A bicycle odometer (which measures distance traveled) is7 b b.5+hyeae3,toj /evg3oa il0j*y x attached near the wheel hub and is designed for 27-inch wheels. What hbay+0,t3eble3ox i7 /jj * .a5o gevyhappens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at cooy*2mmwpm:+ j nstant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocit2mpm:+jw omy*y 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 described by a single value of t dp6vk/bu+y3d ego) q6he angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larvwi9;nf as2 9tger 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 )rtg7s ipn-n4imh*0 r when your arms are stretched out in f)4i- n pn h0tmgr7i*srront of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the.zhgz8*6h- 0 ivx hz0 qnemp7xs1m2wy pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is isx-* qghve.2pihw1 0 0 z7m znmhyz86xt 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 legs wie+*p43*cwnts x-qgik :w7w c -+irvvzth flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of -4 ws-ng*k+:i*vri+zcqevx3 wcp7tw mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, nw5jyoe c8u* s1arrow beam?

If the net force on a system is zeroa 4jdi pn0,.a(pv/gdkee7hp 9, is the net torque also zero? If the net torque on 79 ippak.eap4jh( 0gd dvn,e /a system is zero, is the net force zero?

Two inclines have the same height but make different a 1 0mo ep612ed/faptcql tpm qse95;k)ngles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball atp /f1t259t kd;)0pes 16mqeqm cpleoa the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from res 2jnck m8;rl68m3yla wt) 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 there? Which has the greater total kinetic energy at the botlj2am mwcl 88;r k3ny6tom?

A sphere and a cylinder have the same radius and the same 27cal.4 iusaqj115,m /rnmmkghrv6 ;sd x ( dimass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the grev/id2 m.n4;a,ga(mh m u q1sr16d 5ls crik7jxater 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 cwq- vz14h99lkhm0 lflonserved. Yet most moving or rotating objects eventually slow down and stop. Explai9h0q4f91w-ll hlm zkv n.

If there were a great migration of people toward the Earth’s equator, how woulsf(tk p35q j.md this affect the length of the spjtfmqk(.53day?

Can the diver of Fig. 8–29*7jtnonwbqd 8; h -2ql do a somersault without having any initial rotation when she leaves thelqd7nbwh 2; qo*n8jt- board?

The moment of inertia of a rotating9ghqe0x c d71i solid disk about an axis through its center of91e7dgh cxq 0i 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 holdin/jc:7 q ; ;y5/6xwkxpxwklas ug a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Exp5wcqx7wx / xyu:6jsa/klk p ;;lain.

Two spheres look identical and have the same mass. However, one is ho/q6p 78gpog dqw ux6vxi114ux llow and the other is solid. Describe an egv/6g 6 p pu4wqidx17q81xu oxxperiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In w; 5ud8l j m51 zv*zxm,c6wvjkyhat direction is the linear velocitycm;vwzyudz 81k,jm 5 *lx5j6v 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 standnx0 bo0uf z/(et;khe (ing on the edge of a large freely rotating turntable. What happenske ((0otnub hz0f;/ xe if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it.225mu gnge hdy x /h)*y7dm xlp/(ubh quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposit/2g dy2(bxuh g mlx). yeh5dnh/7upm* e direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discb s) b:tr4jn4tuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to s t j:4n4rb)tbkeep the helicopter stable.

  Express the following angles in rad6q 7oba3,spc.n6l b; sxmm5vb;ti i*c ians: (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 29rmu jfe)rru-3 1itgof an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of gfjui)e13mur 2 r-tr9the 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 k; lbu nldwb2.;)bnykcye c+**wf* n- beam diverges at ay 2cec)wbybb-+d;n n *lkkuf .l*w ;n*n 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 6500k e u.(-zjc wfx5ua2w6 rpm. When the motor is turned off during operation, the bkuewxwau 6(cz5f-j .2lades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another +)lyv i8;7w qh7vggu ,+wfnyr child. If the ball makes 15.0fl +vg wq,;y+u7vwrn)gh8 i7y revolutions, what is its diameter?    m

  A bicycle with tires 68 cm vf5y sqk,va7*9y rj4v in diameter travels 8.0 km. How many revolutions do the wheels makes fykj9yv 5a*qv74vr ,?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculvj:s2 l vkug8ts0kw 4+ate its angularjw4tl+2 svvg8ku 0ks : 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.msg8og (cjf,+mmrua +(*fk 0 o 8–38). (a) What is t g,o moma u+(jrf+f0k(mcgs8*he linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity. -yf ioo7zo;dz8o b3u of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed b2d- au gd7fu8of 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 centrifug*e 5qth jr+phrq5)v-qe rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,00 )r5q- q+ eh*5jqphrtv0 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 2(jfj 64x;q s nhcm,z.f l,(tcv80 rpm in 4.0 sfv6h f( ;4 z xs,tjlcc.nm(,jq. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiu ylj.io n p/-r(7rdm;6h)yq xls $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put ee ;mi mi5.(n/axh dutuq7-i, themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 1-u (nhime /xuiti, . 7ed;mqa52-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 uniforml2 o 1m1qawr8qiy from rest to 15,000 rpm in 220 s. Through how many r q18miarwo12qevolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rgfv;*xk :mq o/pm 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 highspeed jets in a whirli r8l yd/v9rlho6-x 26oyqk1hgng “human centrifuge,” whichl hy dk6q8h ol6y vrxr2o-g91/ 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 36opth i hcl- udb m0(8x dp8lp-0+-5lhd0 rpm in 6.5 s. How farm(puh 8dbplp 0lh5 xcl d--8h- to0di+ will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turneddb4f r5 (0aum*ofq6jx off when it is running at 850rev/min It turns 1500 revolutio *axm( jqofr05d4f u6bns 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 axoq8zhx7 gre6oce9/ c8q2;eo7v ku ) the car reduces its speed uniformly from 95km/h to 45km/h The taueo6x 8)7 q2he 8o;cxock9 z7vr/g qeires 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 (54hk nvdl8 *k)np5tqz0mxjk 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tx(z840 *vndlk5h)j 5nqm ptkkires 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 wzr-1n*k6eeus(0yr b3 usub z 5eight on each pedal when climbing a hill. Tz1yubrur-z ( *60keub5sn3s ehe pedals 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 74 cmgvksu6un+ j df x( :c5y*7nwd. wide. What is the magnitude of the tor+uny nvw k 7cs*5x(ju df:d6g.que 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 axlewjsfn ww)jn()4fm:,b.j e )sc of the wheel shown in Fig. 8–39. Assume that a friction torque os)f4b.fjwcs, )w:jjn e)nm(w f 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 kbe1* xu lz5obi/(d9c opg60d of a massless 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 ofe9/1 ( olbxdu0cgkb6*pdzo5 i the net torque on this system.

  The bolts on the cylf0 fyn+ 16 nou;tr1nisinder head of an engine require tightening to a torque of 38 f+os 61fyin0 1n trun;$m \cdot N$ 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 momentg,l7k(,7:lrh o5ef cab ccyo8 of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through c( 8oly5r: k,7clo,bghefa c7its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyclgxgbeoih w0qtl(wkw3idr 7k ;h52o:3lf). ;we wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub diekw 3l;5;f h bh.qg7)tk0xl3r2ooiww( g w:can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the.n n;em3 o n+ziadz0fi w3e,3m end of a thin, light rod is rotated in a horizontal cid mziw+, mne0f3n eoniz.33;a rcle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on 7 k)n; rahfc2) vcno2ja potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force be2)2fro cjvah)n7;n k ctween 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 shosi)r h/i l/n09jt8stcguy /4cwn in Fig. 8–43 about g it/ /y908lih rn4c/uts)jcs
(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 oxyg8z /v : s.-qadoslla(wwl(uz, n t6-chec3oi4en 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 corr26o 89ukhg*n b5xmh1i5*husi ( lctsi ect 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 satellite is to reach 32 rpm in 5.0 min? ic( hhgkl iit* mhu*sn9s1u525x 8b6 o$\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.5:as t90iuz ar4 ct06 8orv m0zyg*mec30 cm and a mass of 0.580 kg. Calculate arzvo t4s0m8*9 c :ym3gc06ezrti0ua
(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 3 g -d3/nis+bb d$rev/s$ 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-dri r:qs 9ftsroa 1yx36cq g1ig(0ven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s.r6( afo0 1trsyqg:i3 scq9g x1 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 w u)-o( e-j05xvjrah gis eventually brought uniformly to rest jawu5v hjex g0o()- r-by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg balam z 3qe+lh-dokd).w)xr2 d,dl 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 bfbs5wsu/t)5a3m0e2vssc(q ,q bx t*yall is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 bsmq (2,s 5usw 5tefyb ac0/3*)xtsvq $m/s$ 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, a-r:p otei-h f zl6(.bc na*m)es shown in Fig. 8–46. eeli6z)ho:- rba.c tnfm*( -p
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two venfh ts1/( 39 ry:4 *l6b-4nycav -jorlugrmmasses, $m_1$ 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 wit ahsuti,10 -rohin four full turns (revolutions) and releases it at a speed of is-1o0,trhua28 $m/s$ 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 aa5q2d/q2kfv -j 1ehsb 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 deveqda (,7,z6e64s qdciwoafa v/lops 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 without slipping down a hs youadb.pn2z .9k .6g.ea( clane at h(.y.dz egnacs2 69b.o. kpa u3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the y. p:qaowcxubt9o vev x.:+56 sum of two terq c+paw :. obye.o:5uxxt6v9 vms,
(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 kq2 98xa z.7 xj n7suricgj8kwxo2d 5/zg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revod wzj.ux q gx5cni /8s o8k729z72jarxlution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm andig5:e7j wkvj 4 mass 1.80 kg starts from rest and rolls without sli5e4jkv 7 gij:wpping 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 vertically on its tip. It stav ,-9lixoj1xri)2fd6 k; lglcrts to fall and its lowe-),gx d;ori l192vfx ck jli6lr 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 )( hz.sorl*ohof a 0.210-kg ball rotating on the end of a thin string in a ci sz (l)orh*o.hrcle 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 cylindricakv, 4pagrrv.5c+ yp1 9)vs kodl grinding wheel of radius 18 cm when rotating at 1500 rpm? 4kvk 1+ sovvr.,9pa)gpdr yc5   $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at hi sa 2w;axj8xo32yku j9s side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreaaw 2u2ja;j3xykxs 9o8 ses to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one showzm re6, pow4s:n in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1. zem:rs wo6,p45 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotlmnk. h po+jpd6 qm0c1d1o4k6ation rate from an initial rate of 1.0 rev every 2.0 s to a qmmkc+ 6j.1o4dpdlp6n1k h o0 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 itwnc319- aa(b v9: 32 ltjga pgcl2fxzss 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 center alwc jf3atg n2 xbzps9:a3 -29cg1(vl of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momenng as lq17p (t9plvz3/l 9f9lxtum 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 of the Earth 6xvn45u xqo79*)xlw1e jbjfy
(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,w c, uq :g(irwi3kgu4 is dropped onto an identical disk rotating at angular sqk g,iwiuurc ,3:4(wgpeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at-t, f cve-zz)qa f2yi* 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 a rotating merry-go-round platcbrmpw6z/3 k8r zl3) rform o/blw 6 r8z)33p crmzkrf 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-go-round is rotating freely with an angular veloci) e ,q qr+esxeyn6m58jty of 0x)6y, eernm8qq es+j5 .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 wxkym0l): /gb9:leh oi hite dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to m/:l:0io )g xhyke b9lthe 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 1llj. ic ysv76l8 zk)1y20 $km/h$ 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 u b +ferrz.sz5+6xaf+ $ 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, initil 9s wei0gx),z);e tosr1po* vo,ygn/ally at rest, that can rotate freely without friction. The moment of inertia of the perwer snvzti ;s, *,1)xegg0ploy/ )oo9 son 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-m diameter merry-go-round tuh. xlt/x.)k 1tko/qnn j7knx2rntable tnx nnq.x1hjl7t)// .tkk2 ok xhat 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 on the ground with the end of the rope lyinc:ioes :hdg2+ g 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 w2 +edc:sghi :oithout slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth su.8rx 4ive ngvnfts w+l, p(:t/ch that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle or8tw xsgvf+rn :v,t4/ npei(l.biting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest(apoe uy* 8dw6m(/drs at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquires 6pb/zy+ox e5rf8p08u2wo; esli0ofl a rotatis p6w e+8oxy zup2ol0 ol8;0fe5fr/bional 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 cylindrical diskk+8 j1sflcll(s, each of mass 0.050 kg and diameter 0.075 m,l lfcjsk81 +(l joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the ale:e7 hsskyevngjyp3 :83m jl 242sm)ngular 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 w.z2rk fkhqgyi4+hc h u**f52kith 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 q2k 5z 4chh* +r2g*yh .ffiukk11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobilq r2h k wnon9l..(gml2wk8cl8e is for it to use energy stored in a heavy rotating flywheel. Suppose such a 9nqw 8ll8hown2kr.l2cg k.( mcar 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 illustraad25ez* 6-vb- w /yvr/yz bxjwtes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3.2/ gs2f(ws kgf3 $m/s$ 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 fod1a civq. 8k:reely (i.e., we ignore friction) about a hinge attached to a wallv i:1 kcqdo.8a, 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. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is swe j)hsa;kdzp/: )v:otanding vertically on the floor, and we want to exert a horizontal :o; wk)apj z:dhsv)/eforce F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on qb,pdt(** fsaii a(-wr g-ed j04iv9u a flat road is making a turn with a radius The forces acting on the cyclist and cycle are thegwe(vid 9(-t psid *-ij qr4,b*0aauf 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 0r0m( 6h ay* pqbiqn,b;.50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, acceleranm;ai0yrqhbp6( ,qb *ting 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 wd5) mce2o .jhnml 9s xm 4cv4w/1kth)arms as thin rods, makin)ehc)m29k4/ m5cwhj1tno ms 4lw xvd.g reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two.;xo-4 ebfl 6t) a2ymz epwd+b cylindrical plates, of mass dz6b4-pe o l+wm2e)y aft;.bx $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 an6cunyr p/ pal,bu7q g:o4* 1rfd radius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop withoutn76o *yu1lcpugprqb :fr/,4 a leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assuchcmn +8 -mz.c.mbi/r g)p6os6, mkkkme $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car recrl zxxn 3tq6 -:wn+0+g qu.biduces its speed uniformly from 90km/h to 60km/h Thex ic nqt.x6z3-uwn+: l+bq0rg 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