A bicycle odometer (which mea,d +olf;svrb)r4x4 h8cthj 9 wsures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on d,8 lhf j t4+4vr;hsb)9rocwxa bicycle with 24-inch wheels?

Suppose a disk rotates atp. y10 vc27ujt jmo1pg constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component omp2uo gyp 1j.1jtv 0c7f linear acceleration change?

Could a nonrigid body be described by a single vxqq jzbu0kh5 y4w57v,9nry4xalue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a lar*n*6joi6m r dsger 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 g7qj7l0x vqs0 eq . i5udzcv(tn1jm 96behind your head than when your arms are stretched out in front of you? A diagram mayz1n7d qeqc69l 0vxv75t sg 0qj.jm(i u help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel andylij8 -/nr y ousq5xfw v-141y three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large fron / 15jwilvqyy- y1no8fu-srx4t sprocket? Why?

Mammals that depend on behqrn 934rv 5teing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the bhre 9 45n3trvqody (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, pja: t;m7 o:ahnarrow beam?

If the net force on a system is zpe)+ecl 24mzjm* nq37 kj6wwc ero, is the net torque also zero? If the net torque on a system is zero, is the net forcezeq3l2njc74wp6)kcmejw * m+ zero?

Two inclines have the same height but make different angles with the horicg0kl(xd1jbymr 7 86xm.jw:azontal. The same steel ball is rolled down eachjyxkm wmlx8b(d1 .6 g0 jc7:ar 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 incline.2zt7ejxnsb*8 -y8fud 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 zju8yf2s* t8b -dn7 exenergy at the bottom?

A sphere and a cylinder have td8kl.fuv zck:do1*u1vbm (v4 he 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 totvfmd 1uc4 8.k d*zok:1uvbvl( al kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet mos+zs7:yq w m-j wqznj sj6m.5y/t moving or rotating objects evj5n/z- ss7m j.j 6myw:zq +wyqentually slow down and stop. Explain.

If there were a great migration of people toward the Ea 8 xklcde otl.s34d+t5b qzfvw4.-3m brth’s equator, how would this affect thq84- wxtb+4s3blzevk 3od.5ltm.c df e length of the day?

Can the diver of Fig. 8–29 do a somersault without having mpwps :)k(p.r5pd7bv k,f q0 eany initial rotation when she leaves the board?qv, bw .ppkkpd :7(m)50fpers

The moment of inertia of a rotating solid disk about an axis through its cen)di9+l*gta.v r5 fijw r;d-lj ter ofli*fr9j r+j gtalv5.wi- d)d; mass 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 zr8a;ujd a4uw5c;yq5a rotating stool holding a 2-kg mass in each outstretched hand. If you sudda ;yc ad;u4w8jq55zu renly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identid7r,-hi ylj1fxs6s94dew lt 9p-2fav cal and have the same mass. However, one is hollow and the other is solid. Describe an experiment to d-s6fi- p,2lx4d7whv 9a r dt1yefl9jsetermine which is which.

The angular velocity of a wheel rotating on a horizontal axle points vb z(p6ih x-:,l eb5ukwest. In what 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 point at the top of the wheel. Is the angu :- k,l5pi6zub(h bevxlar speed increasing or decreasing?

Suppose you are standing on tk8 5a( fx4;pzt+qrdgf he edge of a large freely rotating turntable. What happ;k5dt4 8f+f px qr(azgens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he th lhh7s(t:,o:- i uog9,n7c c2vqqn odtrows the ball, ug v,q9: cn n:o2scqo7t-h ,ith7(l dothe upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of con9 u6/boen nq7tservation of angular momentum, discuss why a helicopter must have moreu / o67tq9nbne than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 3 b1f1im+ l8(by 83jkz3h w*qo) lb qz3zdrt/vp0 $^{\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. Calculate, usn8vchqo 6;)v ;c8ivvubcvl92ing the information inside the Front Cover, the angular diameters (i b;cc)8 ;9ivuoqv26c8nv vhlvn radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the M8lvw;f. n vk b1miubbld( 8/3loon, 380,000 km from Earth. The beam diverges at an ang vl 8(.1db8lub bfl w;3vm/kinle $\theta$ (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 rat0i z(4txg(+ .y vbmaz/eb5b)qj3fph de of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the b3yi j/t ( p(v04. f+xe5ghz)mbqbz adblades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makesvj.bn+ t) puu5bs2;p go21agd 15.0 revolutions, b; 1up5a2gv +up )tnj2 gsod.bwhat is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. p*g 8x9jt;sxf How many revolutions do the wheex 9;8xgf stpj*ls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates ,l8o*y du rb uf2o,e- luhcno2r+g/i; at 2500 rpm. Calculate its angular v,8roo2*u ihl/e,cu fr; gl no+-by2du elocity 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-roho +dbdm,/o6yund makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m fo, yh+6/m ddborom 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)ndjdi8hnl ;gab1 /v 54 in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spg 2qj+5d* bg*4l dg n(y.lvpmhkdl59seed 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 if a particle 7.0 cm sxzn*q4 q5biq;w-ty5from the axis of rotation is to experience an acceleration of 100,0nxi5bq; w-5 *4qsyqtz 00 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 r )4b/an1u apmopm to 280 rpm in 4.0 s. Deterb 4/u)po1nmaamine
(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 90z/nahk9qv.+ n mqej $R_1$ 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 v ru9dgb(4zt q3 3y;2v7l uicwthe Apollo spacecraft put themselves into a slow rotation ttycv;7iur q3d4v(b93zwu 2l go 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 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 accelerate (qjsaw*8ywz +s uniformly from rest to 15,000 rpm in 220 s. Through how many revzq* 8ajyw(+ws olutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 120delvdl 07h*4i 0 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets i5.8fdcw3my r7wcymz a* 3ng8 an a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its fincw8d y37zaayc8 3rwf mmn5*g .al 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 in 6.5 sa) fte.qtqw65zio) 5q. How far will a point on the edge of the whee)teq)5i zo q6qfa t.w5l have traveled in this time?    m

  A cooling fan is turned oflhx5e.)1 a 9y.gn- fo:glp7vy5ipqf vf when it is running at 850rev/min It turns 1500 revolqe9 7a1:5ylpihl yg ov5gf p.)-vxf.nutions before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as thr5. hu gjri1f.e car reduces its speed uniformly from 95km/h to 45km/h The tireri hg5r.uj.1 fs 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 unibt1 n7-drbo c-3, fsqtformly from 95km/h to 4n7tbrsb1fc o3d q --,t5km/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 weight on eacr pm 2xj w4g1/fq42fdy3lji/ ih pedal when climbing a hill. The pedals rotate in a circle of radii4d22 fxj/qi /1glrw j3fmp 4yus 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 thei q50vyf9 .j8ncjy.g e magnitude of the torque if.g.fye50q jj8cy vi 9n 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 iw+ mfuuu. 0-drn Fig. 8–39. Assume that a friction .0du wfuu- r+mtorque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod whicwhr j;ix0tb0xx(/ la/h pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of rjb w/tlh0/x0 ;a ix(xthe net torque on this system.

  The bolts on the cylinder head of an engih46r )g8r k* ppiwoj 6ofz2v(ine require tightening to a torque of 38 p(v8h26fzgir 6w*ok4pjo) ri $m \cdot N$ 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 fy 3gde9 rkj-a34kthu 6j6 g7vsphere of radius 0.648 m when the axis of rotakd uk9jvy76-f4 g6jr33 thgeation is through its center.    $kg \cdot m^2$

  Calculate the moment of inert2 f tclqahv7ct;7,(x tia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can b cat,(;tflc t2xvhq 77e ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in )t0 x.bp3r z2rjj) l 0:djqx,gqf1ez la horizontal circle of radius 1.2 m. Calcufbj:x q).,jre 0xlp1z32)jtdzgl 0rqlate
(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 const+3 js9sdmrpkd*2e 3 c p4v v4qc4lqte6ant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N totje q4q merpc6ds3349kvt+lv2 pc 4 *dsal.
(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 Fig. 8–k+mlm;h nc+5 :w vrt5y43+ 5rcnwmk+;yl hv:m5 t about
(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 total6sj5 po/qh2 zu2 mfl*pe utm/) 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 sa*i8lgzq3 t5vi +s m9f;eo;8xc tjjn)pinning at the correct rate, engineers fire four tangential rocketivx;q+tzf j5n)s*m a oi;l93 8t gc8ejs as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a unifoi:f 9ac4p7csmr 94+ q9jtqtyeh* m5btrm cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calcc:49 pf5rtqqj*y+4seatm ic 99h m 7btulate
(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 i)t de1ne 1y.jv:rukn.t 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-round anglq8 9dhqq/1 cd 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 gq/ d9c1hl 8qqopposite 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 shur2pb0*gh33jf8hqu (sqqt/v-vhd zb) +b6 hb nt off and is eventually brought uniformlqvfj- hbbt0n vh*hp sru3 38+ b)(2gq/d6hzb qy 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 9x euseqxp/n2 u9p+z0 accelerates 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 t: t3fpmz rngg )ra+;g:he action of the forearm, which rotates about the elbow joint under the ac+g3t r f n:;:apr)gzmgtion 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 coif hwvhw*lzt1 d1hk +0 ;k5h;tnsidered a long thin rod, as shown in Fig. 8–41 wlk1ti*h dwh+fv hk;t5;hz06.
(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 tw;j:w9f)7zdw,3og anrcx7jm ho 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 full turei3n1 v1otczg)na )k, meo.a*ns (revolutions) and rg a.ei,*)a)1ctom n1ozev3kn eleases 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 a0pcbh0 -1mxz o 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 deven;g:l0ryg:jv/ bkd +iap*.i dlops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls withoup07i nswqmk2, t slipping down a lane at 3 pikn72,msw q0.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy o;nchpr mq10ugx 7zn3tyt 81h(f the Earth with respect to the Sun as the sum of two p1 h0t c1nhx7t8qy(;nu3rgmz terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radiq(mnjf8 ;f+pgi 3q1ts us 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+q8(jtpi n3f1mgfqs ;.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg ,(-tjdf b vlb9d8z 1fxstarts from rest and rolls without slipping down a 3- 9,( 1xvfbd8jbt fdzl0.0 $^{\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 tvz7x8- g; i b6mlmz*wgo fall and its lowe gi -bgm*8; lxmwzzv76r end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating ong2f 8m- l-g8ptqg+r kj the end of a thin string in a 2-l+ g8pqf-rkjtgmg 8 circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular+3 y 7vhxpg+ueebnv82 momentum of a 2.8-kg uniform cylindrical grinding wheel vxhey3 8ge7++uvbp2n of radius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hand c(;o uj5gs:hcs at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizonta(c;soj: c5h ugl 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 shown in Fig. 8–29) can reduce he,c5snn hm er pnl2yuf+ 6x( - i(ip*lcx3v7j8pr moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 ry73lp,xc2pusvcj m5r(i*x6nfil 8ne n- p(h+ otations 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 iixd ; evwe7wn.. )qi/jncrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a f/.w ;wi7)idvqex.n ej inal 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 ar(xhhs /rg3:n eound 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 ce sxr/h:gnhe(3nter 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 s*- q0h ny pr2gbaf0rprm76t 8dpinning 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,f3ie07 ugpu xoge ,m ,fs/7io 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 aya 9 h1wwy0-tfb qz( me/+.bgof inertia I is dropped onto an identical disk gq1hw/+0 fayez y-w (bmat.9b rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atiq 7gy8(feq gm9x z h39yd0pw1 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 kjb656.i i saggg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inei bs6.56i aggjrtia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocitt-+ixcj7(rercpv7c s 5 gq0*u y of 0e c7 vc ri0x-gtspc5*(q+ r7uj.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 eventuallybxw8gbg u 68w6:t qs70p(grrv collapses 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 carp0 :gqgt rgu(wrs68v7bb86wx ries 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 12r9uh11 cyde5m.or -iddh+5 ko 0 $km/h$ 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 )+ap06tvwfmo ) vw 6uwbwif:2$ 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, o s nu5p m;pxy:36jy bbu9u9h,initially at rest, that can rotate freely without friction. The moment of ippb 36: m5uxyyuob;j99,s hnu nertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at ts/al()ss7 pk g tauc(t ou;x6(he edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of o7lspcau(sx ; (tsut/6)g (ak 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 k:g59 fv l1hndon the ground with the end of the rope lying on the top edge of the spool. A person grabs l5hk dv9n1 :gfthe end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alway(1/xkzy 1 5dn- emmmeas faces the Earth. Determine the ratio of the Moon’s spin angular (dkm11nx /5e azmemy-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 rewu7w.,snw ot) :y6zzlst 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 ths z5 gue7: 2t1pxskumu9b, ta-e shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculatepg e2ux-7 sbamtsu:t5k9z1u, the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyli-uak,d;6jw e6exo /xu 4yqycie7(f a+ndrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical o;qd eyakcu6w(+f, y ij6x4ea x-7/eu 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 rear wheeln (x0hehq1u d,bbi2+x ($\omega_R$) 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, wwzyyw3bfn 4 gsq7ufv9.9-i8pes4r l 0ere 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 be? Assume that the star is a uniform sphere at all times, and thzw 9 uyr lp98f i3b.74-0ynvgwes4fqsat the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy0b;9k4tl8vmj4n/iof vjx 9b z stored in a heavy rotating fl0/j 4498 vbolzm ;9vtijbkfxnywheel. 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 illustrates . 6z :wvf4ikle7m/bcbj c4i4han $H_2O$ 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 r+f(v rzcbh1m:. p /+j.ly jz xpyp;3phorizontal 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 frictioztb str4u-e a)t(f- yk1q k;a5n) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizonzr;)t-qa5f14 y-k be( u akttstally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor)vh g w,f1m 2ts,qjs8k(gh ,wn, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, wherhf,vwmhgw1ss ),2t kg qj8 n(,e h

  A bicyclist traveling with speed v=4.2m/s on a.h4jps9x n:zz flat road is making a turn with a radius The forces ach.jzp s:4 nz9xting 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 rock9 yx .t4uz cskyf7bv)4 into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?vz7ub) fy4y.9k xtcs 4    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her vt6 i.5 bz1.k ro3ufh2e 6mmugarms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of mh .reuuv.12tb6ik5o3z mgf6 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 platesjxh-pnsc 77 wa938rzz , of 37z8c pj9z xn hwrsa-7mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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),a euirl cs u4kh/s9, along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marbl/,9u4 ha suc,)lrseike 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 no/ 3mes. x ga.1wkg,y*y,ejngoi7jpk9t assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly fr2cor k2sv 56yn wn+wynaumqz 6s)l7(/ om 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration 2r6n mo +2v 7wcs w/y(z ya6nn)5kqulsof 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