A bicycle odometer (which measures distance traveled) is attached near the7f0 m2+afx +xqtub/to wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicyclmx bq2ou/+axt +ft07 fe with 24-inch wheels?

Suppose a disk rotates at constant angular velocity)etk6zvz9y 5 nfp:b q/. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of eithe:e9/z6vt)5yfbnqkz pr component of linear acceleration change?

Could a nonrigid body be +-poc ql 1ve6k qt4ux3i4d nk8described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a lady0d vcl luak:3i(0v4rger 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 a-)gt wsffu cxy+**fakvlrd+ rz-*/61ls 4qabehind your head than when your arms are stretched out in front of you? A diagram may help you to answer thi *fzvqrxrs1d/+ +ca* a-)*y4wfg l6s-k lfua ts.

A 21-speed bicycle has seven sprockets at the rear wheel and threet 18usmy h6)nb 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 i86m hnu)sbt1y s 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 lowervih6;ug;ajit s 2m 38e legs with flesh and muscle concentrated high, clou6ae8i gv ;jshi;23tmse to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (ayl )sq65 z(cl5r: 73awbkyklFig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net u(kz jf 9o-/d 8eqyvo1met7 3ltorque also zero? If the net torque on a system is zero, is the net force zfoku yv3e 9 -8j7(mdtzle1oq /ero?

Two inclines have the same height but make different angles with the hou4+ gv5vzwn(q i +z3s *zqyh1hrizontal. The same steel ball is rolled dqn q1(g+y 3hvs+ *hz4i 5zzuvwown each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from znib (d1xpm (c 9x8v;yrest) down an incline. One sphere has twice the radius and twice the mass of the other. Which (ybi19z(8c;dvx xm npreaches the bottom of the incline first? 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 start o bk4k5y jlwg2 ork3o*,5g o2sfrom rest at the top of an incline. Which reaches the bottom first? Which has the greatk*o lso4 k,y 5ojr ggwko523b2er speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum aredb0zzj0q ne:dipazqh; (/2j+ conserved. Yet most moving or rotating objects j +p/zn20b d; jahq: e0zdz(iqeventually slow down and stop. Explain.

If there were a great migration of people toward tl qmycdubj5p )/i8fj( s-7h/b he Earth’s equator, how would this affect the c 7/5bs)(8hli /jfm- ybuqpdjlength of the day?

Can the diver of Fig. 8–29 do a somersault wit)q xfj ) sm6gh/2h1ooehout having any initial rotation when ojg1 fh / h)2qs)o6mxeshe leaves the board?

The moment of inertia of a rotating solid disk about an -g..i;iz1pqck.wnc l axis through its center n;p1zil.c.qci.k gw-of mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating kzzn( cey gb ;db5p1+frm511w boqo65stool holding a 2-kg mass in each outstretched hand. If565; 5fdgp 1bkr(ceb11noq oy+bw z zm 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. Howevl/trduuov+ )h6 w(b/ yer, one is hollow and the other is solid. Descri(6/vrb/ou+h ul d wy)tbe an experiment to determine which is which.

The angular velocity of a wheel rotating on a dk mv4 6b3ndi16 djwv;q(gj)shorizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If 4;k(1qb6j wgjm )6v3dddi snvthe 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 eyc 0pps.u2of7 - dcw8mnds:n7 dge of a large freely rotating turntable. What happe:2 c0pufn ow7ymdspdc.8 7- snns if you walk toward the center?

A shortstop may leap into the air to catch a b j .agx5w8un0nall and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hw8g .xjunan0 5ips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, dih44f4wgshgp *y r8t 9xscuss why a helicopter must have more than one ryrgf*84gsx4hp t h9w4otor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles y+x, ki3b ja0fin 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 v (a 2z-xlyy)uamazing coincidence. Calculate, using the information inside the Front Cover, the angular diame 2u )vzxlay-y(ters (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 from nqux++ tcgg.o(jz9f2oi5g *t Earth. The beam diverges at an a+t+2jig5 xggcoonf.(uq9zt* ngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the mot-nwb z9m)jy2r)j 2f cvor is turned off during operation, the blade9wr)2nyjv-c z mjf)b 2s 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*dau av6 -me1u.5 m to another child. If the ball makes 15.0 revolutions, what is itsuv6u -da*me1a diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km./7 wo t j7kluh7mgivt1- /n9xe How many revolutions do the wheels m wouge7jn hlt/xi7v7t1-k9m/ake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate i 09ea 4agbqpfr:rln xf;7/e5zchm-4cts angular velocix/fl;e 9nhm7ca 5g0e 4qc-:rzfp4b ar ty 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 lsd4h )zak,rp0 h(:n/ouu7 )q4ekngg s (Fig. 8–38). (a) What is the lineu,qu0kna4nr)/( 4 sh:pe7)kggd zol har 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) cui k,,yccln 0h*xbf(+v75 fo pi +m,tin its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poq*bfjr ytsve ,s*0 0y3int
(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 7.0 ttb8q - lw j-jia35jbwwz:-7d,ez3 +qp *jdfb cm from the axis of rotation is to experience an accelerationtbaj-wbzjwq e8+3j-,q5zlbt f:wdj7i - p* d3 of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center f m7k6c9ijqkm4yhwu6a5r-h z nock) 8/rom 130 rpm to 280 rpm in 4.0 s. Determini7 q9uk6-wa)kn5jk oc48m c/zhhm r 6ye
(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 radiun7mq j47di t0us $R_1$ 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 Apolspdv,e- b dzf k04;ss5lo 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 mid5sbpf4v 0 sk;edzs -,nute 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 rpm inbvka(8 -4 syeg 220 s. Through how many revolutions did it turn in this timebg 8v yaske-4(?    $rev$

  An automobile engine slows down from 450 nkif2i7*gwy,w5im*p:y r2 s b0 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 ofn-vg qhrz qrf;)k4g5e4s98sm flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachh; s)59k-grvns8eqfrz 4m q4g ing 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 diameth5mjfu9t 5b s i9k03jter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edg9hs iub3j55m0tk9f tje of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850revf3r -s8tr0af mpl0 b6v/min It turns 1500 revolutions befor- 3raflrp0st 60f8 mbve it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speeb8mv2b2we de y2zeo )(d uniformly from 95km/h to 45km/h The tires have a dia(yd2eo bm eb2)8vwze 2meter 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 65 revolutions as the car reduces its speed unifo3bn4d )77 ojud25 4rtkkjbxjarmly from 95km/h to 45km/h The tires have a diamete d7b)345 j7j doraxnj4ktkbu2 r 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-w gkl xfevzw3 7p-94o each pedal when climbing a hill. The pedals rotate in a circle of ra f- 3ww7v4zlpe 9gk-xodius 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 wid4bl (c*u8xdm we. What is the magnw(8ml* xbdu4 citude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assume0.yguzf/a):g *nzrk n that a friction to0gk ya. )uf/rnzg:z*nrque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of massxv ri* bvll, .1xizy:d(ibm5) m, are 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 the iydi5.zmv(,1* )ixv:rlxl bbn released. Calculate 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 a0upf:vx 4rn m8p9 mh338 03rm pnhux:8f4 m9apv$m \cdot N$ 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 ac5r+rs zs4*6a9hr eshijyh01 10.8-kg sphere of radius 0.648 m when the axis of rotatiore4j1sz hh0h+ir59y6ssr * ac n is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheeltmvgro ,x 9 .:38vtvbcmaq58n 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass8:5 nmg 9t,m.rvx8c vbv3oqta of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light r*ag eh66wf +iu(icpb -x dtf6)od is rotated in a horizontal *dfu6wt + i-6ef)xbpigca6h( circle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on bd -u ;c xsc+28gih.vda potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands an8 -;bhvscgdi2xu.c+ dd 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 array3*3wewicnmdwrwe: 55h j2.gv of point objects shown in Fig. 8–43 dei5wwc w*rv e5mw.3h32 j:g nabout
(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 oxyge,lp:;rb t9g (b3c7fgcjvlom u 3-3ryqn atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at tt*s9p m2m3 tqg71s kh x8g8ywuhe 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 q kt 82mmwy*91ptu s38g xhsg7m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with /pezzq, fs(s-2* ug indf-8eqbjf 8j/a radius of 8.50 cm and a mass of 0.5808p eis-fz q j*n-/ bfz/euqj,gd28 (sf 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 swings a bat, accelerating it from res)fvmndi lt ,(37rx6wfyhht (4d 6s)zet 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-round an 6(qfnz*-,urk wk,59k/c r xfcchc 6 wdy3.rand 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 child3wy5anh c rfkd.xuz6 r6/,9* c (kf-wkqn,rccren (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 atwm zvlr5r406t 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torqtw5m vrz 40rl6ue 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 acceleri87ugjvufd: ;qn uf al7x;10r ates 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 tbbbf/: 3;eayall f. ,the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accela 3a /e;ltb.fbf:by,lerated 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 hu: ng )lh4gn;Fig. 8–46g lhgn;h: 4n)u.
(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 ofkm s,nm yt-se mfq-a),+;:xwf 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 g;)v1)mvqxn p from rest within four full turns (revolutions) and releases it at a speed of 281vvp; qn)xm)g $m/s$ 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 h( xmi/yg2 ;b cuz4kmp,as 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 a torque of wq// 9p:e gf2:ftyayq280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass i,w6n fv,q oxih)h+5s7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.3qi xvos,n h wh,65i+f) $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with t s8l/ 6e -irji8i54owdfci7prespect to the Sun as the sum of ti5 fi is4c8w8rt/-dje 6o l7piwo 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 7.508jjptl3 vl,zrxe 60 (m3fs idw/j; mi, 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 se 3lm6(,tixmlzj0;/vjd p3w fr8,ij it is a solid cylinder.    J

  A sphere of radius 20.0 cm bd8 l;brb95 ,hoqhbw6d0 zw a8and mass 1.80 kg starts from rest and rolls without slb ;l w,0 abrdqd6bw h9o88zh5bipping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall a+(isz85-i pl7u pqafcnd its lower end does not slip. What will be the speed of the upps (-7+cpi 8z5alf qiuper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum ofe9hnylb: vw c 32hf0xk1ar 58z a 0.210-kg ball rotating on the end of a thin string in whx8n1z 95 k3 ahl:0fy2rbevc 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 7 z1a3 vfjln*dtyueq)sqww7z)d1;: kmomentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 kyjfw 77 *;)za:dnusq1)t3l w e 1vzdqrpm?    $kg \cdot m^2$
(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 .i4m8 x5u*)hytuh6+ug dpgr co 9xv/ga rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the d4x /gcx*6 g.+vghhiu tu5r)m8 oyp 9uspeed 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 sho-ru)num5wbhfg mt hsx( a50.( wn 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 the tuck position, what is her angular spe ()05.nw tguuxa( bmhrm5f-hs ed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.0rskr4v9a bb 80 rev every 2.0 s to a finalb8rskv 40arb9 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 verqz;;iq6 qyl jj,hbal 5;7tz1 dtical 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 j1ztl 6,aqhq;; lqjzbd57;iyflat 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(rp2e 1kui0xpa zvn;iq,-zk 6 skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momeniw67 yd.;) wqiwe/xsytum 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 cylindub yx;7-fif6 i5eh7j vrical disk of moment of inertia I is dropped onto an identical disk uffijb 6xv;57- he7iyrotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns5v)o*( hf)cgnov 3(dzt6 vj wq 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 ;vbd2( w g8 97 sdmcn1,ze.jkaj;hpdqkg stands at the center of a rotating merry-go-round platform of radius 3.0 m and q.g8 d;djmh2dv knswa17cj,p (; be9zmoment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with aph.wu lp7.yoe 5;;iao n angular velocity of 0.8 io.plp yeh.a5 ;u;7wo$rad/s$ 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 intwwy7mc-yj 525yrg 9(0u nyggbo a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assumi(bmy2 y -gj gcug90w 55yrnw7yng 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 ot6bg.n5v odj1 f 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 emhvc1 hz;3ggts2 34y $ 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 freely withoutiti: goik)z ;; friction. The moment of inertia;i t;ki:z) gio 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 standp,nl*7c c.8 fpqqa.y;;shis)q 2q cdw/;hacc s at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of lpayh)2q,qp7 . ;cc*a;.hcn;fq sc8ic q/ s wdinertia 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 ) g7vz+63 zmyt)5sb4c0m azd u+ tvwdkof 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 doesmaz )tvyk++m4bz7 ) ug6z3d cs50dwtv the spool’s center of mass move?

  The Moon orbits the Ea g+xij9 5x.cr/ pt(ygfrth 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 lattxj/ 5rpxgcy9 (tgi.+fer case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates frogiz 5 di1mk)h1bkyc:j0l d 0e7;tk6gzm rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of rad/tkap/(gdud;( 0 zcz pius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the mu c zzt0pgd ;/ak((dp/otor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, ea91- dqy lhdd0; 1 albd1mfas6ffap g15ch 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. Use conservation of energya 1dfl16 dfaadmbdpl 11hfy s5;-0 g9q 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 isvb 0qr9mlwsarbr *n)8 5m 5n)z 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 n hu2jj, od7ydx t-;3ggreat, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational colladtjdh2u,xo-7ngyj 3 ; pse 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 is for it to uean xb 5potnui,.d v00lu38k )se 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, annl8p ,na 0bx 5u)t.kuioe d3v0d should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an hx4lrd2e g0o:$H_2O$ 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 surr vb*dmd7t/jzp* a7 n7 bax3o/face 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 ln lf jzf8:7n-pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of rel7:ll -f8fnjn zease, 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 hatyx):xf5r 2a y2 s9asqs radius R. It is standing vertically 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. 8–55). The step has height h, wher)ryx5f qy2 aax9st2:se h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn wigb6/brjq)2obm taz 6 n 5*gdt2th a radius The forces acting on the cyclist and cycle are the normal forcarb2gt/qmz ndb6g*j 6)25bt oe $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 begins whir9 1eyoc pgsp1s k/ao-4ling the rock in a nearly horizontal circle above his head, accelerating it from rest to ec9soo-41ky/papg s1 a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms lskf.w/l6tzs /lpe*o14 7e hgas thin rods, making reasonable estimates for the dimensions. Then cal*ewhfzs// .gtol4l slp7 ek 61culate 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 whicmbj r x/ut8;:nh consists of two cylindrical plates, ofr : xb;ntuj/8m mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radiusn / 3qpbpp dvo3igt9,4 r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach,/tbi4oppdn39qv gp 3 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 73*: m benaqn r l4rl83)xvnu+kxvqa3not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 cvg jz( 8+d(gurevolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angul8+vzg ( cgdju(ar 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