A bicycle odometer (which measures distance traveled) is attached near dw5.rf8uszz 9the wheel hub and is designed fors5dufz. 8r9 zw 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant anguly1gqx7re /wz xdbuf u.0u1/ y/ar velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component yzb e.1u/7x fw d0gru1/yq/uxof linear acceleration change?

Could a nonrigid body be described by a ;,fcwwu k*r q5u,j*d)(hc rzasingle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a la1ulsyv703 fy zv(lg3 vrger 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 head0+6 acbmilquaqk2) n( *:1izw0 ob n/uekv7bm than when your arms 2 1l +avekk0q/n*u(u:a)oii7 c 0mz6bmbnb w qare stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven;essr r0l y84q sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sl804 rsey;sqrprocket 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 rg5 wscl:hy)6 yun fast have slender lower legs with flesh and muscle concentrated high, close to the body hy:swcgl56 y)(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, kfs p.;d9i9k6s g(h hinarrow beam?

If the net force on a system is zero, is the ne*1gxud fef7/ / msw4mkt torque also zero? If the net torque on a system i*m7eg x fwfmu/k 1/ds4s zero, is the net force zero?

Two inclines have the same height v5stzlzk4 ccqm7i42 + but make different angles with the horizontal. The same steel ball is rolled dowmqc+v4 7s l2cz k4z5itn each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inadjr;;bn ph u;f6ev8) cline. 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 v6u;f nah prde;j;8)b energy at the bottom?

A sphere and a cylinder have the same radius and the same mas:xbgak-dwfc mfh8pg99l6y(d 6cm n7 +s. 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 energy at the bottom? Which has the greater rot( lx9m7hdkwm+d68g9c:6fy fanc bp-gational KE?

We claim that momentum and angular momentum are c,ptf9*- c d;bvsiqof :onserved. Yet most moving or rotating objects eventually slow down and stf9foc -svb :;t,q d*ipop. Explain.

If there were a great migration of people toward the Earth’s equator :y,.ywi 1dyyc, how would t,: wy .yy1ycidhis affect the length of the day?

Can the diver of Fig. 8–co16b0h uuj)k td u1i529 do a somersault without having any initial rotation when shukb1 j )tidh6uu1c 05oe leaves the board?

The moment of inertia of a rotating solid disk about an axism48 lq49r+mfoj4d 5bth-bjy u fju 46z through its center of mass+j4u44j8q z db5jr9ol hyf um 6ft4bm- is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2lh o u)y6 *aiykc0)q6o-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay th)0qc) k iy6ol* uyh6oae same? Explain.

Two spheres look identical and have the same mass. However, one is hollo v7vsyeu in,0:w and the other is solid. Describe an experiment to determine whichvey: ns0,7viu is which.

The angular velocity of a wheel rotating on a horizontalhp4xxux1n x r,a433 in axle points west. In what direction isx hxi4p,14 ruaxn3n3x 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 standi m wro98(8b)kz5c cgfdng on the edge of a large freely rotating turntable. What happens if yowb ozg9) 8k cfc(rm5d8u walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As ha3n,,yke yz aufs01.u ;(uygf e throws the ball, the upper part of his body rotates. If you look quaa1f0,; ue.u ,ukgfsn yyy3(zickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the lafc(v++h :iqi3ijb;m,+6edkdoxfpb 2 w of conservation of angular momentum, discuss why a helicopifqdcm+b 3, 2jio+:6i+vh; kp (xedf bter 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 rus5: pr;h/2:d3sht8hprqrba ce o1u9 adians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculatepoy91*nqpdzc n 7.cuw;3p5y ws df .0o, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earthfq 901. yduow3;p7no*swz .ccp pyd 5n.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directedgd h k-0izs9d:ymxz5/ at the Moon, 380,000 km from Earth. The beam diverges at an angle gi9 zd- m d:x0yh5zsk/$\theta$ (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 af 4a2l rk(sl6yt a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is th lal2fk6 y(r4se 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. ; 8)o8ye gbtaauhp)s 6fr+oz ou9.x:nIf the ball makes 15.0 revolutions,spbt 9oahea6uf) :+uooxr gyn;8 8z). what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels9uj0 mg ag/; i rtmg5,2bfke:f 8.0 km. How many revolutions do the wgtfr 0b/2e:m ,ug9mijf5 ;gakheels make?    $rev$

  (a) A grinding wheel 0.35 m in diamet fj 05.:qhbz,t/ fzuhscw93dwer rotates at 2500 rpm. Calculate its angular f dwq zhj 50.ws,h:fb9uz 3tc/velocity 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-rwkt 2fuidd(1f30zt:u ound makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the ce ddt : 20u3fw(1kuzftinter?    $m/s$
(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 aro j/ g(m4y d5r0ntu(wdpel2s7fund the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point jyig2 pw)+f (my,*imt
(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 uft*(k).hbdn v* jh(cif a particle 7.0 cm from the axis of rotation is to experiench tdh(kb* .uj fv(n)*ce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centn) h-bow8axef 5a2r1i3bs f+rer from 130 rpm to 280 2)+aar1s 8oe 3bfr5-bi fnw hxrpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius y79k 0v zjyi)o$R_1$ 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 th- c*kanx*n9,t;qzlsdu* by:me Apollo spacecraft put themselves into a slow rotation to distribute the *;*znkmnbu dc9ayl s :-q,x t*Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. 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 15,0;k)lr hi ge0363o w87s/swes gbuc )zf00 rpm in 220 s. Through how many revolutions did it turn in this time? hb 0 r 3owkc7)fug6sezsg3 ;ew/8i)ls   $rev$

  An automobile engine slows down from 45lup .j,pu7dw5)5diy h 00 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 fl -apgs p9kl+.2iy bfj-ying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions bgs i-9ay2fbl+j kp-p.efore 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 unifor9csyfkf2 ed28 di,t 0gmly from 240 rpm to 360 rpm in 6.5 s. How far will a point onk9dgydie0ts2cf 2f8 , the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when itofu8roc/r; 0u: w y .ny)jum+n is running at 850rev/min It turns 1500 revolutions before it comes too .wu)/:nrju 0m+c;nf ryuyo8 a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniform8 9ki3)ugzkc 7j3vm6n 0qwk wjly from 95km/h to 45jvk3wkc z9 7ng)8juqw0i6k m 3km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformlytr53/hbsb f5e-swjb )e q1hv1 froje515-sfber ) 3th/ b1 bvswqhm 95km/h 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 bike puts all her weigm*z/ :nga6iic1 s)c nfexa6;aht on each pedal when climbing a hill. The pedals rotate in a circle of radius 1g;nf)s e6xaicmn / zc6 a*1i:a7 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 wi4o0zug8f6. h a vkw:obpjos52pp0ja0de. What is the magnitude o ajbswzg k8vhop j6p2fo0:.u 0 4a5op0f 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 axle of the wheel shown in Figttfdm5pvw g( 07 zxz... 8–39. Assume that a friction tor(ptz0zt mvgdxwf.. 7 5que of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attaqzu: 0m/r;w;4mse ndx 8 f45c.n txmwmched 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 releasqmze /;n f5xt wmu4rwmnd:m48; cs.x 0ed. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head ob.-i ,ij0g ahxf an engine require tightening to a torque of 38 j.x ba h,ig-i0$m \cdot N$ 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 ui 08es6du hvr0 3h9bc0.648 m when the axis of rotation is through its centercsr dh3u ehu6iv 0b809.    $kg \cdot m^2$

  Calculate the moment of inertia iyny9a7,+wnxof a bicycle wheel 66.7 cm in diameter. The rim and n9na iyy7 w,x+tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod blyb ,wd0u:*sdcl5myq 1/j( nis rotated in a horizontal circle of 5ljydb(q0y bc1ml*d/n sw ,u: radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular sv)a3xg19/j c+w8o umlo;bp hz-zx 9hcpeed (Fig. 8–42). The friction force between her hands and the clayo/ vb+lg-m)zhcz9 pu3xj 81wach9;xo 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 +3zk 4fkwwu*jof inertia of the array of point objects shown in Fig. 8–43 aww+zkf4ku* j3 bout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total my44)-z4auw eri mu.p2nhm hmm i0b6c(ass 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 cydj p ./xrymd bf*:7k5llindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a masfl/dm5 yxpkd.7br *:js 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 8.50 cm and a mass of0ms5 mcng ba vj5buu1r/qq-g94 f-n k: 0.580 kg. Calculat0qg mg--n9 n5s /u :5km qfur1vjabb4ce
(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 tofzt q-a8q .5xt 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 haq*tm2t q)9vvd nd-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mqd vvq*)t9mt2 ass 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 is shut off and is eventually brykqp31/k-q uxought qp3qk/u-xyk 1uniformly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball ax x4v 0nb09wtyb,om.et 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 the forea2damq;ay.ljd w 6lw .1rm, which rotates about the elbow joint under the action of the tri 6daq. ;1djwlwaly.m2ceps 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 ciildt p(a9.5 fan be considered a long thin rod, as shown in Fig. 8–46.a9di 5lpt.i( f
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two mxt x:x 1otk8ivxfryv:l:f .p4f5-u0iasses, $m_1$ 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 acceleh c5/7muf y+2fqgks :zrates the hammer from rest within four full turns (revolutions) and relg7fz2f+k cyhus: q /m5eases 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 moment of inerlk,xao 2)v6 mqtia 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 torqd3 xb(zl:8h3p: g3edhlw3 nh zdd8t.9x a pc8oue of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls withoutq1b0ih33 j(8lc(qe)meq f 5pxm pk-lw slipping down a lane at 3.3 1m p80cq )jk(qihpb xel 53(3f-wlmq e$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to thf f5t9+o 8vjpqxejw 0pt56( cre Sun as the sum of two term5pfj(t5wtjr 8 xcq0o +e9pf6v s,
(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 masj;5 jlo68 losfs of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.0joo6ljsfl;5 8 0 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest 97etaud*bw pt5o3c+mand rolls without slipping down a 30d3tm eco9b*apu+7 tw5.0 $^{\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 8aiz y *up7 , xwirw1/(ypye-j ejz3g+to fall and its lower end does not slip. What will be the speed of the upper end of the pojp + y -xyie*1z,y/8 pgz i37ewjwra(ule just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum omj q*bcf2+smy b ))u2if a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m au* m+2)jqymf ci) 2sbbt an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindricj xws.j4yz 88u pe4j2jc+jygtv6 +z4hal grinding wheel of radius 18 cm when rotating at 1500j6zwc8e4.uy2 +8pv h+jjs4jz gt jx4y 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 sidwml.4 u*ne/yzxb 9v x.e, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed o/v wl.m u*x.4e ybn9zxf 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 Fa (oy q++x9vdhig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If (ao 9+vqy d+xhshe 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 siou-goc -6s 9cpin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final 9i-- ougs6cc orate 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 cente5 /ggx:pps .egr 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 5:e/ gp psg.gxof the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum o5mvr;dljq9; hf 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 the6 ; 0jike nordgm,:k2e 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 dropph. y.n 3:z(sp)sjfqpged onto an identical disk rotating at angulaj)p. g3f:y.q(h nzsspr speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.5z)9.+iyepbtxj.n vy u:+ mw u4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of azxen c sj,o., p,a ,ln:v*6qgo rotating merry-go-round platform of radius 3.0 m and moment of inertia 92 ng:q,pz6l x,nso,,v eco*aj.0 $kg \cdot m^2$ 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,pm r(3s+mo 5ahral4cwogm, *ely with an angular velocity of 0.sam4( pr+,* 3a mohgc orm,l5w8 $rad/s$ 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 about half its 3ld(ytj ya68r +qx;ilakn( 4gmass in the process, and winding up with a radius 1.0% of its existt; 38jl y64(rng ialy+ qadxk(ing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 120kf rypvem s)ty;,diw m/ w6g.it 6g,8. $km/h$ 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 /ia7yiihhed (yal6f7wj1x:9 $ 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 rotatero+:h.wm moe- freely without friction. The moment of inertia of the person plus the platform wh m+o.:m e-rois $I_P$ 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 stanx p6hvd2g k0w/ ck.p3*jedaml6 pdc-i+fr+h -ds at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of i+e3-ad2*pxmlw 6k -d/dj+kvh6ircp f cp0.hg nertia 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 en ls 7/vu+obvm+d of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a di7 uovb+vsl m+/stance 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 faces the ui5dl8 z:)ixih, (i2fwje-fo Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Mooh(wflif, izoxju -i id)e85:2 n as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerateshz29nh07 ksob 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 a uniform cylinder of radius 0.20 m acquirhf , 0 hl.)skx3pham(co 5asza*)d, o3:jrerzes a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate thaaos r,fh(c:hzh5,)j s .ar3)k*pe z 0xd3olme torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eac/ w1eqt5zj1i ya(.4x aplqujap.:4l t h of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mas jyw 4qau qt:ait pzl1je1/.p 4(xa5.ls 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular zj3s ) uuj6-zh91rvlm( o:rqd speed 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 m1hp -hxa7ch e5ass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed xp a1e7chh-5h 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 /a2ofmlq n2x/;f.f 9 lc ;ju1fdkin/ nlow-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total masaq9 /xiu;2n2dcjn f ;mfk1fnf/o ll/ .s 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 illustram4uno7zygds *b +ppg, dzr5+ +udj1eici,6 d0 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 surface at speed v=3.3f+6hgm jl6w sgb8 1j3p $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., wgv/e;de sb (ghe/x2 )ne 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 acceleree;g2g/xve)n ( hsb /dation 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 rbxy-u b f/sc06(tux+dadius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axly/+xdbf0-6 x tsu bcu(e so that it will climb a step against 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 t qly crq:-h64nurn with a radius The forces acting on the cyclist and cycle are the normallh4q6qr: -nyc 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-h 08idjc kdwl8 5qh7):f0co hhkg rock into a sling of length 1.5 m and begins whirling t8ih8hl wccfd07 :5h)hqk 0odjhe rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is 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;jn8f+ m 9a3zg,enft i solid cylinder and her arms as thin rods, making reasonabl nf9 jmez,+n; 3fa8tgie estimates for the dimensions. 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 ass+0m7xl 2qehpm20 uuv: fl3uv.*i qwjz embly which consists of two cylindrical plates, of mass 0+lpv.mw 2m2 uqu3h7f*vju x :qe0ilz$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 raj nsjm*ws854e-g (aeydius r rolls along the 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 s-y48ej m( j*s5awegnof the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ass;-vpajvyvq -7c j:h.zimb4fo.+h7 v bume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car redy7uceu3e e /(b+ ct1jfyql5r,h:wuf * uces its speed uniformly from 90k cf+/e3yw,:uyu57hf 1q j*l(tec reubm/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s