A bicycle odometer (which m91nu/ 0zpbob yb(ni-+ kyf (nleasures distance traveled) is attached near the wheel hub and is design 1ki yzb-bo/9n nn+p(0l(ybfued for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constan puct+5bpbln(6v di(5t 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? Forp(5lcuv (t6b+bp dn5i which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value oyu4ha3x 1zv 5gf the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Expvm2i7 ajtgpmd *f202qlain.

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 behinh56b2l7t eg; hb xqaw/xk(4fs4*tycod your head than when your arms are stretched out in front of you? A di/ 7t(x;4x sa5hk6lch *btfew2qoybg4agram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at tgz* y4z.eox/. jmjas76 fx r;bhe 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 fj6s7rzx . xo* mj.g; 4ayzebf/ront sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slenderd8ejl(cjijno)l / -.u lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mas l o8leu/j.n -(idcj)js is advantageous.

Why do tightrope walvkv0u(2kghy;v b uy a3-azia4u9 +xm ,kers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If tewcqbr /z s7p*3w5xq .he net torque on a3 pzrbqweq.* x7cw/5 s system is zero, is the net force zero?

Two inclines have the same height but make different angles with the n3* 1 czd9bd.mbbkr+h horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explainhzk3d*cb.bn+m br d91 .

Two solid spheres simultaneo7 /zn/)wgqkj8fm mj, 1+kdapt8j*nr nusly start rolling (from 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 therzkj1ng*), f/tkdwj+78/qan8j pm mnre? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same rad/2,r y6ep*w0asgrl yw ius and the same mass. They start from rest at the top of an incline. Which /yypesl60 grwa2rw *,reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most movingdfe.q;pm -2 zlge mp pu*2;x3i or rotating objects eved2;q f e*3ezm pipx-g.2u;lpmntually slow down and stop. Explain.

If there were a great migration of peop -v*zd xq,0xd)x0giycle toward the Earth’s equator, how would this affect thxz)gd,xidc00 -* vyxqe length of the day?

Can the diver of Fig. 8–29 do a somersault without having any ini(kvo1z(ilm )u3sup m;tial rotation when she leaves the umsu ko 3z)v;1( ilp(mboard?

The moment of inertia of a rotating solidnl*-fe;pccc83u/8sn ub rd6 v disk about an axis through its center of mass is 8/nfnv e cs 6;8d3b lucuc*r-p$\frac{1}{2}WR^2$ (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 r9jy et7:hixnf9dp7,p,q wemp-;wez 9otating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular 97peh ; 7dmw:z9yej qiewft9,-p, pxnvelocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one v jd*dbjs2bl/y mwro;g+-3a7i,qg5 wn : g/bcis hollow and the other is solid. Describe an experiment to-7a w2 ;b:ongbgjm/ bvi/c*gql3+d5srdyw, j determine which is which.

The angular velocity of a wheel rotating on a horizontal .g liqwt:w1 g/axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angula1.igqlt:w/ wgr acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freew:wm z8+3 n0pel99rzql phot0ly rotating turntable. What happens tpl83lwmrqph9o+z0: 9 e0nwz if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. Asq(r.yyih ib+t:2e7mxqrl 1 qsjj;8o: he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate rs:ih7i(btq 1ly+2. q:jmr; yeqj8o xin the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation 0d9bg bdl*g* jn9k:* cg/ aqdrof angular momentum, discuss why a helicopter must have more than one rotor (or propeller). D rj/*l qddb:kn0g**adg9 9g cbiscuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following b:94ez d8 ft8) lixhpi ertn9 svt963mangles in radians: (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 fol*e4j2 -bekrkfc-5rp.g0 t, dj gd7 of an amazing coincidence. Calculate, using the information inside the Front Cover, th rlb*7 je2gk.ft04 rjdo-c f depg-5,ke angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at theo +-(ogfsim 9n Moon, 380,000 km from Earth. The beam diverges at ansfm-+(ogi9 n o 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 rate of 6500 r k-):i4 qvtbqwpm. When the motor is turned off during operation, the :-q 4kv )bqitwblades 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 o. lzz cdp(tjto u(2,i4n a level floor 3.5 m to another child. If the ball makes 15.0 z (2(l4t t ucd,o.pijzrevolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolu(7jp/h3e -,eggo lz21e ;tzqhb tpmv 0tions do the wheels mak37-zg m2o, ht/ezelj;pth 1 q0(egb vpe?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates *0 q 0rhiaa:emat 2500 rpm. Calculate its angular velocity in m0aei 0: aqr*h$rad/s$ $\omega$ =    $rad/sec$
(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. 8–3j+h2.p)valb t //tlmm8). (a) What is the linear speed of a child seated 1.2 m from the chvm m/tt)+ 2.l/lap bjenter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Ea/ 4bj1m1d htfji6w cj8rth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poin snc(0: xw ui-cyqs.+yt
(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 vv c mf+e7*/tzp 9o6)f alwf tz-8/gdeif a particle 7.0 cm from the axis of rotation is to experience an acceleraf fl76 w)-/9t/cmo aefe8 vd+zv*pzgttion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 ft*qnj jz( zwqj6f.)*rpm to 280 rpm in 4.0 s. Determin q w(zn jfqf.6j*t)*zje
(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 ;kx6v b ufl6a5,op ,kk8kqfd7 o:i+wk$R_1$ 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 abo.xtp* vp1dp.uard the Apollo spacecraft put themselves into a slow rotatvp* .d1 xp.uption 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 thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly fro gydc(kizmb z p, ,0pb*g:+kv6m rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in thi k ( ,,0dpkp:ggy z*bcmvi+6zbs time?    $rev$

  An automobile engine slows down from 4500 rv606 - y 1dod2dfwsatrpm 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 a0jxv8 :p ga9p(fs- fothe stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to tuf a9-x0pf8o:js (pg varn 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 oh43 ptoxc 9q6 hetuj7 u0cbi4t(i3 xzh:,g9o uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the whxj0t73i3hte 44qgcit oz u, 9c:o9huxhp( ob6eel have traveled in this time?    m

  A cooling fan is turned off when it is running;i8v)xvz +(a fzw-rew at 850rev/min It turns 1500 revolutionwv)f x-i (+zv8azew;rs 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 speed uniformly fj+lg: wi:f hfc h;*+f .o/lbvvrom 95km/h to 45km/h The tires have a diameter of 0.v;fi:fobvlhhfc+:. +w gl* / j80 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 revolut ay7(dwretyo z/6rn 12ions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.8o rr2ny1 7(dayez6/tw0 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 a4jzs8sb p(-tmdbce 3 :(mj/5ez bf+w tll her weight on each pedal when climbing a hill. The ped8tpfj(5c/ebb34je(zs+ t-sdz:wmb m als rotate in a 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(lxps- c.1-ml)k8esgh:l t zjk1tj5y 74 cm wide. What is the magnitu5( xl1- z kgyjts1j8s:l -hltecpk).mde of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the whee; i9*vsdoee21 .j wvhvl shown in Fig. 8–39. Assume tha92eidh s;vvjv ew*.o1t a friction 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 mah ,5:a7uvrzue(jf qq/ cat3j1ssless rod which 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 the net torque on this system. 7hr t :uzaqfj,aeq5 /(c3v1u j

  The bolts on the cylinder head of an engine require tightenq0j)i8gpekv 0jno;* c ing to a torque of 38 00e*ocj)vg pknq;8j i$m \cdot N$ 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-kk 69nwob(q y1p3m ae8d r57ccyg sphere of radius 0.648 m when the axis of rotation is through i1e 6o8 c(yd9aybc5rqm3n w7kpts center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The r+gvh t(znzvp9yal2t(* 6mj c40vp . hnim and tire have a combined mass of 1.25 kg. The mass of the hub cnclj*v+y(z249 .6v(ttmhg0n hppzva an be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end 1oiug4z4z t6b 4;gdhaof a thin, light rod is rotated in a horizontal circle of radius41h4;o6t abug z4di gz 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 o:v2,tbbcvxk98au ;nfus k :;in a potter’s wheel rotating at constant angular speed (Fig. a9i,s;:kcv; tuxbb 2fu kn :8v8–42). The friction force 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 sm)+* 3hcjncsqhown in Fig. 8–43 ajc*q hm+cs )n3bout
(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 tokg/ne d5/mui4 3w 38t)lcvgswtal 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 satellite7/ zm;raayh9i spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. Ia/a ;9ry 7zhmif 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 uniform cylinder with a radius of 8.50 cm and a mar ls nsx9p1 qstb2 6t9d/k295-azlwkgss of 0.580 kg. Calc skrl6dwst1nq z9p9l xt5k /g2s-b9a2 ulate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest to 0 3ie7nwp;ktix);ac r 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 tangentiall*/.d k3kt ocl:e6es ehy on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk .:k eehle tk*/dsoc36 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 r lq8f,abor*c a-4zk j)otating at 10,300 rpm is shut off and is eventually brought uni4c )laj-ak8b*zfo ,r qformly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates aa0+8zkih qb 4ah*z xf3 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 svx+de2rn9ss0w; b 04fizp .z( dgj.irolely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. Tgb di ne i4p(s.rf; r 0sz0.+x2zjwdv9he 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 thin rod, as shown in Fig.vw8sq 7c5gl3/ rjm 7kr 8–46. 5rwvrl/77gk 3 scjm8q
(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 two4.c s/4nldej p 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 ha).won3qz m t0ammer from rest within four full turns (revolutions) and releases itn m .)0w3tzqoa 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 iq*7 b b b vaj:sbnf5zc3fzv684nertia 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 tort1zb3b6 *0jbl5wpm s lnf)7 jwque 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 ra/yb.c ks12ub0r5 8 x9,dqdam sfrl9srdius 9.0 cm rolls without slipping down a lane at 3b as59rcmdrx.l9srykf 0ds/ 8, 12 qub.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect -8q ;dwb9(vdyillew9 bbve i/ 7*ymmq5u0 zr/to the Sun as the sum of two terms9umv;l/r /bdibmqey9bqwlw-0 ez8 5yvd( 7 *i,
(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 mugph, adq6x9* xch net work *9h xxpda,6gq is required to accelerate 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 witho(2-e /p svyvgvut slipping dy(-ge/ vvpvs2 own 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 verti.c2 akpnl /h;wp os3-kcally on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end;cpn ol.23k /k -hwaps 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 thvgsxr9d) /rx tw(i.j3 v (bzm)in string in a circle of ra9 x/jrdv3w. x(bszi mr(v))gtdius 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 co3 .jl0)qylq/lvx: +v7aqcl)gf p v (oylindrical grinding wheel of radius 18 cm wh )flqxp yqlo3v0gv+ v7l)cal:qo (j./en 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 platfos1)t3-wn q qktrm that is rotating at a rate of 1.3rev/s If he ranq3 w1tq)tk -sises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inertia 18 thq5ammitbd757l;mgly z- by a factor of about 3.5 when changing from the straight posimll-; 7qy5 d51tiz7ma8 h mtbgtion 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 spivjka;a +:te.o n rotation rate from an initial rate of 1.0 rev every 2.0.jav keta+;: o s to a 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 around a vertical axis through its center at a freqbgu2 a85mo,15)b d kg1jwcezduency of 1.5rev/s The wheel can be considered a uniform disk ofdebou12wg)8cazd ,k j 5 gm1b5 mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular *gavy;9b ll p2momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum+cubhx(,g p /fuat ,:e 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 fi6r,1m7i- jqzlbf.j onto an identical disk rotating at angula,rmj7if lqbi 6-1zj.fr speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 3-fjp6 bvh 0bj2.4 $rev/s$ 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 a rotating+ fi b z50lzkl5z(zqdxl(d2 u6 merry-go-round platforml(xb5+ uiqldkd f602(lzz5zz of radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-gr 9 f d3crp1neic0;hn+l6.2fpfvy. bpo-round is rotating freely with an angular velocity of 0yd 036env+f p .;cphbf .r12lp i9cnrf.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 eventuall;zeq(ywnfo4ew3 8go-q;za: au (ck 2sy collapses into a white dwarf, losing about half its mass in the process, and winding up with ga48 y e -(2k:ao;c;u (nzwqzfswqo3ea 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 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 fbuk 3fdo n569in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass d,32oxitdhsxa x:a93$ 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, initiallhlyyp1x t 11y3e*eux :z4xk7fy at rest, that can rotate freely without friction. The momenkx x31yzflx1ey 4uyp:h1t7* e t of inertia 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 the edge of a 6.5-yyud8 8a,c8m j. (bjstzzgn.2m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 170dj tuc j a8yyg.(bmsz828. ,zn0 $kg \cdot m^2$ 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 gdf; ,j*lle;x lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move? *jx,fell; d;g

  The Moon orbits the Earth such that the same side always faces the Earth. Determf+ dx.fjp myk6k1u; m-ine the ;jxdm-k 6mu+f.fky1 pratio of the Moon’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 fzh xf:,1 hjt2g/wf8rhrom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of ftwxq+0zgmae26/e- d7w5hl w 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. Calculate thwxtw0e+2mq6h5wg az d7-e /fle torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kgz0t1ht* *8j g cof.nhf and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg z tho*hf* f8ncgt1j.0and 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 (m9kvri1zhz/c 9wdxe, n8q i5 is the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, et) l k 93 /dzh)ruq;4wo.cbynwere 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? A ho )tlbd/ewun9zqc4 rk;. )y3ssume 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 us7clhiwfa29ydc z t5nyv x/,- ,e energy stored in a heavy rotating flywheel. Sun2ht, v,7dw- l9yc/y afxz5ic ppose 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 g:bmosv f; +c+5zqw +nan $H_2O$ 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 (hoo*;v5ya izqneqhl ont p4a8.+*p) is rolling on a horizontal 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 g zc92gu03/ty.0yq jtlg/yhi l 2kja7pivot 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 the rod, as shown. [Hi /g7k 0y3/ 9ghc.0q lluyatjj 22gzytint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vf *w1777df sd bnqvk+dertically on the floor, and we want to exert a horizontal force F at its axle so qfdw v+bs1dfdk * 777nthat it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2k u(8y l8t,omf1b z5 +cqn1offm/s on a flat road is making a turn with a radius The forces acting on thfc85 8q(fz,bl ofo+1u mnyk1 te cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and b)ewd6 j,v 7;y) h/d mixc y.fdusz(n)kegins 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 t67 z,xsjwk( )/ m ynudc;fve. di))hdyhe torque come from?    $m \cdot N$

  Model a figure skater’3g2 3wqk6arc hs body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with cg3 wh2akq6r3arms held tightly against her body.   

  You are designing a clurowsy9lq,lr,,qy k hn:uyho1 n(j5(3tch assembly which consists of two cylindrical plates, of ma lor(y q r3nh,yyw, s l:hn,15kuj(9oqss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 track of Fig. gfg7 1lrq4ppb1qi mm,6j5n /s8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the trqpqb5jr/ gn4s7p 11 mm6lfig,ack? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assu8yui jf2 m0iihlun/. +r* lxpier7r))me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as tnn u+b.o u//pfhe car reduces its speed uniformly from 90/po uu .bnfn/+km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s