A bicycle odometer (which measures distance travelh710rd o ox(m;jd ii:wed) is attached near the wheel hub and is designed for 27-inch wheels. What happens if h i:j7 ( od10mr;wdixoyou use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on thec84ddbv6ax 2 y 9iawu/ rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential accelew4v/9 6u 2x 8daydabciration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular*;qpj jmfz ug8pc abv4x r((xbs7i**4 velocity $\omega$ Explain.

Can a small force ever exert ap,-;bmoo g-ph greater torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sbs.nrxfs 8 51u3swrso o)rn ;qee- 8pk8 a4b5sit-up with your hands behind your head than when your arms are stre1e5xqw8 s4nka;o 3)o8s8urepbs bs5r -. rn sftched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rearr- 7 o+jza mi uk-;ih4g5m,drr wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? I rk iurd,5j gz-;7 4rh-io+mamn which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on beinvg-4x -zx37bq rp4tp rg able to run fast have slender lower legs with flesh and muscle concentrated 4rx - v3bp7qpxz-grt4 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 (Figchl * (dno-5sb(dke b). 8–35) carry a long, narrow beam?

If the net force on a system i l5il:x7yw*, cogdy0g s zero, is the net torque also zero? If the net torque on ag: g xdocy75w0 i*lly, system is zero, is the net force zero?

Two inclines have the same height but make different av3+byjsp 1 nfe; 3z5o:e8 ehxwngles with the horizontal. The same steel ball ispx: y ;of8+bejh ne33zvwe1s5 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 (froma,0 v bs)gsg6b)vayrldrwo/fa4 e*c ;2q5r0w rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed theres0a;l*6gs2/rfr5cbq a ) ,bwdo0 gv4yw e)arv? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have th*zqif 5f 5cb;de same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total q5 dcib;z*ff 5kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most movingi5f jvlzbn823(yy 3d d or rotating objects eventually slow down and stop. Explaiidd3 fzb2y( nvjy 53l8n.

If there were a great migration of people toward the Earth’s equator, how fwlqfe: r 3 fa/zvt,q+ezzt; 531t d+awould this affect the length otf /wretaz5qf:1 qe3 ,3fdza+lv;z+ tf the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatk 6 b,4qiiaua u(clh);ion whiua u(ab)l6; cki ,hq4en she leaves the board?

The moment of inertia of a rot1vd / dy1idl(yating solid disk about an axis through its center of l/id1(1dydv ymass 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 rot*53gb3wxaq /jm o xl(lating stool holding a 2-kg mass in each outstretched hand. If you suddenly droplbl a( xjxo 3gw5/m*3q the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the ypl.y1x m()(bqb1iyi same mass. However, one is hollow and the other is solid. Describe an ey1(.1 yblbm( yixq )ipxperiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In /ork5,go; rn 6vbk9bh:g x w1lwhat direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this pg; 6r/:1xh gkn,vbr 95kloow boint 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 turntable a;-k9iasshz ,. What happens if you walk toward the cente a,a 9sihz-k;sr?

A shortstop may leap into the air to catch a ball and 6ka*h kx.i2g8b7qs j othrow it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that h ob*jk 8i26.gsxak 7qhis hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the b/ewx0*;vgc8 i t0zex5,pifh law of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller 0cefi*xg e5t px0/8;i w,bzvh). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: +g7hmz5h xg3t8u1tb m (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 Ea8 nu*e,ly: penfj;*xrrth because of an amazing coincidence. Calculate, using the informatnuf*,n e;j*l:r8 ypex ion 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 directed79;arl 0 z/hb7yc kzt8hznc ou6llrk1/.7krb at the Moon, 380,000 km from Earth. The beam diverges at an anglhn 9 /kbr1l;/t6.k0orcka hl8ruc 7lz 7b7zyze $\theta$ (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 j.g:y-jyd uk/zny 7z3 off durin j dkjuzz/yyy-7: .3gng 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 ballk5zjm0k.v jswzog6 76 makes 15.0 revo7v6 ksw50mgkj6zo .jz lutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels674m wh,t +s *qvjvz(gxl ug,t 8.0 km. How many revolutions do the wheels mus 4,tqhm(,v7 6l*jvxw+gz tgake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Cakin l3t a iivb98t,m37a (0ddqlculate its angular velocity inii qdbl 93t0m3da,(8 nkvta7i $rad/s$ $\omega$ =    $rad/sec$
(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 ghn +a8u//;odgu j0da 9b /t(eutd:q wone complete revolution in 4.0 s (Fig. 8–38). (ajndbe//o hadq +uta g dut:8 w(/;ug09) What is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbitn35vczt) zo6 j e zxqp*c73t5f around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed-ozbzn0qu m 4s2n.+y +y8lij v of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if6xy k42zitewax4r n12a4v5 vz a particle 7.0 cm from the axis of rotation is to experience an acceleration of 2 41neyi4v r46azwx2xa5ktvz100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniforml4g-.lp lf rh-ebjt 0p(el f,q4y about its center from 130 rpm to 280 rpm in 4.0 s. Determi lt-.p - le,ql0(frg44pe jbfhne
(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 radiuhln 0u8 1zx. t/ekbm 6brkwt38s $R_1$ 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 spac hq18;*wnu4t9ty.,o tcra0o8sxdy enecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be th0ry;dsetn1q y4t,to8a .9oh8* cuxwn ought 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 unifostg ln1(kh 09mbqsb 57v18c1efdf4wg rmly from rest to 15,000 rpm in 220 s. Through how many revolutions sb7 9(0c1lf fh4ts 85e dbn11gvwgqmkdid it turn in this time?    $rev$

  An automobile engine slows down from 4u7cm, o- gbo4z500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeedi b.he80 yutchr: +z-eias .r0 jets in a whirling “human centrifuge,” whi 8.eh:ry0 -r.cse tiz0+bai huch 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 rpm in4on6u7t/b hacl1 bx 1j 6.5 s. How far will a point on the edglntaohc jbu1 b/1647xe of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500z,)d y5.bxvqt revolutions before it cometz,5)x bvydq.s to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make (e ub;1 0j xu91 pal,tbnvi29se64 on /czscty65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.l(xtn69cusi0; 4up,2 so9zy /ec1jvb b ne1at 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 , a+iougtr(g s5zr0u 8car reduces its speed uniformly from 95km/h to 45km/h The tires hav 8(g arrtoi,z0gsu5+ue 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 pedal when climbingzlup6.9g x pey q0fe -40q.fex a hill. The pedals rotate in a 0e x0.g xzylqf.p eeqf94-u6p circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What asxg127 i3(:s o y*jbik4 iyowis the magnitude of the torque if the force is exyskwbaj (y ox*s472goiii13: erted
(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–bba,dlzsp +p9nhh) j h -6+pdp6z)l 4s39. Assume that a friction tod+nhh +d)zb6lha9sp-lp 4pzjpsb ,)6rque 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 ohh* 4rjhf t0y1f a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then fh1 t04rhhjy* released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an enqcb r0n1-w+z rd9/gn omr0 g,bgine require tightening to a torque of 38 rg0m+nb / -qr,rdobz1cn0gw 9$m \cdot N$ 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 of5x a ik0l-2ejmh-+wk(e omd rlry3w7m/;:tc b radius 0.648 m when the axis of rotat7kba c mrrmeld023ox -;:ty5k (/- ih+ewwlm jion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66. vrb)bkuwx6::7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).:rk6uv) xbb: w    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod g:mlmr;y4cjxs/6x: -l tnpz* is rotated in a horizontal circle of/6yg*j :m: ;pt4c-lzmrxlsnx 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 pe (e,(h) .klb. qzuqly0) krcf-z n(mpotter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hand ).n -.h0fuq (q eekzkcplyr( z(bm,l)s and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fiio c;fds-qsdoa/. g(0 g. 8–43 abq0sdo/gf; c (.-s diaoout
(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 consist :fv-bv+3gx s6x+tcj,c4mdl+s 9 nbbes of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rawntsgx b( -;nf5yr :s*te, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reachbf*; s(t:wrsn -x5gyn 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder wi1 ryy ;b8mpn-xeyz2 :v9b shx*th a radius of 8.50 cm and a mass of 0.580 kg. Calculatn: hxb; 2 me9z8yybx py*-v1sre
(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 swi;0j x -b8p-u+xo xjmytngs 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-driven merry-go-rout8o)5 kur z5t loem8lir62v5w nd and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the al88l 55tok u 5 vrzoriwt2em)6cceleration, 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 brougv+oels i )+ ns:5y3otxht uniformly to rest by a frictional torque of 1.+lx5s t:)yni3o so e+v2 $m \cdot N$ 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 accelerateqpp y 4qt)j--is 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 act-ynif0n n0tgg )mo:5gion of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly frfnn t5-g:m ogg )00nyiom rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long4809 uwfc xs bcu750ekbl .:ecgr4ywc thin rod, as shown in Fig. 8–46clby.uk0 5ubfrccc4g 0 s8:eew x9 4w7.
(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 ofdx9l-egox(iszr3/ q4cqv *0f qk)ct) 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 w-w8s., vdamvdithin four full turns (revolutionsd,vsvm. - dwa8) and 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 moment ofwc/kslpkn-b 6oj/ 2s( 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 to.5)phxjgu6jl:z )inew,ja 9z rque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of masb nr;-ho0/z c is*x8mdg:3fnys 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.3 g b hzn8 i;0yf c/mon*ds-3xr:$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of t: odva4e( mntr1*ha8the Earth with respect to the Sun as the sum of two termsa:ervdh*m1 a(tt on84,
(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 4fsp5 lpv:ib:kg and a radius of 7.50 m. How much net work is required to accele : f5vp:l4bipsrate 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 from rest and rolls w z(m6b )tk )pji su;4 koq-w;+alz-fuwithout slipping down aaz4mbp- j qkwi;k+(uu))sf6z; o-l wt 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 balanaltw*o5zd k+86 co8ma m21,q uk:z wquced vertically on its tip. It starts to fall and its lower end does not slip. What will ,2 6 88alkuc w1 k*:zqw5qdzaummt +oobe the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin s 51bii mhybxddob legr /h,*j/w1w774tring in ajh4b7bh*l5m/ ,gw/b7ywx1ed di1 ori 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 aj q:vn u88y2pw 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm? 2ujq ywv8n p8:   $kg \cdot m^2$
(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 p/ayjqtr wt6u( s.h16 rotating at a rate of 1.3rev/s If he raises his arms to a horizontal 1 rp h6sjy/ wqtt6.ua(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 one shown3h3l.5q f4vvo hr cft29zsun7 in Fig. 8–29) can reduce her moment of inertia by a factor of a.7fnr o 235hz3v9 thqlcuf4svbout 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 angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin r 6t .l2bsr q;sq1 aptb*cj3e(rotation rate from an initial rate of 1.0 rev every 2.0 s to a final ra*clsb qba16r;3tr.2 tpes(qj te 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 b zo p9lt saj+(lnr(tw 72xk;.around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the centekops2 r(;zn.tw9 b ljt7a(lx+ r of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spicilz8ilyt(0 v pb*7*lnning 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 Eart o7z4hdxm6 )..6gow6j sjhleuh
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an ide0ini6hty g)*w shk/+ yntical disk rotating at angulay)w hykshg +i*6t0/n ir speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns rpw9t: t13e.+dl bphbat 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 standsk qz.ayn .8u-j at the center of a rotating merry-go-round platform of radijqk..-8 u yzanus 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 any4k2 er8eze2kgular velocity of 0.8 z2yerke4k2e 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 9zmv +qkhj9-:qn))tv sea6e ocollapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nevehzaone + :)tkjqs q 969m-)varest 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 120w(vfm(jpjb 6* $km/h$ 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 zf3rb l-ce * 2;*5jqmsf7w orv$ 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 roen85,0dru 6h1bqmt aqtate freely without friction. The moment of inertia of the person pqq6edhb18r a 0n5 m,tulus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m p6z3df h7u-;kvh2eogoc m;ydu).-p hdiameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of ;uz7hpo; ckf v2degdmpu)h.-6-3 hyo 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying on i5 5)spwh-7kj kb: p0nhysoo.3w as)k the top edge of the spool. A person grabs the end of the rope and walks a distance L, psno5s5w a)i3 - 7shjh k )wp.boyk:0kholding 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 alptj,;0- zm/uy caur p-ways faces the Earth. Determine the ratio of the M0j/;y uuztr,apc-pm -oon’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 accelerates from rest0).zo 6+zpecajup q f 1autk;0 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 o:(p+jdn vkkk7u.9 hu2jnu4zw f radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angd:k.kp+9jun4 nuk w v(2h zju7ular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two so1fcg (snifqb:5eg- 1kegu +p; lid cylindrical disks, each of mass 0.050 kg and diame(51us- ;1infcgb+gf: g kpee qter 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 from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the r dt:ivcpz zky, f0)/y(pud9z+ ear 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 Sr, y)i c:n5q a,kebm(cun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of rak mrync(,)i:b, 5qacedius 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-pollution automobile is for it to use ener beq)r* /f6i6x,j 0xcjcw ox9*itzz y-gy stored in a 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*/ztcyxoeji6x) zx*jfbq ,wc-06 9ri 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 illustrate.u pww6l38 sl ep tq9wmm+;ra+s 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 horizontalzc3mhoh (:n y;7nnz .y surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frics v)a ns05x7hqtion) 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 that5a)0nv s q7xsh 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 ixtds(rd9d 2ss 8aao8:.eue g2s standing vertically on the 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.a.s8st o9dux(ered2 2 a sd8:g 8–55). The step has height h, where h

  A bicyclist traveling with spee twu,72 ymibb7d v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist,7 27umybtbiw 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 sl* j:; ubimrvjo u4ng78ing of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it frombmjn4iu* 7vou;: grj8 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 iv4b8a.p5vgh body as a solid cylinder and her arms as thin rods, making reasonable estimatv4 b i.avhpg85es 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 assembly which consists of two cylindrical plates,+akt/hz- 5s ym of ayzs h/5t+m- kmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 hux3d ( 8,mguy m7byhl8i9u*i 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 highest7d* uuy3h8umhy, g9 mb (xli8i point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume s:,e gyl )zmgm68 u *mfv*n2xu$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car redu. nk kdwz)b7 5 1yxd6kdmfkids42*a-eces its speed uniformly from 90km/h to 60km/h The tires)1zi7kkwmsd 4 5 ak n-2d.exfyd6bk*d 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