A bicycle odometer (which measuresdngu0,s- zo8z distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it onu,8 0nzzodsg- a bicycle with 24-inch wheels?

Suppose a disk rotates at constaaqew :5q/r,h int 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 tan :/i5q,ew rhqagential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be8dp.k*rxpzl :2dx2 t vf/c:id described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a gd01 va(cc2; yifgb5fj reater torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind your head than wzv+f:o-:j gr1natj37j g 6sum heg6j+ oatfrz-un:m7jg j: 1sv3n your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has sevjt ;,jhpea 8k)y0m8 mrl /bwz:en sprockets at the rear wheel and three at the pedal cranks. 0 ;kblp)jm8w eram8zyth:,/jIn which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being ableeowlw*r2lc3z- /z/mj 4d(pw g to run fast have slender lower legs with flesh and mz2 /delmz o 3pcrj(4wwg-/wl*uscle concentrated 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. 8–35) carry a long, nar:cjff:2js 86x9it+hsz smbg i .bb8/lrow beam?

If the net force on a system is zero, is the -kgz 8liy437ehjx .8xx 2mgjknet torque also zero? If the net torque on a system is zero, is . jxg4zg3l8x- kkm je 7hiy8x2the net force zero?

Two inclines have the same height but make different angles woemuga)u6y l c -;;+vfqc f3h2ith the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of c+m)-hf ec vu3q ;g2;yoaful 6the ball at the bottom be greater? Explain.

Two solid spheres simultan+ey z3nwuz8 x,x 4gf)qeously 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 greate fn)qe,xy 3 4uw8zx+gzr speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have th-hq *tfmku3 dg9z-0k se same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the grea kug- 9mtz d0q*h-kf3ster rotational KE?

We claim that momentum and angular momentum are conserved. Ye5a)2nh.6 / ffnlu- ;ikvxrmqrt most moving or rotating objects eventually slow down and stop.q5an.26 /) n lk;fm-xuivrhfr Explain.

If there were a great migraksgs6 x8no6g9lzx h uw/ (p3h2tion of people toward the Earth’s equator, how would this affect the o8nh6hxl wg 29 3xugk /ss6pz(length of the day?

Can the diver of Fig. 8–29 do a somersault without hav k5l:g/t/b8dil1 ju lsing any initial rotation when she leav5g b1i/k:lljs/l8 t dues the board?

The moment of inertia of a rota poy0de/t,l) sting solid disk about an axis through its center of mass it),epd/y o0lss $\frac{1}{2}WR^2$ (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 rotao.b (fjq: g. 0rcw7d pmk(ccv-ting stool holding a 2-kg mass in each outstretched vb.-r(pdw0c( cgq7 :.c omj fkhand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However,hyt g l1k(u+-w one is hollow and the o+uy-whgt1 l( kther is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizon /cbg6h;c 7hen3ffsn. tal axle points west. In what direction is the linear velocity of a point on the top;. h7g3fs c/henc fbn6 of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turuz*, r tzfo;zn-)y8 m1suo:jbntable. What happens if you u :jnbszzoyr1*f-, to)m8;zu walk toward the center?

A shortstop may leap into the air to catch a ball and throw it vjco7yltg(n 4r-hk 65quickly. As he throws the ball, the upper part of his body rotates. Ih(4o c5-ln 6gjtvky7rf 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 hel8m bzt-tpy :vd2ru v)-icopter must have more than one rod-bvrtty m:puz2)8v -tor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following anglesgk/:pyb(ra f: p:b o9l 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 of an amazing coinci nv 65; dbty1jui4cm(gdence. Calculate, using the infv tn;d ub56(iym1gcj4 ormation inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km fi4pmzt:r.xjq4 n+l cl8 c+q)0 cl cfl0rom Earth. The beam diverges atqcn+.+l 0 cp4i l0):c x4zf8j qmlrctl an 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 a.lm9 b8laeos2nu4 -i snj r: ze8z8t6mt a rate of 6500 rpm. When the motor is turned off during operation, tme9ia 8uos .ltrn :8zjslbm n8e6z24-he blades 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 child. If the b9: dk+idtsf)3 dhrir .all makes 15.0 revolutidr:3s dtif.dk i 9)r+hons, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutior xh0o9p,m8tb6x z* dans do the wheels m azd80hx o69mr,tbp*x ake?    $rev$

  (a) A grinding wheel 0.35 m in diame0t ds53htx;v(zk ,uo hter rotates at 2500 rpm. Calculate its angular velo ;dohzv3tk 0s(5txhu ,city 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 w1m b.hhnuf3-jghc h0:1h6igs (Fig. 8–38). (a) What is the linear s.h h0g -mciwh1 :6 hnhj13bugfpeed 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 onn uum7w ndgf33g5-+ 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 of a poi g, 5dcy3s*vkyrujq * +-q2miwnt
(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 roa )6/ 3w /-dssfkpzq ttcj(ya8tate if a particle 7.0 cm from the axis of 8fatk3/pjtsca6 ydsq/w-(z) rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acce tigjg5ny7.n 0lerates uniformly about its center from 130 rpm to 280 rpm 0 itg 5g7n.nyjin 4.0 s. 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 radiuslhuf*t;zp uw( uex1:/ eo7m/j $R_1$ 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, astronautssh2myirrd/1+s2 h3 :jl(dm at aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval.iys a t2/j(h2rhm+3lddr1m:s 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 from rest to pzqtj2h. hnw8b y0ke32t.h.j15,000 rpm in 220 s. Through how many revolutions did it tur2b e.jpht.0yt2j zwhn h q.3k8n in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in )q,qk7vm c)ey2.5 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 “h+qdw 6i:7x,gcaq fy56yfmh 8vuman ce ,g7hmq+fy:w x56qcfyvad86i ntrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerz pd n-l1 qaqkca p0ma/xj 0v5q76*e9wates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a pa/- pq0x56pvwa qn0 kj cla7*9de1mzq oint on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1 3wh,jp hh*4it500 revolutions before it comest4*3hwj,hp ih 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 d.lbl0u --mt-.x d qhcuniformly from 95km/h to 45km/h The tires have a diameter of-.qx -tb d.0ulldc-mh 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly frd *sn 2bkmw9j:o9dmk:bnjsw* 2m 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on*gbhf+ te6oaj-1y cft p+./)xm1a gva each pedal when climbing a hill. The pedals rotate in a cv+ etx-h tmo1j./fbpa y a+gfg)1ac*6 ircle 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 p h v88ngrv kl9psa 2qn 2oe)c0an-+o)magnitude of the torqv8) snap9qvkh r+0pgn8 a2n olc-2oe)ue 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 wheel shk9,ebnzhn8r+-vao xp y6k 4-1pfyw * hown in Fig. 8–39. Assume that a friyxk-a - p8h 6n1*nkrf w4e,9zoby vhp+ction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless--o,krak8, gy hl /z-0kz e0pvl+ptf s 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 d-0/s,-tkv kpgz+y l,fez p-hlrok 80airection of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening t itp6) bq( .pu8ykok5jo a torque of 385qj(p it k8yo)p k.b6u $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius //pfa8+2em9ca+tang3b g q-e t jbnj8 0.648 m when the axis of rotation is thromj n - 38epnect 8+f2/gaaq9/tbajgb+ugh its center.    $kg \cdot m^2$

  Calculate the moment of inertia4 pjrcl kqa0j3d 3phm7mht1yec4/:ll*0 * slc of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (whlyh cd/073 trs:0pj4l 1me4cpqmk jhll*3a*cy?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotate)) hdx)/cpohk aw* x1d6c1mxz d in a horizontal circle of 1d1xcm/ ) d)c hz 6p)akxwx*ohradius 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 a potter’s ;i6 g uh6zkxeie )(qk22tp egpa) v5a/wheel rotating at constant angular speed (Fig. 8–42). The fricagv 5;p6 izgxh6eu (/p2i aeke)t2q)ktion 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 shown in z:+fzlcxn1 md/ *uwc- zcke-/ Fig. 8–43 aboumnl/czz*x-z e c /c-k1w:du+ft
(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 consist3 0hhwz, 6vsbw(vu1e xs of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical sattp(q tkzfs5j 2 *v(x6 9eaxt7mellite spinning at the correct 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, wha6f*( tt pkjts2a9v m75xxz( qet 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 ):rlqm5 knprlmew,l 46 m7vv/a uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculmkw:l enqm 54/r7 ,p) rml6vvlate
(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 k,c5 gt .p b.l7s4di.mqmm 1p:7 tcxyu$rev/s$ 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 handjpynj( unidk/u; l:6j .,2db n-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 d: y.(blj/, ndnp6n;2 jjukiud isk 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 adoaocbpzf0g2qr. s++ 1/4zzot 10,300 rpm is shut off and is eventually brought uniform p.cr sob dz41o+/ 0qfgzzo2a+ly 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 accelerate 9ypw+v3wswh :s 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 balm b4b vaeh 9hi-tn83y,l is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of3-by ,vb 984ehimt nah the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered aap/+x t:fcsb, long thin rod, as shown in Fig. 8–46.c:+bx,s fpt/ a
(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 cze59yy cuq+-0vqpj t)ya ju.) onsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within f 3i7z pvz z*)t-ejvnx0our full turns (revolutionszzi xvvn 7jez-3*0)pt) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has zuu g)1rv ga6.a 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 develops rv r,ymb(n.0ia 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 xa.1m /72u ptbb:6 cxssh9tav,8n agcdown a lane at 3.36ag7s . t:cx svhu xb19/aa28,tcbpmn $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to t 3gbs/)ro1* nee7p/pkwc +8isobqu1zhe Sun as the sum of two1/e7*+k3cr1 sgou iqbp)zsbn8ep/ w o terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much ne( *s9pf9cigl3 ( ahsudt work is required to accelerate it from rest to a rotation rate of 1.00 revoluta(g*( siu hsl f9c9p3dion 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 r1i7 gi(szvw h6olls without slippingigzh 7si6(1w v 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 balance 885 :lfhxurn8ru1yuyd vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper en yxluyu88f:rn1 ur 8h5d 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 fcjop4 c:bco (e* s.+fpp6gn/the end of a thin string b 6cfoecf(pgp/pc.j*: nso+4 in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momenfwv,h64kz4 iu tum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500k h4w46ivzfu , rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a b n c/kdibzvs5x0x21org: 3n4rate of 1.3rev/s If he raises his arms to a horizontal position, gx3zsi1nxo0: rcnb4d 5 b/vk2 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 show( n6yt(hdr hi8n 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 1.5 s when in then(8th ih dy6r( tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotatuh1 5zmwf1y e8.e y)qcag n zk.ua.z9g.u.p2dion rate from an initial rate of 1.0 rev every 2.0 s g5cwyuf m gp.ue 1 ua.yhdz2n..19k)q8 .a zezto 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 cendo296jmo p re(ter 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 thenjd6 9 orp2(emo throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at up9v3s;vaoq -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 momentue5) ju p4w0iqrm gt4e5m 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 br8i oszq4 (mv4yz93te0jcg5 inertia I is dropped onto an identical disk rota4iveoys5 j 3 (9 zct48qzg0mbrting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.c-6z4 u.+qs/laoi iqk4 $rev/s$ 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-rounu:w*l54fpa u xd platform of radius 3.0 m and moment of inertia 920 ua:u lxw p45*f$kg \cdot m^2$ 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 velu:yjeersmu7 jb:h 8 2fg3-.opocity of 0.8 jpy-82 f o jh.3s:reum:beg7u$rad/s$ 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 evenlr+ch:,/.0.c2t(jqt lig kcuq ieeq0tually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rat.qlqcic ike.cje,h+u:t0gr 0( /2lqt e 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 uc- ri ;6hv1t0ng)tpcexcess 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 *ky6z7be9t oo$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can n8:(.oc q,cl5zqih y.o -sokwrotate freely without friction. The moment of inertia of the person plus the platform yqn z s, 85c.ioo:qch-(kowl.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 merew h8ysj qceb*4y 5cf.-jz53m f8zpo-m5d)nary-go-round turntable that is mounted on frictionless bearings and has a moment ofc5 -sfy)yn 8pe wom5z4bjd-*mhjqcf 5.ze38a 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 r(c g1xn keyq6:ope lying on the top edge of the spool. A person grabs the end of the rope and walks a dis(g: eckqn61 xytance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth suci)o8caa;; :v ;yngto qh that the same side always faces the Earth. Determine the ratio of the Moony:gt;oqcna); 8i a;vo’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 from rest at is2tnu;r 02 srwi2d2va 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 cyli*ot2uo; x/p sonder of radius 0.20 m acquires a rotational rate of from rest/o; *xo2 usopt 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 disks ;4ijhzhkkl96v ;we n(4r3 df l(2ouog, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter ndjl;k h(l4u;of k(z4iehvgw r9263 o0.010 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, hok::os7n,4b7cj + yhtxvho jy6w 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, were rotj u)2 mfwewq0;cx s lp/or+y81ating 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 wos;ywp82e/x0 co l r+)j1wfmqu uld 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 a0 jhqb7k: :mba low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able toqbj m: :0abk7h 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 ab3yee ad6s zri 8:hr/:n $H_2O$ 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 surfacephdtu0dbtc0/*fgn13su e : ; zl0o )mo 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 frs u+dvg1y y)2yepk(xda ;:z4seely (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 mome1ddsesp )z2xyk(u + y4v:;g yant of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verticalmirp. 85piq*j ly on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–5.pm5j i8 *qipr5). The step has height h, where h

  A bicyclist traveling with st)npi ;du fq/gt6p6jx 4wo2: mpeed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the wxqjptu pg4)t6md: n; f6/i2onormal 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 w ;edwe7v3 z8(xt7fqxx: io6j begins whirling the rock in a nearly horizontal circle above his head, accelerating i3j f e;wox: tw8(diex76zvqx7t 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 bod 2f/batg0 ksb.y 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 arms held tightly agask/ b. t2bga0finst her body.   

  You are designing a clutch assembly which consists of two cylindrical platu3k. nijx-ifjw )l. ;mes, of .3f i);xjj lu m.w-kinmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radicijewij- j ;ri ix6z).sy54r(us r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h th.ywie)4sxjrj;jzir c 6-i5(iat the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not as o2gv:k4-(.dh zf1ubuwi .jr gsume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car regad2f-g.k9 5cq (6n vzfjq-z 2no6mrnt v0,odduces its speed uniformly from 90km/h to 60km/h The tires have a nkzo g5f-q-v 0cjfrt vaog n,n md96zd.22(6q 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