A bicycle odometer (which measures distance traveled) is attached near the wheel1s2if lo.9 ecjl5hug+ hub and is designed for 27-inch wheels. What happens if you use it on a biul 1fg9o52. jlehsi+ccycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a pftw/ lz9 x)zmiki) b7o4-lt-toint on the rim have radial 9l o-z/tbi) i)mtwt-x7k4lfz 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 either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular vtu7g6cmv*x 1 )xebt6delocity $\omega$ Explain.

Can a small force ever exert a greater toe en/6qbje,;k rque 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-upalkk9n(+zm j60((cm6rfx* wpqp wc ;m with your hands behind your head than when your arms are stretched out in front of you? A diagram r wcqm*06zk(n(mfc pk; ma69+pl xw(j may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three qk- 3kbn4j5eju q-g5v at the pedal cranks. In whi j v3-4q5nq j-k5ubekgch gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend o v og23ghahe+rp+1 ( 6sqwky9mn being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis ohpmvw 96++gkgah r2oye1q(3 sf rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) cqx:8 o)m( q-q j j(jtfoy)ez0yarry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If3qi x6v kopa;* the net torque on a system is zero, is to ;k3qxipa* v6he net force zero?

Two inclines have the same height but make different angles with ;ayx)e ;768ndeciw 5k sge gu0 5*xwgwthe horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the e* ;)gega 5eysw 0iwxx d;w6gu58n c7kbottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inclin ir+tjh:5bp woj1s8,ee. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first?j8j:w bsr1eti5ph o, + 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. Theyd slkcw558 8( 5 q;szrlkvddp1j ;e4uq start from rest at the top of an incline. Which reaches the bottom first? Which has the4ds1el(dk ju;l8rp 5;vz d8sqkw5cq 5 greater 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 are conserved. Yeti/enu9ls7+f2u y h) :km zz0 lwy3snu/ most moving or rotating objects even7:2 +kl mn/9zl03nw/ )eyfushizu us ytually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, hoqw5y .:za1fe xw would this affect tfezw 1a5.y: xqhe length of the day?

Can the diver of Fig. 8–29 do a somersault without havingq g8htf5z- i,*( gco(ofyo:ld any initial rotation when she leaves the b -iohtqfo,o8z(y*fg:(5 g l dcoard?

The moment of inertia of a rotating solid disk about an axis through itsksqj05g-, cpb 9kpb7qh yuh49 center of mass is - p7by9csp5hhgjkuq 0,9kbq4$\frac{1}{2}WR^2$ (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(s4zltp *+ngj rotating stool holding a 2-kg mass in each outstretched hand. Is glpz+* n4(tjf 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. Howedgdz r2c-bujmf5dc45f5t6ndw5 5: hy,9pn gever, one is hollow and the other is s5en-g5d fd 2,bgd 9cn6z55r5p:t mychdfj 4wuolid. Describe an experiment to determine which is which.

The angular velocity of a wheez d2fjtq4+uwc bnk0 *2ggr*n7 l rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of t2 d7nnf+ctr*qgjz2k0*bw4 ug he wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a rlg, grvz u* uld982a7 y5y*q:ujh7h ilarge freely rotating turntable. What happens if yo,7u59u:7a8q zlr u*rg liyh *dv yhg2ju walk toward the center?

A shortstop may leap into the air to catcrbu ;+bq4pud 85ff3suh a ball and throw it quickly. As he throws the ball,uq5+s;ur38bp ffb4d u the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law v)v9j qw1+k tzix.8 xeof conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the hejviw 1tqxev )+9kx .z8licopter stable.

  Express the following angles0rl5s+l1 hh3zko 8 5zobalysk; :t8ay 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. Calculfi dik( 47b)vwate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen bf i7d4vi )(wkon Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon,+epg,w*0jfv1eyb i fy8 cy9yuo0 w/*x 380,000 km from Earth. The beam diverges at an angle8f*b* pvy0icyu xofw0+e/1gjwy9, ye $\theta$ (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+muj;a sf(/0nb.kd9ky7ewrzrt+da . of 6500 rpm. When the motor is turned off during o(njty;rd .k.sm +a0e k w/zbr+7afd9uperation, 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 level floor 3.5 m to another child. If mu6o+s 35 ashj8ci tjv;0s.wf the ball makes 15.0 revolutions,s s5ia8h+ostm6f uw3j j0.vc; what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutionk 8wxf 0yimqpy,4s 7,0i qlk(ss do the i ,qyfxsq7w 0mys8kkp04(l i ,wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Caunjwvt c ln9s6*ciwo )98)5hrlculate its angular velocity iolu)5 i9 s )ncc wv*jn69whr8tn $rad/s$ $\omega$ =    $rad/sec$
(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 msl4a q92ypiu 1jldjb. 9 f/oq4akes one complete revolution in 4.0 s (Fig. 8–38). (a) What2 j9s9 dyl14l 4q/iuof.ajqbp is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (nllg(napy :ie ;n *+f9:6k bzmr9go/ibl vl6)a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed+189acycf in p8ookt. of 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*og8: ynd0xt ck q7o lhn)zr6( if a particle 7.0 cm from the axis of ro 6n lrx:z0) 7odnkhyt8(c*oqg tation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm ttqp1yi2w;x yto g + 0k;kdcz;8o 280 rpm in 4 10+gk;;;otx 8yq2wiy c dzpkt.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 radiusyy13/p0d msi arig 1ib ad320j $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themsd-2 sec rsiwr js2ax7ly7o,6l.y ::kc elves 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 spacec l2cseor l k6s r7i: jady7.:s2y-cw,xraft 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 x31.uis6v uwnkv9p 3r5 rniu -from rest to 15,000 rpm in 220 s. Through how many revolutions .u srpi-1v wu9n63 k3xivrun5did it turn in this time?    $rev$

  An automobile engine slows down from 41oyj7pc+k7 *xlp +xw f500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirling “e:w+5o2v qzoohuman centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final soo+ezw q2 :5vopeed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in x:sb 7wrt;l y5(.1smmt rg1 yc6.5 s. How far will a point on ty:r.tg(lc w;7 tmr 1s1ymbx5she edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running 0aklv n(aj5 1qat 850rev/min It turns 1500 revolua1jl(k5vanq 0 tions before 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 speed u;ojyu-wl(5gai. tny (niformly froj.nt(uyg -aol;5w( iym 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 the2sifj 63(af hp7uxn0q car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of ihq76u(3xsp0jf fna2 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 each -i jubdaffa2k(o21edj2 6,lz pedal when climbing a hill. The pedals rotate injbo ,lik12 fddafa( 6j22ue -z a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a doo3p d 6 ruz3uf;-+li w +egnnsuzzyn-eu5,8 cv/r 74 cm wide. What is the magnitude of the torque if the forc-8yuu3ew svif-nr,gu3ezp+ dul / ;+5zn c6n ze 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 ohz6vs,a 9sjzdkwx1 4tw / k:a)f the wheel shown in Fig. 8–39. Assume that a friction tor wzhs4d: 1,6v tkak)aw9/xjszque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attal /gfi 4c+,ewuched to the ends of a massless rod which pivots as shown i4ie+ lfc,gwu /n 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 requ e0qu4pxas81vah ;p.i ire tightening to a torque of 38sq 0uva ;hi pap8.14xe $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of rz47yxoj+s8r zadius 0.648 m when the axis of rotatiz7rz+ox 8sj 4yon is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter.iu -ubz9fx mzj* :-a*u3t agj7 The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignai- 3-*f juzgt :uxjuab7zm*9ored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a hoc a.p(+q; hj fsw3zd,lrizontal circle of radius 1.2 m. Cal qjdzfaphl( .c 3+s,;wculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular ;n7ybfuh z9h3558w+ykmvm mf r u( q;tspeed (Fig. 8–42). The friction force betweenvr7muhwt qyb f h3yk;85( z;+mu5nm9f 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 po51f l68 giqp e),jxqmaint objects shown in Fig. 8–43,8j5qfge6amx i p)1q l about
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose totai q0wf *be,h-2zk zc8dl 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 coh 3-vnf9f; ;/meg /zqhawgs2jrrect 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 rocket if thesfw;v3gqg-9 jm h /fa/h 2en;z 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 8.50 cm and a mass of 0.,xa3f9q f/p br580 kg. Caxqf/ 3rbaf ,9plculate
(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,rej1z 8xsh2v 7 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-ggv.9h+el ygs 3 p)e)vto-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produc3 9)vypg elehgvst)+.e the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10n.i. z0bwjt-;zz tf-o ,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torqizz 0j-t-zw;b o.n.ft ue 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 ball tfphmo5ed jnc- ly 5,9*q.4q mat 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 ball8 5-c na76,ysgzkx fil is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–5 cf kas-i6n7,xylz 8g45. 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: /j1m /lsi5hhhix1l8of.r fa be considered a long thin rod, as shown in Fig. 8–4ix/f1 :h /jilsoa .rm8lh51 hf6.
(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 consisj. v (kzm+fvfj1t(6q wts 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 th96ctevapf* -io (-s cte hammer from rest within four full turns (revolutions) and releases it at a speed oft(va -scfe-itco9p6 * 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 hwnjihl)s(kul* z. rt 6ybw- ds52ni21 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 2*)/n 1v6t+d 1idlf5s9c wr n5 kvunexj80 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and4 extm4k8 xv9 h/rrs4ufg,6ah radius 9.0 cm rolls without slipping down a lane at 3.,e/r u4h 9gt4axhvf s84 rkm6x3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with ro9y,lsw9 l;y iespect to the Sun as the sum of two teryoil;y , w99lsms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How )6up ksxj+r w(much net work is jpw)r6 s(x+ku required to accelerate it from rest 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 m(awkn.uvj za4;0 wqmqv)z4 s:ass 1.80 kg starts from rest and rolls without slipping down a kqsa;za( 0jnq w 4v4 )u.:wmzv30.0 $^{\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 iti:(7j :to e9gupiih1us tip. It starts to fall and its lower end does not slip. What will be the jiiigo ht:u 9u(71pe: 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 momo1:n6p9gvx ydentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speedv 6ny 9dg:xop1 of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum 2)k p i2./m6tk y6 d4qalqiwg)z(qfezof a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotatie)2 fp 4.(zkmdqagyqt6qiz2 k lw/) 6ing 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 ag d/8 diapt flxr6t.9,t his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his/txf9iarpg6td .l ,d8 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 sh5bs-53vtnlj j-cv:l town 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 2.0j l:5 bjn3vtv-5ctls- 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 ;e *x(f b1otnrincrease her spin rotation rate from an initial rate of 1.0 bo;1x *ntf(errev every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotatino)g8cg ,ist 1og around a vertical 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 tho,s8g1o tc i)ge 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 x1e .k bu: zmthd:y,v.xm:uc) figure 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 momentum of the Eartyw0j:zgaucf) vkum5899xw5b k +zr+th
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment o,2)ncj q(0 mwk e/fh0x(xotthf inertia I is dropped onto an identical disk rotating atfwh0jq,te (t2) nhc/xkxm o0 ( angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a3p ;dm hk7a-lx rk.6(lcf)l ba5rteb2t 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 platforws6t3 *:1 f jh3ot0xa f (bzmqc9spqz8m of radius 3h8q*3:3(sp qmt9z6zb swf t10acf jox.0 m and 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 is rotating freely with an ang+qa x5aeh5 lt*ular velocity of 0.8 ht5la + *aeq5x$rad/s$ 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 aboe))yj)pp z i,g9l7lf,bl 2u4*as pytuut half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the ne)* uy) ,gylpi9e,l2u4sbpptz f) al7jarest 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 exg rsj,4sz mrgdl),/9 ccess 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 ex9hcburc 0 7 r2u:1kt$ 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 platfo3 jwxr4o5chs - o)j+k 8kc:iqzrm, initially at rest, that can rotate freely without friction. The moment of inertia of the person joc sro8q3j+ -kwxk:4c)zh5iplus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-rounimc)+v3xba . wd turntable that is mounted on m.b +axicv )w3frictionless 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 the tuzm7dh66o i1 xas1ng e/t-ec:1 4anptop 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 6gmitats- c 7uz hex1: 41/ea onn1dp6rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwq/)9ins9 u.vf5- nkc 4pzxl; z etzwzj z)5k+uays faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentvt pn) lzuzf+k. 9z4 s)-q/5zkzw ci 9e;ujn5xum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates68j(z s eugfrhk,(/e h from 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 unifw 1-q)n22f ylua 6my7+ tzxrweorm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered r1lwfz+en) w72 uyymq26txa-by the motor.    $m \cdot N$

  (a) A yo-yo is made of 1;s ykvr6:wd xtwo 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. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its xv s;y 6:r1wkd1.0-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 angu*2.9aec spqr- l.vq xdm(ly;h lar speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were rota56g2e/c,p,b /p qcr qhf(jj+ io+s4 zp 6ehsjating at a speed of 1.0 revolution every 12 days. If it we+hqe /p/crg6jf,p,ezo 6+4jhcssq 5a( 2pb jire to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution ay(fsfu) d2-rxbg,o/m utomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be m-fxs ur/,dgb o)2(y fable 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 illustra0l0jj 3ryp:bzqhi,bn 70 fx5e tes 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 speed v=3.-kfskdht +f.(p ;wxd7 0rzukiaht.48 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 free mhru*bcvhb;e4/j + ;v8 tyn3xly (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, det vt4ehnc+ y;b urh*m/ vxb;38jermine (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 r):8o+m,qdt z:uq uhj jadius 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 st z:8qd+uu,hq :m)jtj oep against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling withpxupqb*kx*yw v- ie-1f dy5fm( 5y.g, speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the ppim yqdv-uew- fx*(kf 5 1g*bxy 5,.ynormal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins w p,4ieakks,6ynkh( ;:+je arn hirling 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 t ;ejp:4kerki,(ahnnsy 6k a +,orque 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 herwu7wu4p gh,*0v5g7 oig qme p4y1olk5 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 til7wp*oe4hgku7i5oy g 0wg4v 5,m1pq ughtly against her body.   

  You are designing a clutffg j5a6 vg8)xch assembly which consists of two cylindrical plates, ofg)v8 ffa jxg56 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 radius r rolls along the looped rough ti+ dhrgce fsp5)*ap81rack 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 loop without leaving the +idecsfh58ap gr* 1p)track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assc,ax7n3vp2 kis cfu: /ume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speekdr-jg-) k3 rkkrg80icky.3sd uniformly from 90km/h cy jd83i-r3gs0k)kg.kkrk r -to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s