A bicycle odometer (which zs6.lq1vs0 y8l kl)ju measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What 8)slzj. kvll1u 0qs y6happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim rq t8)7lgdb k5s9d8ll4 vt1lx have radial and/or tang4gllrk58 l 9d1t qlt 7vdb8sx)ential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angulh-r8f; .*lxm gxjl jd/ar velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a la1fgh *tkllmbo0o ; 9g6ynry(p - 1fe1wrger 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 1 7h1kcjnzs2a vixn+/head than when your arms are stretched out in front of you? A diag/+cxh1s 71ia k2 njznvram may help you to answer this.

A 21-speed bicycle has sevent l,t k+vp3w03w4vpyt sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sw+03vv 4wltt 3kyp tp,procket or a large front sprocket? Why?

Mammals that depend on b 59a 0na2xwf.kyy: zpleing able to run fast have slender lower legs with flesh and muscle concentratedw05l . n pz:f9aykx2ya high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig 0d16oqjgcim ui7cff+:7gu5 q. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If o geyrf13dc:+the net torque on a system is zero, is 1 fe3gry:do+c the net force zero?

Two inclines have the same height but make different angles with the ho-c2w wll 2/m, xlp++cgpevltccbf. + 0rizontal. The same steel ball is rolled wcxbfg-wpv+2,t lccl l+e/lm+0 2 p.cdown each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simhr.zq9 l)y1dv ultaneously 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 fildry 1)z.q 9hvrst? 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. They stardk ;3 t:xfu-hyt from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed:fhk 3yx d-;ut 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 j *pvwzj5v 102amitk6 most moving or rotating objects evei0m6 1wp2 vk v*5zjatjntually slow down and stop. Explain.

If there were a great migration of people towb.l( voed2au. d6bt y/ju1x8xard the Earth’s equator, how would this affect the lenga2d e.6 lv( jox/b8u.t yd1bxuth of the day?

Can the diver of Fig. 8–29 do a somer :)iiu/ncar 97nbh8ro0a y1 svsault without having any initial rotation when sryhcrvu: 9 o8ia 0/i7s an1b)nhe leaves the board?

The moment of inertia of a rotating solid disk about an axis through its centezqa ) ;hij48w *ui6jpsr of ;8*u64hj sp)zw aqjiimass 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 hoyt:qd4a+t .:wzw4t zw lding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease,dt44:q .wat+zw:yzt w or stay the same? Explain.

Two spheres look identical ancu5 n7ke(,2 n5afkxl* rztvv+d have the same mass. However, one is hollow and the other is solid. Describe an experiment to c 7a*z2ntf(veu5 +kl , kvrx5ndetermine which is which.

The angular velocity of a wheel rotating ou+*aazqfsf 1fm-+y 6r n a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed inz qmrf-f*1 as+6yu +facreasing or decreasing?

Suppose you are standing on thex uhh mzk+790f edge of a large freely rotating turntable. What happens if you walxhz+h 7kfu0 9mk toward the center?

A shortstop may leap into the air to catch a ball and throw it n vwk u76y:ey8quickly. As he throws tk u7w8evy 6ny:he ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a he* mom9 idv(q).a nuud/u)w;o blicopter must have more than one rotor (or propeller). Discuss one or more ways td(b)o)dmwau ;niuv/uqm o9.*he second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (ag:y: nr:lhrg3bh*v r zz:9r *e) 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 coincidence. Calculate, using tr d v-2psdyr-k ,71ndihe information s nrypd7,d2rk--1 i vdinside 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, 3/cjvix+9a/,yw 8 7, cirlzxtd80,000 km from Earth. The beam diverges at an anglajc89dwr/ ix/i, yzxv l tc+7,e $\theta$ (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 a, ((w w5v twpq8ibdrrod4 0kwvyu99q,t a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular accel0(yq9,twd q9ko(4i vuw8r, wd5pwbr veration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If thlun) a8m62d nce ball makes 15.0 revdua2) l8n6mc nolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How man9m *:elx,i iovsn 22sby revolutions do the whee2blo2 sxiv*si :m9, nels make?    $rev$

  (a) A grinding wheel le.: lcfcy sjdfq40v6;vx45g 0.35 m in diameter rotates at 2500 rpm. Calculate its angular velcq4gl0j4f.6 v l5ev:cydf;sxocity 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-roun f1jme6,yi jg,gng. gt.s./0qr +etmf,*oywkd makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is th6,0 r,m, y+j.m jggsn.y*toe/e1 .wigtfq kgfe linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth ( 3h9qx,2bj wmgbhn w.9a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of b f0. ;*;iiud n2eefsv+ft a)7pg/tnca 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 cv k9ld*t4oyy4 so8o:7.0 cm from the axis of rotation is to experience an accelerati9od 84vc:ysl y k4o*oton of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm tsnhg*4+ f oj7j(hwe p/o 280 rpm in 4.0 s. Deshw +ojjhn 7*/fp4(getermine
(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 (px q adv9ujt21knw/+s $R_1$ 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 thd0wobt7 v8e;sjg5 +zkemselves into a slow rotation to distribute the Sun’s energy e7t jo ke+dg;wsz5v0b8 venly. 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 uniformlyjwzev z. 6c-g6 from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in thizvj6e zc.w-g6s time?    $rev$

  An automobile engine slows d/p,jlo moyaq(s l4paviv9 ;99own 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 highspeed jets in a whirling “hux 8kwf r*c4fwa:mx0x xe8wh/)man centrifuge,” which takes 1.0 min to turn through 20 complexfw 8mw*rw h:fxc0x)/e k 84xate revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm innrtj )j w:k0dtc y,1:d,qq s(c87rekq 6.5 s. How far will a po:dj:)k,cqr jytq1qdner7ktc8(0 ws, int on the edge of the wheel have traveled in this time?    m

  A cooling fan is turnezybfld.)ty l b5u/(1 ge.o lv1d off when it is running at 850rev/min It turns 1500 revolutions before it comes to a sto.y.dbof51 bu te/( lzyvgl1)l p.
(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 a*/ qbdsa8u)bos the car reduces its speed uniformly from 95km/h to 4)o*a/ ub8 qdsb5km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 600mj jpn/) srf5 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tirf 0)/rmj0npj ses 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 weighz, f *hwdurb/*,k thc1t on each pedal when climbing a hill*r1fh,c zt*bwkdhu , /. The 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 cm wide. What is the gkcxz58*5 o l7w90ghqe .xk(qzjqo 2cmagnitude of the torque if the force is exe wh.55z70 qkqqgloo(2zx*c c9e8g kjxrted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel/m )gztg+ s9jo shown in Fig. 8–39. Assume thj/gg t+zmo9s) at 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(oa0/kiz ;r q/) ia1)uwwdwpq attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculat;) ikwd pww01)q /(au aozr/qie the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of 3:lxmwm-ana.cvq cuzg 4op(8;t8c*z0 8 -cmzop v;04cw*q8:lazt .(g uc 8mnax$m \cdot N$ 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 inej 6ba:jnh/g )irtia of a 10.8-kg sphere of radius 0.648 m when the axis :gnhia j6/j)bof rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 nk k5pi wasxo6i5)+1-jml* odcm in diameter. The rim and tire have a combin-p1kwo*ldj)5i6ok 5mn si+ x aed 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 e, maqylq j.lg/ 6sx.g-nd of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculy/ qq6.l-,a xmg jgs.late
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular 0riqjpd 1pf 82 59ya r:wdyyr/speed a59pdyqyfp yr 1rr 2iw/:0 d8j(Fig. 8–42). 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 p/y.0pgljp ii up17d ty(j -4neoint objects shown in Fig. 8–43 aboy 4jlgd.u- jy0(/7et1ii nppput
(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 w oi )y y.qwkef;qmi*1q15i1evhose 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 corsdjunm0 i( 2y,g+ .vnd,gef.jrect rate, engineers fire four tangential rd,d2sy, j(gifgm0e nu+ .n.jvockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a unifoj ,0tm: hn q1oc 1vjfcgy-s):k,xf d8orm cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculateffk-)0n,:tx 1cq,1dj: c8y gsjmovho
(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 7qkia.;x9ozorest 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-.uod(poy7i1zki 6+ h7m0fpnw h(; ocjpi2u,ddriven 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 of radius 2.5 m and has a mass of 760 kg, and two children (each with k+.,dp(dpz owm7(;20 o o7niijp1cu6 hfi huya 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 ro bq6.rko// htstating at 10,300 rpm is shut off and is eventually brought uniformly to rrt.6k/bo/sqh est 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 acc1fnl.y:zi11+gjvz l +fkfti/ elerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the fore mg abg979zq .cnslr4;arm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. 9c9 4m.sazl7n;qg rbgThe 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 Fiylz. ,rs 80uaig. 8–46.0l. iza u8y,sr
(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 joa7f:g ce7,bp(bke. 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+zchi+l0xw w,uz q)(z ; 9-meuy3saa o rest within four full turns (revolutions) ande ,- uzm)i(a9h ayx30qc+wlu+ sw;z oz 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 qxj8aup(q w6a)hwy 449l(ascinertia 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 of kf7t.ti* d*e e280 $m \cdot N$ 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 withxz zw, 14wkdwen,6p) c8mzk0eout slipping down a lane atxk c8zm4 ew1kn z 0d6eww),p,z 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respectikqv*:f:u yn9yuqw 7+ to the Sun as the sum of two teuf79n vyk+*wui:q y:qrms,
(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 o8 it2dlnu ydw3asa.e4a);cm / f 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is/ 2a dd;mt8e.siuywl43aanc ) a solid cylinder.    J

  A sphere of radius 20.0cp:)mn4b2j 9t4v ugtc ws2sa- cm and mass 1.80 kg starts from rest and rolls without slippin 2ntt4:ac)cusm bgvjw - s942pg 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 vertzpc+: j(lmm8 iically 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 pole just before it hits tjz 8 mm:icl(+phe 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 two.xim ogs g5pq-zk4zr;qd+ d+0 -h5j hin string in a circle of radius 1.10 m at an angular k-ogri 0d5+z m5+;w xdsh q.-gqp4ojzspeed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of awo+m:2llrmhdozf j pgd9 onr1;1 87e. 2.8-kg uniform cylindrical grinding wheel of radius 18:lmpoolefj 1 w d.r98o72dh;gzr +mn1 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,,4h lrix5+ayv7z 2oca on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–487vxy2, raa l+zh5ico4, 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 inertia8pet4c)vp/s nse:p 7f by a factor of about 3.5 when cha ss:efepvn) ptpc47/8 nging from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rayw9)1 h,ykf54wto5 x9 pa ic-3qi iy6p jhc2oqte of 1.0 pj9yi, f9wq 532-ioy k4htoq15cyhcpw 6axi)rev every 2.0 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 ve6owo l o- 6bd0fxqmvv(8 m)i/yrtical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. Whab8mm)(xq o iv/6woof0 v -6dylt 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 3.5 ) vxcxb ;e sm vuwxka1b:y+2.*$rev/s$ 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 moment-xf.(vsg1agiz l+huv1z sm x-/3)rtn um 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 droppe (t;gyc 3vyz 9szqw9q-oe; uype(gk2)d onto an identical g9q-tz)s2yv(3kc ;z 9 o(w;uqgepeyy disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns)ji *)mh0gudk at 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;u zr9 .00yazajh9 4m:lxdte z stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertie9aht ;r0 xamzyl0zd:9. zuj4 a 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 velocith(ur4qwud9c1p og79 ty of 07u(9o cq1duhgr49tp w .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 col6 -uiev36n rfs+6tlxgn; oe b;lapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carrte-uslxngn6r oief b +63v;6;ies away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in ewa6+c*m+ jz ;namw, sixcess 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 ewaz:4 gc7(4txw+ga z$ 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 plat;tk8 en tr33kav d k jiw99kp0w ;sm81+,vgmzsform, initially at rest, that can rotate freely without friction. The moment of inertia+szknt8 kdakvw;80,w99i 1 ms3j; k3emrg vp t 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-m diameter merrkbr ( /t/m -shfrfob(9y-go-round turntable tho9rkh/r(- ( bsf/fb tmat 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 grpc;1c5qqh1bi .nn*ev4 vc 0p kugr6n.ound 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 spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spon16vq.0 .n*;pkcrgpnb4u 1qvie cc5 hol’s center of mass move?

  The Moon orbits the Earth such that the samtudc138x .rzn d0pna5+ c zqf,e side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orban3 ,utpdd + 58zc10xqrcf nz.ital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1r8:p i. g8v aqfuu2gh-drm3j4 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 ofcy/ h2(0prbxc avusm5g7y4) pp sor.lx mk, -2 radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered/sha y)24,xrprgx.2b5po(m 7py vc l0mkcs-u by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and d ,x vr*bavj.0 7c5svpdiameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to csr bj 7a5d0p,x v*.vvcalculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed ospo, ff3*eh *;h h9bk.v lsbx-f 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, werz/ev:9bp3f .iw p d-fwe rotating at a speed of 1.0 revolep v3-df/p :if9.bwzwution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile ison z5epl6zk r 4 m49nn33gs+wt for it to use energy stored in a heavy rotating flywheel. Suppose suc4wml9ksn+6pz 3 4ez5n 3gt ronh 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 illustratesyqt26 u7v57ag jfd7kv 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 surfacx2i4flzi+cfjw9. td ;e at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freell4p3hpkdr h7; y (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 thhp;34hrdkpl7 e rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on cn8:+).5d7hjibxe cbbhxu 0emqf25f the floor, and we want to exert a hori x)5d7.imbjxb nf c5cf8uh:qbh e2 +e0zontal force 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 a flat road is mau/p he80 / 6kzywuaq(gking a turn with a radius The forces acting on the cyclist and cycle are the normal y0u g / az/k6puheq(8wforce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 slingtq6ll2qgb f90s(hbg ( es3be7 of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.qlqts39gs0h6 7e2lbbfeb (g (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 4o l; b3)lliwa ik2r,uas thin rods, making reasonableo 4ll2a,ib3kl;u)w r i 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 cylindrical plates, ofof-1mwg:zik z-q1j + e -1qwkz+jzi1- : mgofemass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 theq 1 pcg- (ng2fc8y.c8tpyl88ltny l7 x 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 rgfl8-g18nyqt7c2cy8lynt.p 8 cx p (leach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bu +1yckg4wjorjpdr7 j51 lt/1jt do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 rev8 /fwotv3h gvnsot: 80olutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires havfg8nsh8 v3vo0:owt /t e 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