A bicycle odometer (which measures distan6) gx fvzo 3a2suo(ft4ce traveled) is attached near the wheel hub and is designegt3ooxfz )6a f4s u2v(d for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angularxiy;rxd.lz64hbb8 fx1x(c9bi p 8i)c velocity. Does a point on the rim have radial and/or dp4yi 6ib8 xibxcr8 (b.9)xfc;hl1z xtangential 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 angulw9)7ierr:vm g4qr,w zd,v. c yar velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force?rmfn q1c0qrt,4h ,vm 3z 0we+g 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 behind1bapd, d aj1h. your head than when your arms are stretched out in front of you? ,ah.dd ajb1p1 A diagram may help you to answer this.

A 21-speed bicycle has seven sp r4s4iiac 0zjd9mhhe:12vqa-ov: d(grockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a smal:dzoe-r0 4shj(v avq9 :i4 imd1c2 gahl rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with fles8ofb5r2xv v+y h and muscle conc+8bv5yo fxvr2entrated 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,ny:4,u -ect8 cqv,ncn narrow beam?

If the net force on a system is zero, is the net ur l*v h*p*3o0o3bpo ptorque also zero? If the net torque on a system is zero, is the net f3ppv*l 3* hu*o0p oobrorce zero?

Two inclines have the same height but make different angles with. xrogbe+b6jm9/f i4 rgc).tz.7k ifb the horizontal. The same steel 6ofb+.f gc)gkt.bxrrmze4b9 j./i 7iball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rold8)gw7gkj6 ( (7)zuooho othnling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the btwonk)jd7gz ooug (( 867o)hhottom?

A sphere and a cylinder have the same radius-ptcjj77rwuhv v41 ,x and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greatev-pj7th,4xjwv 71uc rr 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 ang-s0tg sepm d/ 9qh.3p mpu0lo:,) amxmb7ab(yular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Explai la7 bm0uaysemd xhpp s, pgmmb0:)/3.-9tq(on.

If there were a great migration of,eb0s cw * 6cw5bm,:l ijfjqd. people toward the Earth’s equator, how would this affect the length of th bimfjqe,jww6 ,lcscd: 0b. 5*e day?

Can the diver of Fig. 8–29 do a sold n2dln:wrxj9c 464xmersault without having any initial rotation when she leaves the boardc2l4 dnlnj:dw94xxr6 ?

The moment of inertia of a rox+i p- v,zfl+atating solid disk about an axis through its center of mass is f+xv i+a -pzl,$\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in ea0od1homw-v-ny5 n b )-nqw rc5ch outstretched hand. Ir)-mdncbn- hwovowy n-0q 551 f you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identt s mbf;0k j-,xd8b+yxical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine b;- xxb ,sj+d t0k8fymwhich is which.

The angular velocity of a wheel rotating on ap+tx:lt9ns /fy xsql+zzt .2t :q6d-s 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 a.:6plq9 sn qt:xsf l -zz+x+tt dtys/2t the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a lj.3 3 3i/ng0dw haczvryds-+ garge freely rotating turntable. What happen0 3d dghg vj.cwn3si/+rz-ya3 s if you walk toward the center?

A shortstop may leap into the air to catch a ball and7xm)x b :rjpia2q,ld gp3cn/ 7 throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips ani),n 7pqmcg axx2lr/d3p j :b7d legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum,g5kf 51 u.tzum discuss why a helicopter must hamkut51zf5gu . ve more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following anl p.8dn82baopgles 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 coincidence. Calcu16yvqixgf0w t r ,i,ccgj 5n2-late, using the informativc,n x jc1 2fir5-tw0,gqy 6gion 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, 380,00;po8fmn2ma.yrpwe ;;dm :;hf 0 km from Earth. The beam diverges at anf m;rypemaf h.;dmw8;;:n po2 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 rots z7,,yv3c;zn+5 mnk ,yp ailoate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to ra, ,,scz; yyz n7pknm +o5l3ivest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on ag xq(ns5j s:guc,+ud- level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what isjnsucs, :+ugq- dx5(g its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revocop3 k.j1r+ 6y;oju,a8ku juanm4t8hlutions do the wheels make?,ph4jkc8u8a oj 1 ak+.6o3u mn j;tryu    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcr5 jw -b u(pn(ja)v6dkfp1qs9ulate its angular vefa 1 (5)wv9-( spjn6pqrkbujdlocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38)../gjcec,k p sc4van4/ketavmty5 ) 7 9 (a) What is the linjc)474pnvcvsaae/kck 9,tg m ye/. t5 ear 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 (a) in its orbibj+ 6g.rvmmq s)/pb+v t around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a p+lcvu : x2o+/wj g1pjqoint
(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 rot0z5lropfjgupk :/u n7d/ g,y)mkx7 m3ate if a particle 7.0 cm from the axis of rotation is to experience a7 /gpuxk 5rg,fmlj/du0 :no3pyz)m k 7n acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its mfvvh fhtt3dy* *i9l+8o,7ydcenter from 130 rpm to 280 rpm in 4.0 s.+3d tohy9 dvv, 7*ym8tflfi*h 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 radius u2zay4 dhzas8*l-th8 $R_1$ 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 gvi h ;4 fvuu2zwg:m2*(y2mv baboard the Apollo spacecraft put themselves into a slow rotation to d2ybu viv(mh wzgv2m4 fu:;2g*istribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpmwxa3nzv- -dm db5( 3ib in 220 s. Through how madz3bb -vm5i( an-wxd3 ny revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcula6, ks(v(m+awl j,c;zwkbsrb6te
(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 jev;+3+q(ewfbx mqoo x2 ts in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 oqq 2;m 3fex+(+xobvw 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 rgr5o rh//osc ns 1:-arpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have travel-/1 nsro5scrog a :h/red in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turv. ,ahr,)f m uj3(l8i1wdtwqdns 1500 revolutions befo3,uw i)h,t w dfqr1aj.dm(8lvre 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 a6; mal2fpn)omv 57e*tpmsp) us the car reduces its speed uniformly from 95km/h to 45km/h Tpef 5tpm vl*n7a o s2;)u)mmp6he 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 makea /bfq9wk3f)xhuq7k :;so zyxj y:iw6nt.f81 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.)9j .;n/fh1zkfxytwis6::qy8xo ak w3 f q7ub
(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 each pedal when cli9h nu1 jdc k35d-fj*wnmbing a hill. The pedakd5 wn9f-3*d1ucjj nhls 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 N0h2j a-yjzro igf92 l 2w4g+tdol:gc* on the end of a door 74 cm wide. What is the magnitude of the torque if aiylg- c2r 9jh: j2lfw+4 zg*dt 02oogthe 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 wheeft*ev5 +;2i2/hvrv sz5 ksnail shown in Fig. 8–39. Assume that a friction torque of 0.4 t5*isfk /+ha5es2;nr2ivvz v$m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the endlj :h0s*tjuxmf5 9j( ts of a massless rod which pivots as shown f tjs05(ux:* m 9jhjtlin Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tig9rql1;m c /7 ;vaqsv b2g)lbnahtening to a torque of 38a rlnqb71ml/vg aqcb;2) 9;sv $m \cdot N$ 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 sg u2c e3ofw* 9ba)b+ig7-ssaa 10.8-kg sphere of radius 0.648 m when the axibs u)c*7w9sb+sa2ggif3 e -aos of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The 8nbo kd;9xox4 u1ck4;8vbuhj rim and tire have a combined mass of 1.25 ou4b ;1xocdku b x;98v hkj4n8kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram bal) d(g1zcvcpl 8l on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculap)g8l 1zd(ccv te
(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 whewyqj f1*k/q)dvld3jtyo k / 1yd;yv 04el rotating at constant angular speed (Fig. 8–4210yj*tf;kvd dkqy/4)q1dlyy3 j/ vwo). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown h7ez )7dpwk6w in Fig. 8–43 abww eh6d)zp 7k7out
(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 fz0-+ mglz9 1hgu/tt avl:x.koxygen 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 q h6djq24*vvcn,fo l7m.r oh7 wn3xf) cylindrical satellite spinning at the correct rate, engineers fire fourv.2 jv 7 qowq ohhdc *nx4)fnlfm6r7,3 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? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder withyj78c 5z - . zo;ytaamjjd+kas5r/w 4h a radius of 8.50 cm and a mass of 0.580ah5a54-dk + ;y jc s./jtr8zojm wyz7a kg. Calculate
(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 swingt u/x 68jut.xss a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-rous/ npmi(2ra/yak,:zr /,bk:6yxx hs end 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 a mass of 25 kg) sit opposite each other on the edge. Calculate the torque requ bnr6/m2/yp/: : zesia yh,xksk a(,rxired 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 sh *xd:-q xz/frw7jh0zd ut off and is eventually brought uniformly to rest by a frictional torque of 1 dwd/q0f- x:rxj*zz h7.2 $m \cdot N$ 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 ball at 7 m.mc8bi/rd:ct 1.sqn $m/s^2$ 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 forearm,-c99yiag/ ckwe qft -0w0rnu5 which rotates r0-atcnww 5 y9q-fik9g ue0/cabout the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor bn5: msw vr r +)pld*rublb:1,plade can be considered a long thin rod, as shown in Fig. 8–4b5vl )r :m: u ,wb*rsnrpdpl+16.
(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 consistygzif */0ptgsq/3 f5g s of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four fully8a)qq8y 1kl)+g4eb9 xgfqce turns (revolab1 q y)+q4kxq8ecyf9g8g) elutions) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment 0(k mfp* 8cpy(*xk n*aroh-vtof 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 a2,vm s,q knqegr9ec 76 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 anvll:/ xfl-: t qjjj*parz;4 6md radius 9.0 cm rolls without slipping down a lane at 3.3 4ap-q;6 rjtjxl:jz*: f/ vlml $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as;k )yxzg49hnh9szz +b the sum of twobxz shngz 4;)+ykhz99 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. Hoq r2yp./yz urt61oq+j w much net work is required to accelerate it from re ruo2jy61y.r tpqz+ q/st to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg sta )wtk2ndisgc k1 ,/v3igj 5fl-rts from rest and rolls without slippingd2ftgis/53gln,c- )ikwkv 1j 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 vcv3 ay:5g;6zj zqqj9 tip. It starts to fall and its lower end dojvz 5jzq3 gq6;9 ycv:aes 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 of a 0.sw3yo zek8lekm)u7 (o0.sx h6 (blpn)e:l.j y210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at ap)(yxuo7 yn k 83s.)llz j emhw0sb .kle(:eo6n angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg *qv: ()dvmxw+,r nbsjdjc -(yuniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm? dmj y (q-v(*jvn xrcsw)d:,+b    $kg \cdot m^2$
(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 7-1:wr9 wm(3rcr.e rh cbslp es5j7rqis rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the spee-ec3r rcer:5 .rblmw7 sr7q s(1jw9 hpd 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 shoa y23t1cccss)ll(0kj wn in Fig. 8–29) can reduce her moment of inertia by a factor of )t2ckclsy a3 l(1scj0 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 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 rate u/);birh;uh 0 0viv adof 1.0 rev every 2.0 s to a final)0 ar/0;v uh;ivi ubdh 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 axfc/ nmbx-3(j jo,j0 luis 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 sha3jmjuf / jc0(xln-, boped 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 atj ykz7:.g 8ipce;rl;l 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 momentuvh(tl x;zr , ws:p9dt6)6 ofq7pomnf5m 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 cylindricf wo (dld.*e.,jt7zi eal disk of moment of inertia I is dropped onto an identico.d ei 7jdlzt.*w ,f(eal 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 at zx)hy11t kf43du s8 ps(erh +)wor* jv2.4 $rev/s$ 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-u 2 euiy)5( 9c c;lojhvnad/k3round platform of radius 3.0 m and3lvuucdoan2iye5k j/ ( ;hc9 ) 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)isg( yl1b*r-zozfo0 is rotating freely with an angular velocity of 0. sioz)o( -yzl 1grf0*b8 $rad/s$ 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 whit) skiyqm9g,a 2)djzd,(s w(+z w jzx+ue dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of iuw+) 9gakz2s(x ()jd qi,m z jws,z+ydts existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of *h a/4gz(uevr120 $km/h$ 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 oq x76dv 6mti)$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freeqq(s/n; wwiolzm 5-+dly without friction. The-now z i+q(ls/m ;dw5q moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge 7p 5(x+bho qcs beq(n4of a 6.5-m diameter merry-go-round turntable that is mounted on frictionqoeqs4b+c n h7(p5xb(less 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 lying on t2ldfeafe36o(m0nd r rq n0sp..0 ak4vhe top edge of the spool. A person gsepqed6a.a 3 o 00.dkf0nl4m fv2(rrnrabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the E f/0tnb0xxb6parth such 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 fbxtx060pn/b a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m//9svp q u*sl1o$s^2$ 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 acqp a(i3wn :m/ cdqbto+;uires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivere pod+(:w bmin3ac; q/td by the motor.    $m \cdot N$

  (a) A yo-yo is made (+c qw,09dma k8dky cgof two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. +yd(am kqc08 w,9gkcd 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 angular speed of tae2(qmm3 czyf(vgf --)nt(gnhe 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 8j.12vbfovkq/e(k6/m* m c hyj .0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what woucf(j1j6vm yb2v* k/o/me k.hq ld 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-polluu3v4ks6 )lvbption 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)3vpv 6l4uskb 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 illustratesn1 q apl t*0w,i4z) 7/acqterv 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 h s/ cg(h3be b5o9i3xyxg:3av 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 freely 4 bdzn7() crfe(i.e., we ignore friction) about a hz4 cdnef7 br)(inge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically oz71h)affwakw /.cqtm:c 2hc6n the floor, and we want to exert a horizontal force F at its axle so that it will climb qz/fk a162mc hwch c)t f:.7awa 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 making lluzeju97u7)my /.i lma n31 ca turn with a radius The forces acting on the cyclist and cycle ame /zyuimull3j 7c9)n1u7 al .re the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a slxygw vz08n 4n:*09vl9oipd eving of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rezx gv0nli*ewypn9o89vv40 d:st 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 cylz oa,)u,*rni2 ss v5i(7dvgwv inder 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 againavdgsn,(vu soiw,7 2z)*i5 rv st her body.   

  You are designing a clutch assembluq ixt.e9d :: sn3l e5w-de/zqy which consists of two cylindrical plates, of mastilesn::/ 3.zqd quwe-59 xd es $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 y:x5epv .plqcm1;m+)hu ,3s8q ndjlp 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 lo.unlp qy8v,1s: pmd)hqc;+m p5e3lxjop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do bthi68 r67vnfin5ho2 j n1eg9not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its qw(f zj4q h4+02mxy grspeed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wa0zj2g4 y4xwh mqr+ (fqs 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