A bicycle odometer (which measures distance travel t 6yzrai-14prz*u76thart. wed) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch whe4.a6wtzh-ryz r* p ua67r1titels?

Suppose a disk rotates at coifj fy 2iu -t0d.w+z os/4n4qtnstant angular velocity. Does a point on the rim have radial and/or tangential accelera44f itnzo/iqt +.0w-sufj y2 dtion? 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 of linear acceleration change?

Could a nonrigid body be descv ivimy.;3pluh pldz:*151 nzmzp9+wribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than pp-fs0p r1yrbj/zyi/4-t w wt+a: /l ka 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 sit-up with your hands behind your head than whito r;z50guc.en your arms are stretched out in front of you? A diagram mt0 r u5z;.ocgiay help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheeldbjcfh.9kyl 2+-hu (* u,g y)em vn)vh and three at the pedal m)f2g,n dk+ jcl*-u v(). 9evb hyhhuycranks. 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 front sprocket? Why?

Mammals that depend on being able to run fast have slender lower 0 jh4edef4l0l legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this dis4hfe del00l4 jtribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beaqhpo, s t1jogz.e-x1 -m7f/2slss2 qmm?

If the net force on a system is zero, is the net torque also zero? Ifdokif(mqf 7b mh zp9v05i-x 51 the net torque on a system is zero, is the net force zero?-(1f0vbim fihdk7x5 oq5m 9zp

Two inclines have the same height but make different angles 8yqy2 t0layvm/eg :21p w: rvnwith the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of0ytn lw:ramg ey8y 1:2v q2/pv the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down z0 q zuh7h8lr:z175bsrnii,han incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the inclineri,qu n7z57hb 18rli hz z0hs: first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They50k3s7fh7q 2qspw*k 8fbp huj start from rest at the top of an incline. Which reaches the bottom first? Which has th2s ufspkfqbhjwp k3*h0787q5 e greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angulmb bnzy8ilxs3y)bb5am0o 0gr4 88 a, nar momentum are conserved. Yet most moving or rotating objects eventually sb8 b0 xm8380 )rmn lzny5 b4agobyia,slow down and stop. Explain.

If there were a great migration of pedi2dhs x5x4u- ople toward the Earth’s equator, how would this affect the length of the d-x2ddxui 54 hsay?

Can the diver of Fig. 8–29 do a somersault without having anz-uoj n3/g e)gyo/k r-y initial rotation whe/g ogyu z-)oj3e/n- rkn she leaves the board?

The moment of inertia of a rot g w;/80u vcu ds4;g;kinlcvj,ating solid disk about an axis through its center ofsg wj;ui 0c ,4/vkc dl;8ng;vu 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 a rotating stool holding a 2-kg mass in ea n3ks1b :z3pznch outstretch :nk3sb znz3p1ed hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identicalg6u8s d/4vb rg gc;t t,+7ij 9kvvxdl/ and have the same mass. However, one is hollow and the other is solid. Describe an ex7tt4irgvgd/s k;9 8x cldvu+v ,/bj6gperiment to determine which is which.

The angular velocity oiakjo3v /h2 /vf a wheel rotating on a horizontal axle points west. In what direc k2h //a3ovvjition is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edg-xbppcb4s+3ilc; 9toe of a large freely rotating turntable. What happens if you walk 3bi -94pxl so+;pcctbtoward the center?

A shortstop may leap into the air to catch a balym)rgy5xejb7 (n ,5+g 0xuyiedn :y9g l and throw it quickly. As he throws the ball, the upper part of his body rotates( ee5im)g0+ b x75r : jg,xuydg9ynyny. 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 conservation of angular momentum, disck h sadzgc ,)o;gm9.b +.wyt1quss why a helicopter must have more ;ycb1k. )mh,sdg9w +o aqz.tgthan one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in r:nzv,5c w0ky)dfluf)+x x ez n1o(gt(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 coinciddy.rmyu gbw2.1) cmr*.hi j0hence. Calculate, using the information i. dum. r.ghy)ybwj*crhi1m 02nside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam kbac. s 8k8h9ab(/abw diverges at a wk.sc88bb/ak 9( haban angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a ratfa 7fs koz/l(kth ,* yf95mj,ie of 6500 rpm. When the motor is turned off during operation, the blades slow to rest iljk, o f*fst,myh(57kaf/z 9in 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.-5frhhbi *n 9k5 m to another child. If the ball makes 15.0 revolutions, wh5nfhh- rkbi9*at is its diameter?    m

  A bicycle with tires 68 cm inr 2;2y+ ti0wpkxac3ca diameter travels 8.0 km. How many revolutions do the 2ck;c03aiwap2y xt+rwheels make?    $rev$

  (a) A grinding wheel 0.35 m in dl p;sgitkjr :3s10u: fiameter rotates at 2500 rpm. Calculate its angularg ji:fpus;1 :sr 0t3kl 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-round makes one complete revolution in 4.0 s (Fig. 8–38). 4f6e:mgv (m llv*v/ f4 9gp6zi jy,xj+z6kuyn (a) What is the linear speed of6yp + 6zum/lxv:g,4fknel gv(yfjmvj 4 6i*z9 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 orbit a gwo-7a mk-jv:round the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed s -kcfsofg;/( 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 fro1rhvl4* of/kknzq( )ym the axis of rotation is to experience fv/rky* (qnlk4 )z oh1an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its ckhym:;a r m*m,qi/-t3 pjxz nnq;4 sb;enter from 130 rpm to 280 rpm in 4.0 s. ;x kr4mz/hamn bt-qn*: ; ;y3qpsij, mDetermine
(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 radiuscsu7; :-sj1iwmdy be4 $R_1$ 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, aoa0wd 67 y)y2f6jdwao pd8u+ fstronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no 7a)d+ dd 6a80j6yfwoo 2ypuf wrotation 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 accelerat86mx ztk: :zqxim 6p7ses uniformly from rest to 15,000 rpm in 220 s. Through how:m8:m7x kqzszt6ix6p many revolutions did it turn in this time?    $rev$

  An automobile engine sloi55m,q 3*ggr k ff3mq m44ms1vuyq*ox ws down from 4500 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 highspe0 t9 izpx4;5vedcvlp/ed jets in a whirling “human centrifu 4d/vc0li9pvxzt5ep;ge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 3b3c2hzf 46kxo4z (voe60 rpm in 6.5 s. He4k 3fb cxzzov (642hoow far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned ofe1dq,;s z8-8lr6 my5pjvwear *rj0 o lf when it is running at 850rev/min It turns 1500 revolur8ar,ve 0les lo m dj*6pq;8j1y5r-zwtions 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 the car reduces its spegikw(bgc 5tq,e,ia:4l,f3l ded uniformly from 95km/h to 45km/h The tires havegdg4(qlw,, i ,a3 cl 5k:tfbie 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 revn924cru x97 y+dynsty olutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires ynsnc92tr y79du 4xy+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 h8y8r0 zq+0ar t+ ,qwtn1v/ dobwdybo 5er weight on each pedal when climbing a hill. The pedals rotate in a c1 8n vodow8/q +ab ,yr05qbtrzyt+dw0ircle 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 en, ev4noh.c h qmo40lp.d of a door 74 cm wide. What is the magnitude of the torqp,homh.o. l40 v4necque 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 tor f)iwqg hsj-,2que about the axle of the wheel shown in Fig. 8–39. Assume that a friction wjq)- gsihf2,torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless roj7mj+ v1ja:9emx 6qcot q,)n sd which pivots as shown in Fig. 8–40. Initially the rod is x9jsjcqo,t a1v:mm)7 j+ne6 qheld in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tight,ah k/vctnb v-v;kqiv:k2h*0z c 6f6nening to a torque of 38 nq2/kvt: 6 hb0;nvzkv6*k,va f- i hcc$m \cdot N$ 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 zg)loo m-o jt lvff)vt4117af 9c9af710.8-kg sphere of radius 0.648 m when the axis of rotationltl aoj 4a)9 fgf1ov)79zcmtvff7o-1 is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 o. k onzzui1, 6ha a8kv8ic3:(ead +vocm in diameter. The rim and tire have a com ae +oncd iuoiokzv vk8a683.1a:z( ,hbined 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 is rotatw8k79cm bb 3lulh5b2 aed in a horizontal circle of radius 1753w9c ulhm8kbbl2b a .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 v ajery5(ztb /nk-;r mhj*j,1 wheel rotating at constant angular speed (Fig. 8–42). The friction force /brj1jht;zn 5 (amv,ky- *jre between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig. 8–43y 0igrrw e6qd0pe+(6h ab( erd60ghw +qe0yi6rpout
(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 conrbj5yn0 4k 1acbzx:u7sists 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 correxj q0zf*f5dlw (n+4man;ijk: ct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the sad*zinq fj;+ kwj0 n(5xm al:f4tellite 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 +0 qw(eelsg6n7s j d;pradius of 8.50 cm and a mass of 0.580 kg. Calcul(7p+dsq;ng j 0s6weleate
(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 frho-u,t 8y vvt1om 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-dria;zpznwz:u4:; l+ixf ven merry-go-round and is able to accelerate it from rest to a frequency of 1;+nx;:zia4 wlf:zp uz5 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brought kcjd ip4e;3x k+;x 9rfm0w 3tu.5zlwz x0tk fpuzj;k9; + cz4rixle .35mdw3w uniformly 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 acce)9v x73rg2yd *mp aq7qn /8mavdq0wohlerates 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 bd ,.6gz, .whee ;*fa jnxuczo.all is thrown solely by the action of the forearm, which rotates about theu h,fowa;.* gezxzn6. j e.cd, elbow joint under the action 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 considered a long thilxglr) v /- rf0yt)6uvn rod, as shown in Fig. 8–46vl-)rt0x/u fg)rl v6 y.
(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 x8uqh : .dy0yyconsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full - kz)aei 5hmxz c:y5n7turns (revolutioc75m-zknaih :5z ) yxens) 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 of inercqn qy; 0hq).pov)z 9xtia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque ofl4f; k1 , f7r*szdfaom4bz,cr:u(lf t p)pc1o 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 3ygzbbo b0pqq h;670irolls without slipping down a lane atzy0o hgb7ipb3;b60q q 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic en7yzqcr3i(il4r r z 2,f+ cxlk:ergy of the Earth with respect to the Sun as the sum of twokrlf24zri(xq c i:3,yz c7 +lr terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much netkk+ /gd f byl5nb30pm1 work is required to accelerate it from rest b5glkmp0 +f n /b3yk1dto 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 dfw vf3r-c2r dc9x39gfrom rest and rolls without slippircfx w2vf9d-g 9 d3r3cng down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced verticall(x-7q aic+5+d 9dqdtyc0l7yl ug i f*ay on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the po9i7 g 7qu5+ -tflcdy(dxi0d ca*al+yqle just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum w87t sh:swp qqrux* g1av/b73dl6oy6 of a 0.210-kg ball rotating on the end of a thin string ia7h :63 ldqw py*w7q1v/s 8tx6go rbsun a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheen;7 wv)gf1gx wl of radius;xnfv) 1 ggw7w 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating atudx;lun1kzis 64) n0;e d e1ao a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed6ndn)ldus41i kaz;ueex1 0 ;o 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 k 5fuvz1 lfty-*9h ;mxreduce her moment of inertia by a factor of about 3.5 when changing from -fz9 klxv ;y1*u5 htmfthe 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 in)in,u1c w4fl gqcr68t9hvah hqvp (57crease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to g9wv,tcuc h vqfh 4a1 i()76r8lqnh5pa final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating 3xq )-urwz o *qexu9a6around 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 0.40qu oe6rx) u9z*a w3q-x m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at qqs(/e( p g2pocqn1 e75i2ikt 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 mom gav5swf2 ,h0 atx8 .qje:j.hdentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an .laed33gz te-3a i3ju identical dg3e 3du3a3j. ti-l eazisk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns fv(sueoo m ,s7hs9)x 9at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of ae:30 bgzl6v kr rotating merry-go-round platform of radius 3.0 m and moment ofz:0 ebr3lv6 gk 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 ang3m nm ;h w11ga5vjs 243lwibblular velocity of 0 2sg 3vin5 bmb413;l1a whjlwm.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing aboug, q2l(-o*0a:lt qf *od ecoiut 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 a2i-auotll(d o:ecf 0* *g qo,qngular 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 excessv u *j zssqxu(gu, psri()6+e2 of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass pkd9,oj.;xg 3czx69okltey5l g ;6 b8fnb4mp$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate f5. /9 cfhjf .ad parip:k9y qlarvs+*:reely without friction. The moment of inertia of the person: vralj.hsr.fia*pfq9 ya95pk/+d :c 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 the 4.ws gix f-+cier0a1cdp k-;kedge of a 6.5-m diameter merry-go-round turntabl1sarke;i.cd-0w gc kif x4p+-e that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls onw:wq4vs(t rfi +c1*p y the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The sp1w p* fv4t(yq wicsr+:ool 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 saml,) k55gpvalhm 9c.ope side always faces the Earth. Determine the ratio of the Moon’s spin angular m cv lpp5gmho).a,9k5l omentum (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 rel jwy8ydwzr5 m(q* qxx62a):z-td :w rst 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.e//wo2wq;ngy)ps o o ;zzl7h120 m acquires a rotationazpw/ w oz;onyh;7/)1lo segq 2l rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindricacl :ihrlux*.. wmhc+3 .lp9x wl disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of ene.w9humx .c: i l xl+*c.hr3wplrgy 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 spp m1ha;f5dsoh 0g:g44p8 v.lw *bk mfceed 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 Su: xh:jzaz 7st;x40q kan, 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 radius 11 km, losing three-quarters of its mass ink:0 qz4:xt aaz7h;sxj 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-pollutionsu d (u)b,i eg7hts *p4l6kml1 automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flt)bs se(7 ,m4l1hg ud6i*pu lkywheel 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 an 0y3nku*kzzcrr*8d4d ,5aqn t $H_2O$ 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 sp h:(hher 11zdzrq(mcn;r6* :bwr3tyaeed 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 pivotrtg)ib8 m1b*7s w4q;m(p 6tes t0ipas freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of b4b(msr7g s; se)81p ttp*m06i aqtwithe rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has 3pqhulof5pkzwl6 :2p t7o(s/ radius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axlel/k p3p(o2pw5qsluoh 6t7:fz 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 lq+jg5b4vj f/lsbj 57 speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are thv j4b5jjbfl5 l7s+/gqe 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 r7;l8 t;bu(eltyf1 uqrv52ng vock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerarubg8vf y; 1(qlveun tt;5l7 2ting 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 solid cylinder and her arms as thin rodu7:yn+rpdee-m( c, e ls, making reasonable estima 7n(e-yr+l dc: umep,etes 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 oth.uw 1a t0+xv;klza9tmf*-consists of two cylindrical plates, of ma- +*0tkhw muv1t9xz afa.o;ltss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alo8 ji8) qlckp)*7lcog wt17sfdng the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h thatcj)qis *p7)o 8fgld8wc t71 kl the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do noth0f4 kjxt1 s*h assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the wsv8)p/3mwll ybm wx6xp+f 3,ye03vfcar reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of w,fl03sm8pp f)w ylv/ eym3xv bx3w +60.90 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