A bicycle odometer (which measuredg bs*oy7 g)u2pvn19as distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you useoy 27*s9gdpvn ub1a) g it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doe6l1 rwo9b( ;;jaayhgp s a point on the rim have radial and/or tangential acceleration? If the disk’s hpg;w1aa9lob; ( yr6jangular 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 l;sqvx 34 e kdfd7wv.*angular velocity $\omega$ Explain.

Can a small force ever exert a greater tv:dcsv r3 /(6 wr hf97xktni2zorque 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 you l(x/o: 0:9 hk)7nr )elbvlclufunc/sr head than when your arms are stretched oull9: x(u0)evlnoccnbl r7kushf// :)t in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprocketsd0hf 8(suap d; at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it hardpdf d8au0;sh (er to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on qdw2po+m9 ..vqq v 5.bwrk 4mmbeing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). Oro+m9qq2 mp 4mbq.5 vv. .dwkwn the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam?*o,j n;kz*foj7/oxt l oz *w.d

If the net force on a system is zero, is the gxm(byl4550uu8ma g5akb2 tpnet torque also zero? If the net torque on a system is zero, is thebxb(ulp ytg amg205a 58k54 mu net force zero?

Two inclines have the same hoc70aekukl yfdc 71a0gck49hw8 6ho ,eight but make different angles with the horizontal. The same steel ball is 4a18 7c oh0ga09c,ly ek7wuf dchk6 korolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneousfhzv 7ugd79* v-gr uvig8 m10vly start rolling (from rest) down an incline. One sphere di7gv8z7g-v vf*0hu vgmu9r1has 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 bottom?

A sphere and a cylinder have the same r*veue1n6lbs z spy3 ,jq: .r8nadius 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 greater rj1npvb6:u elyrqs8e. nsz ,3* otational KE?

We claim that momentum andqv9gcggkt1/q*uj ; * t angular momentum are conserved. Yet most moving or rotating objects eventually sloug;g* cqt 9kv/ q1tj*gw down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wou77fu+(1ybgp 4 (+nivowgzvp kld this affect the length ofinb+kg7f1 w(yvv og+u7zp 4(p the day?

Can the diver of Fig. 8kkinj4/kv ju xj8b,3b,lfv;,2 + ocxj–29 do a somersault without having any initial rotation when she leaves thekock j 2,kbu fi j;l84jj+,xxvv/3 ,nb board?

The moment of inertia of a rotating solid disk about an axis through its center ufr 7n -h)k0qrpblm;1of mass if nkpb 1 0;7q-mlrhur)s $\frac{1}{2}WR^2$ (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 ij5 c09bawk.lfii 3 ab6n each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or ibl 0i3. k5bafcj6wa9stay the same? Explain.

Two spheres look identical and have the samemzsp ( 6;f+o*z 1 g8nvgzgb.er mass. However, one is hollow and the other is solid. Describe+*( epmsz6rgb g.z og;z1vnf8 an experiment to determine which is which.

The angular velocity of a wheel rov(hx: 12+sf mf6tfa wgtating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?x(+ f621sfht:gmfaw v

Suppose you are standing on the edge of a large freely rotating t a; ukmij-(d6durntable. What happens if you walmd a(-u;d6kji k toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. 8hcghk ue5adr,gw47y zn5i 8*As he throws the ball, the upper part of his body rotates. If you look quickly you wilhy4c g87 z5rkaw ugn ,*d5eih8l notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law.x 0)yaerxm-im0 r i6a xw:m.z of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or xx0rrma..i xm0iz:)6 ya- w memore ways the second propeller can operate to keep the helicopter stable.

  Express the following x + ze* 1n g0, 3jodbwwl/c(diaj/)lhtangles 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 amazinq 0*lml 8xbvg :h7gpr1g coincidence. Calculate, using the information insvrl g7 1qgl8 bhpm:*0xide 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 from Earth. The ben1 9j kqg;j:hln)*byd am diverges a1qdyl) 9hjkj;:gn n*b t 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 at a rate of 6500 rpm. When the motor is turnrxoyy9oc9/*fk+z wq. wy2k:wed off9k.w 9oc wok:fz*wyxq y/ry+2 during operation, the 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 l aze ev7-e 2bw h55zpecqxm kw no150c0,te42sevel floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diameter? -oxh1w5pk0qbt2v5awee,e n z25ezsemc 0 c7 4    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions0.i gbtkw (ysl imqwc65+;d;f do the wheels mabk;ml;6w.+0 g(ti5cfdi qws yke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angupj(zw-. j uayk5hav v: (3vmatbxen72/1*vxqlar vjvh/wma: e21 k b z3((avtajx y*uvnq5pxv7.-elocity 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 y dplx74g7:bjzh wsw 0:j*bg6 s (Fig. 8–38). (a) What is the linear speed ofzgsl: pxwdw 46gb7by:0 *j7hj 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 (acih(-i;p,f qtl ,nj*o) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of mmf 5+:y v wkk*hj4jt+a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a f4meev(. x0wjparticle 7.0 cm from the axis of rotation is to expef jx .(ev40mwerience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel m fuo8l)t9rz+ ibu*2raccelerates uniformly about its center from 130 rpm to 280 rpm io+l mrb)9it zr*f28uun 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 radius yimb+ o,1y,) cqaqch7$R_1$ 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 aboakxgu bq4x )19trd 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. The spacecraft can be though)19xqbgu xt4kt 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 acceleratet.f9r vbd53rli3 p5p 5hwdg) as uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this timea3dr5 ip3tvf p w.rl )b955hdg?    $rev$

  An automobile engine slows down from 4500 rpm to oqbks432s2d( 8dxkei1200 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 stress71.sz6s low vfes of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions beforefo lsw71.vz6s reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 wx2bb)8*ur1 lon pcu3n)ykb,rpm in 6.5 s. How far willk 3 l*p,u8b)by n cx1wbnuo2)r a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned c7svwlx/p7nkh fae6g56a la x k7+07d off when it is running at 850rev/min It turns 1500 revolutions before it lsva +5n6ehaaxgl7d7wfk 0 6p7 c7k/xcomes 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 re-ah0xva / 9pv ec13hcxduces its speed uniformly frhv/a0c 3h axcp9-xev1om 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

  The tires of a car make 65 revolutions as the car reduces ;rypa1eyd , i,n gm9z-its speed uniformly from 95km/h to 45km/h The tires h1r -amd,ngpe,zi 9y ;yave 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;n)q;oqaq 0ys bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm;nqq oqa0y;)s.
(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 ip .xv)r0.pa has the magnitude of the torque ix r.hp)pvaa0 .f 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 thgoym +.w a5x(voaj b40t6/vtw e wheel shown in Fig. 8–39. Assume w+( 6 wmob. vgv5ya 0/otaxj4tthat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to 5tk6v(xzw9 idthe ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal pt9wz id(k 6v5xosition 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 tightening to a torqxmob9sy;e ;d k )f3uy.ue of 38 k39o ;xfm; eby.dy)u s$m \cdot N$ 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-kq,sr i--0d.wxnu0kzdg sphere of radius 0.648 m when the axis of rotation is through its cens0i. dr -uzkdxn0q-,w ter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle t9tj9dsn n 33ewheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignor9 jsd3tntn9e3 ed (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, *mwm/; dc+h emr;cb n6light rod is rotated in a horizontal circle of radius 1.2 m. Calculab 6 ;r/mcdc m+me*wh;nte
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating ay e 2i(w8hp(ggk1wub-sdq w 5)t constant angular speed (Fig. 8–42). The friction force between her hands(pbhwyd-gw 2k) w 85ue(qsi1 g 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 shownhvzgt o3 aei,09)7tc lnte v;. in Fig. 8–43 abou3otvz;lt 90eev, 7a nc. ti)hgt
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen af i)v/ g5dp: 7gfcz:5dzvllkv*jpy2 .toms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the bu7r i)u24 1dlawsfm0correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each roc bwluri2s d4u107 fam)ket 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 a radius of 8.5jqbg3q(zsbg9bf-ru r/9 ha+a 99nm: r 0 cm and a mass of 0.580 kg. Calculatrh/jb s b(n+a -9rfu3g 9gaqrb99m:qze
(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 v.yrq a)/x-xwoz 2; ix$rev/s$ 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.4 yt wbsilotki/o 4l *.v:.rj small hand-driven merry-go-round and is able to acc:wto.k4.r yl *vs 4tob.ili /jelerate 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 required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventualt6g3r*c dthy/oo:; t wly brought u6chw *tt;/ gr3o:tdyoniformly 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 accelerates a 3.6-kg bd y;2pvtbz rv (*4e opw9*ds0jall at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, whichclj1*4(quu t t rotates about the elbow joint under the action of t tuu1l4 (tqj*che 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 c-lil2r 0eyzkxqsm9r.0.9 ji.(vkm k k onsidered a long thin rod, as shown in Fig. 8–46.-0iyri.. v90kl( m skmkjrzq.x k9e 2l
(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 consistsi g0e 9d: ryf,6aoao1q 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 g*jw;3 roe.fe 17* ujdkqg+ebt v/ j3ffour full turns (revolu31*v b ;f 3d.jgk7geowj*t+/refu jqe tions) 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 hw.,jt g2 h)qg2b4g1omv*4ztey m pk-i 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+sv:fgu mxb9 / x3+pmf 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 f;kvc+i cvkq i* 4gny*dt/s;3radius 9.0 cm rolls without slipping down a lane a dkqg;t+vi4cs* 3*cy; vf/i knt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of t(k/l g2iwsmp 5jdi27x.a j;jnaijj7x; gj5ni .l2 2m(kp/dwswo 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.owmbcp2 6wd .-:ixo 0a7 vvs;l50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution p w .-:x 7i6ospdo0mbl2vc; waver 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from ro c1b-1hgkqy*3ag*3 oerhgmr gb;)vmbaw:.6 est and rolls without slippingh6 y *3g.*r1 vm3hmwg baac )b;goo-qbk1e:gr 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 vey:sg+aeth 1+nd-z5l hrtically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper endn5:-hl g1+dtz yha +es 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 the dua g/dsqb*wds) hl;d 2/.s*z0f,f gjend of a thin string in a circle s/*fdgl .hz/s s; fdg bj2,d )*du0qwaof radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of z7w 1 2vrps(rja sxmxi1m7:k.a 2.8-kg uniform cylindrical grinding wheel o2 sw i(zx .sar17p:r17mjkv mxf radius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his ae4d(c hp6 u5hside, on a platform that is rotating at a rate of 5adpeu (4ch6 h1.3rev/s If he raises his arms to a horizontal position, 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 shown in Fig. 8–29) can reduce her momy5 hms tt hosm90 z.o01j,kcu:37ief b/a*v cment of inertia by f 5,m h 7zm3h/ve oa1uo0ib 0st:y*mk.ts 9ccja 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 speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin q f(-srzu3xq,rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of 3xfq r,s-zq3u( $rev/s$ 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 veyns m n,jq9)5 ux,otd(3 kaav;rtical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after t(3;qm)uvayj kdnos5 ,9 ,x nathe clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skate/fyj ml7lb6 ;dr 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 angularhoshr/* v/y3 n momentum 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 xa 6jh*tbd 5z+moment of inertia I is dropped onto an identical disk rotattjz 6db5ahx+ *ing at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnswyv(n c1, vyf:7 aea2f at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round pth94)os t jv6hlatform of radius 3.0 m and moment of ineo6 vj9h tt)h4srtia 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-roupvb8(m-1 d+)y qo)zucuuc )hsnd is rotating freely with an angular velocity of 0.8 ) h81cd(uupz) u)qcos yvm b-+$rad/s$ 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 white dwarf, losing about halfuft w g2*/a,mii2 *nlz9uxin, its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer,2t,zma* /li *gnuu2nifix w9)$\approx$    $rad/s$ (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 inu e a ob.aebys81/a59x excess 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 *n2lsh1zbde t;xm;t nwqaro k6,)i4 0$ 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 withw:ytju-.(uf vah ir3- out friction. The moment of inertia of the person .-ij3wvu-f: uhytr(a 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 ati s,;d2.jq8l pbbf:ct the edge of a 6.5-m diameter merry-go-round turntable that is mounted o :,tcf;d8bsq.l 2jbip n frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the -,acd(p(w q aiground with the end of the rope lying on the top edge of the spool. A person grabs the end of the i p(qwaca(, -drope 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 0+i/x wcnw -eg/9rty oEarth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axiswci0y ng/t/oe+- w 9rx) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from res75 lrp)tzr: ijt at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m ac 35,wmuv jda xb v17s81.anycdim h(+pquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque ds+ v d,b5(dyvi aw71 mc.3uj1xan8phmelivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical di,q001zw p)nfb1m ml g-mjy4k1.vz xuhsks, 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. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-mnfjqk-ym. )1zg, v bxmu4 m01h1lzp0w-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 the rea9lgqw,f*e3 o lr 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 :rwrj 5xr (6d+j8juy xqd/lsc5w+f -vas great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational 6j5x:dfr swjr(j5lv+y/8 wc -+druxq collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution auea vyejwl78:9r aw7rxd79at 1tomobile is for it to use energy stored in a heavy rotating flywheel. Suppose su7vwr e a1 ly a87dw:ea97jxrt9ch a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an a/ltk: 8r kyiydsc +3*$H_2O$ 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 horahptjd84l(juu zbt+ b:6wd.2 izontal surface 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 freely (i.e., we irhzjms0uq;w/. e;bg hc* hl8 1gnore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration om qjles/ h 1u0zr*c ;;gb8.hhwf 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 on5u/4ea2wgey6r5ihfz b u 0, ky the floor, and we want to exert a horizontal force F at5 uye/fzh6ag ryu ,254b0w eik its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is makingd2p+/hzlnju 9, wnj5l a turn with a radius The forces acting on t pnd u2 wz5jhj9n+l,l/he cyclist and cycle are 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 i q9v.dr5jjk*y nto a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat 9jyqjv d.rk*5, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin ro jx;f 9p)g5lk4uzn0 je4jri,s ds, 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 tightl;,0zns jj jpx)4lr gfkui954ey against her body.   

  You are designing a clutch assembly which consists of two cylk-n/y,mlo8ouaxehyx3 wd ./, indrical plates, of my/x wa3 n,8k oye.hu- omx/ld,ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rollsla0qvr 0hh8 1u 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 poin0 uqa 1lh8h0vrt of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume1e0r,za k qq3u jj;)2lwzg8wz $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speeuz wfj+xw1y5*bxo /3g70g)vbzp*m ned uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire+uz gzwv35 xypon w e)*mf701gbbx/ *j? $\approx$    $rad/s^2$(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