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 illustral bq f.cr) 8ek7iowe i+5 24bppves,g:tes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal suru akap:e h:y32m90 toyface 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 : dcc3iu a+cs0M and length 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 of grai3d0 cc+ scau:vity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R.gr: :mo )wjwp*sy(ilz f 7v--r It is standing vertically on the floor, and we want to exert a horiz( rw-*l7vjg)p-:yw imsrf o:zontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist travelink-lp ,zlb v s0x+0fn+lk 8+n)rwl*pzrg with speed v=4.2m/s on a flat road is making a turn with a radius The foln*r,zs-kp 0lz flknx+vw 8p+ +0b )rlrces acting on the cyclist 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 into a sling ohnt6l+m1z kg1 f length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate oh1zlmtk+6 1gn f 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 thinrdgf6o c;im 2bo;md 4wz58e.; teq;nz rods, making reasonable estimates for the dimenet ;ew6d;z .im8b2on c;4o fg;5zrd qmsions. 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 of two cylin7:6 x:)gnhig 56npjz(hdalwndrical plates, of m nxwg7pg(d5 hn 6l jnhi:)z:a6ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 a5/30dhvjm*mkd qz:yk nd radius r rolls along the looped rough track of Fig. 8–58. What is /vm h: d5yzqj 3m0*kkdthe minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not auj 4v9 iot s5g/v8i1abssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly f ) bgs63dml5fpeirx+b5e .qz(rom 90km/h 3e( edm bg)5 5xf6qpb.lirs+ zto 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