A bicycle odometer (which measures distance traveled) is attached n:7(fw mw:6lffe2,zk) k k qvpgear the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-im2fkzvw ,qlgk:)7 k f:e6w(pfnch wheels?

Suppose a disk rotates at constant angular velocity. Does a point *j.l dgsj u1e8wxz j0.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 acceled* u 1ljjsx.w8z j0.geration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of n 5be -3(ndxxrthe angular velocity $\omega$ Explain.

Can a small force ever exert a grnmg -oet7a 5r:eater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your headajfidg 3 i3e*/ than when your ar3e *gj i/dfa3ims are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three4yo. iq*ae9m6 x,fhbl at the pedal cranks. In which gear iy x 6.i,mh4qa fl*e9bos it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being at0 h g4rei-(k1mz t0rcqd+6ehble to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distrk+gr m-1t 4e6zdehi0th(c 0rqibution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long,6 l/ 2vge ya(lcd49xn)n fld/e narrow beam?

If the net force on a system is zero, is the net torque alwd-u+nv34t *5nc xzqmr4w v7dso zero? If the net torque on a system is zero, ivm-n3d t5wnq4d c74 wrv+* uxzs the net force zero?

Two inclines have the same height but make different angles with -m0zzjk4b je+(s ( ortthe horizontal. The same steel ball is rollejz(rteo-4b(0 +zjmsk d down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. Onei s cgi) 8wwlgrs3c 72q h;3q.hsr4q1t,zfly2 sphere fs,g32csg.s 8 zrhwt7y2 lq)qq3 ;wi 41cirlhhas 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 cylindernei9.nqhs p;6 have the same radius and the same mass. They start from rest at the top of an incline. Which reaches tqh6s; 9pn.nie he bottom first? Which has the 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 m8f1 epxz5p:h58mio0t1e mh h aomentum are conserved. Yet most moving or rotating obje5 epe 8xifha1mt81omh 0hz5 :pcts eventually slow down and stop. Explain.

If there were a great h j(y;,ooyed-:t 9sa hmigration of people toward the Earth’s equator, how would this affect the length of the day?j9 hdy eh o:-oat,s;y(

Can the diver of Fig. 8–29 do a somersau hw.qkw,,de2os ,p 8nllt without having any initial rotation when she w,on. wq kse lh,8p,d2leaves the board?

The moment of inertia of a rotating solid disk abx) pkky5 gf:j ztmr(a++aprm5/ptib6mr1 , .kout an axis through its center of massk+5:.t jmm1)rbgka5xa ,6k(ztr f py prm+i/p 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 stool holdin;zzn7lajy -0nav2-o9 uf t.ba g a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay theln7;z-a0av jy bnfuaz 2. ot-9 same? Explain.

Two spheres look identical and have the same mass. Hoh-ann3 lm /*eswever, one is hollow and the other is solid. Describe an experimentls-e /mha3nn * to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle poin,- j+vg qvq3nets west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points eas +e,jnv-v3 gqqt, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. gn8/8+f, ac8 mibe7 vw sq9xap:psh9aWhat happe 9,9aivp 7nema:cs8ags8q8/+bpfhw x ns if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it qui lv+ 0+yaqb1b,icd,*oewx+w fckly. As he throws the ball, the upper part of his body rotates. If you look quic+b *xei10++o acl,y d,wwfbqvkly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentun8 jnm0) ;l7a phukv+im, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the)l0 mjnp kv8 i7h+nua; second propeller can operate to keep the helicopter stable.

  Express the following 3;bi 8jce;.vgwgqu. 2nad,cc angles 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. gp kfckh9i;nly-e)0j: zq e(/ Calculate, using the information inside the Front Cover, the angular diameterzh9j/pel-) y f:cn gqi;0(ekks (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,0cuaiv(l., q v900 km from Earth. The beam diverges at an a a,cv9.( iluvqngle $\theta$ (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 turned c8wovqa /;e2eoff during operation, the blades ; 2q8/ oceawevslow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makes ;w5-og w k6u1** ajwb+hwifq x15.0 revolut*iuwq -6w1; wkw*oaxj gb+fh5 ions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How m6qordf:-cei6r(z*+mim b(u r any revolutions do the whro+:ce (6r (iifu mmqb-r *zd6eels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rot80:s u)akd hud 5c5aa,zi9mbu ates at 2500 rpm. Calculate its angular velocity ,:cab)duhs 8ai9a 0ud55kmz uin $rad/s$ $\omega$ =    $rad/sec$
(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 comp3ircb+hir1bx vh4o- 8lete revolution in 4.0 s (Fig. 8–38). (a) What is the linear spebc1b4r-3 vii+8oxhhr ed 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 Emxk/4(( :i jsybi9dt2fygc( , a 4uetfarth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pg4n h(qv;m *dgoint
(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 rot x;0tdfjo eq8d221 cnqanmo ,(ate if a particle 7.0 cm from the axis of rotat2 mdc(f1o , 2et 8;0dqqajoxnnion is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about it3iwjyftrq 1d/x2l 6 2ds center from 130 rpm to 280 rpm in/6j23ytdqr xl1 idwf2 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 8j1fqv 4cm 7t*s7qlim $R_1$ 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 the8snz 2(t zgz1s4oms 1zmselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated fr1z(gssn 1 t2z4sm zzo8om no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Through 0ml8d yzc1 nrb8.q2xjhow many revolutions di. b2qx8jlnd 8zmr0c y1d it turn in this time?    $rev$

  An automobile engine slows down from 450v947-rph0unyh ug1t+t;fm ld0 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 sjx ryi-m9of(( tresses of flying highspeed jets in a whirling “human centrix romj-y(9f i(fuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly frvef3nan. -qr, om 240 rpm to 360 rpm in 6.5 s. How far will a-va, .e3nfqnr point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 revozuuo)uaj 8dm.b j3-k/lut-m uujb)j8kda 3u/zo.ions 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 red88 opt2ys.(bya mr eb.uces its speed uniformly from 95km/t2y8op s(.8 mbeaby.rh 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 its speed unifvy+k d ddrz:+(ormly from 95km/h to 45km/h The tires have a d krz+dydv :d+(iameter 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 herazwdm8 .0 vg k;x7n*qa tg(myq5+w/gk weight on each pedal when climbing a hill. The pedals rotate+(dqz ga wg.xm tkg80ynk57wam/qv*; in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cj1pijd b(1x: km wide. What is the magnitude of the tob 1:dj1jk(i pxrque 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. As7m wvy*;vkn n8sume that a friction tovy7 wn8;m *nvkrque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a m2uh/sh-h 9xb kassless rod which pivots as shown in Fig. 8–40. Initially the rod i9-xuh/ h 2skbhs 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 enginehq9f:h kb:,xf7*nvw bm9fw-1md bba ; require tightening to a torque of 38bb:d*ffbha,m w9;w b9x1-fv mk7h n :q $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when th ,89 j5dxinzcre axis of rotation iidn9 z,5x rjc8s through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bi4fc*iab:p .p( o:dlt pcycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass : p4opfcdl. a:*tipb(of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horiudr* uq.1lb o,r e7t7dzontal circle of radius 1.2 m.uq or.ut7 ,7eb*dl1rd Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotati bs-gux:n p7 pc-ctk *oq+w7s+ng at constant angular speed (Fig. 8–42). The friction force between her hnkx-+*po +qsg wtuccp7 s7b -:ands 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 i.+5b tkqa e8)jdn ul7bnertia of the array of point objects shown in Fig. 8–43 ab78jd )a 5klebub +nt.qout
(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 cohvl4sf d+-6 . 8wra uvud3xi8ensists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning atb1s ,wfvlse9 q.r+5my the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0b1q,.fl rm+9 sw5vsey min? $\approx$    N(round to the nearest integer)

  A grinding wheel is at 5 w cyfio4w;6pqj40k uniform cylinder with a radius of 8.50 cm and a mass of 0.5800;yc 5kf6 wo wi4qpt4j 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 b c kjahj+e8y7nx.;ll7 at, 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 + .,xbc.g qllu mtg7u:hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m : t u,+ gqlmxul7gcb..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/))zib7gt b;btbi7h 4h**a8t8x , sdrlaw shj at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.2wsi8l a b);,4st ghtjbbibhzx*hd8t7 )/7a*r $m \cdot N$ 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 vq+-rhz5i m7fball 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 thwrys t+5o)y (brown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated unytbor ) yws(5+iformly 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 beoe5sf/ziw 9-s7xc yq8 considered a long thin rod, as shown in Fig. 8–4qc98yo7isz 5/fw - esx6.
(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 of bdky e*k7ek7af4 h7i8 ;5st nttwo 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 ;v5a c egu4,guhammer from rest within four full turns (revolu,5au;ge c4 gvutions) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment ofb-skvr 2u,( v9: nowag 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 developn* 8smzfvhy23ol rx(yf--9av s a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg be 892l73ry 9mdrm85yxt kadn*rft y.and radius 9.0 cm rolls without slipping down a lane a 9 k.r3ydbnytf re5am *l829ytx7m8d rt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy ofln0ap 2o 22xfexeo*p* the Earth with respect to the Sun as the sum of tw2*pa eefo20x 2*nolxpo 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 ks/axdr g6n op9hv9v2 5g and a radius of 7.50 m. How much net work is required to accelerate it fro9a6nv /95xhs2g dp rvom 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 mass 1.80 kg 4 eab5v51t8dv bcv,abstarts from rest and rolls without slipping down a 3acv4v5 t,e avb8 1bbd50.0 $^{\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 a u a jlz5jk269etv6y.)b 6hdmto fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground?5 javd)hlt6kjb ye6mz6 u.92a [Hint: Use conservation of energy.]    $m/s$

  What is the angular momen4nep abfar8yu3m08 .7rv igf8 tum of a 0.210-kg ball rotating on the end of a thin string in 0pgi8narrv b8 y4ef37fua .8m 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 momentum of a 2.8-kg uniform2m4xu 7x z8 frcca.0ug cylindrical grinding wheel of radius 18 cm when rac7g u4x c2umxz8rf. 0otating 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 side, on as.h uafdc90r2 platform that is rotating at a rate of 1.3rev/s du 2.9facs0rh 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 ++shgtwq4 / (*yn5d jo dww0vgher moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck pog 45s* w 0 h+twonvdg+(qw/yjdsition. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rat rb,kk 4pm/nuzzwoc3pox697 -e from an initial rate of 1.0 rev every 2.0 s to a final rz/rxmpwo-cu37kbp694z ,k onate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a fren8, ,- xrlqmlfn(1j agquency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The pottergn8lj,a(q ,rfn m1xl- 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 the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a 93rz)zr 4hv mefigure 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 angula8j2qx8h 9hork r 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 moment of inertia I is dropped onto an id v(gclsu2uwm+3; .r1xcps h;d entical disk rotating at an urg(vu;c+. whc2sld1psx m3;gular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 25(ltu gbe1p9gh.q9g b.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-rounduwjwax di ty4(q ,8:m* platform of radius 3.0 m and mtwjd,qu( 4:x m*ya i8woment 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 angular velodd c hng,p7,w,city of 0nw,dpg, ch ,d7.8 $rad/s$ 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 dw4sq 6sbc-zk7 aarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the l4 kzs-c67asb qost 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 ub3ed h7n*,aw(s/e-al8p g 8cqzw5vj in 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 irbg 58a eu 7jc9m4*lp$ 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 f z y 4/saqke2 gs.73e7h(qcfwereely without friction. The moment of inertia of the person plus ta32eq47g/s qc(e7 z syfke.h whe 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 diacatygms g9w0,7*j+8vgdr8 h4g e0 rm zmeter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1700 0y8mwgra g4sg gvdz 8 r0cm9h7*j ,et+$kg \cdot m^2$ 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 x4accjtx/p.b 8 q/:9ddmz q4p8r8 zmx ground with the end of 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 roll8qj4:/ 9qmpmx/z 8cdzpbd4 .cat 8 xxrs 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 always faces the Ea*3csxn ei o/ng,e0 lq.rth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a lx* n si0go/.q e,cne3particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of l83yynrkrfni,md; 6igk5 :j4 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 ox/3+,zg g9 rfnjq+t pof a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval j z rgp9t +/,3xqof+ngat constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two so+zz1*kqo b+gqxqm) m uwd(-s*lid cylindrical disks, each of mass 0.050 kg and diameter 0.075 mxk *g +m+zd- (b1)qqwzsq*ou 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-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed oft gi-izr*q 95b 1u5raf 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, werfkl / yba13qg hpx6t45f)x/sy v(d4lz e rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star oalglsdvbf43 ph5zxq1yf)kx(4t y6 // f 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 use energy stored iiu ,lg )1b9au:q2 qu6 lfr9tfsv1(xq tn a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel2 txu f:6r( uqgiq l1su19),fqb9vlat 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 illustra7ic*.p .bd x5ivfj8c )xlg(ettes 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) i) oc cz;xy 4v+z4nib2ns rolling on a horizontal 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 (y7d6;(. r)kjjs u9qrzcjsd hcan 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 release, deh jq97 scrju.;sdrjd (zyk6( )termine (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 r7b-r561. rnuhs9 skuc ah)eieadius 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 ik5.crheu 96e u 1as snhrb7)-it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turj 9gy6igkoyy+*x(dqvv 3 -di591garx2y g ca/n with a radius The forces acting on the cyclist and cycle are the nov63kia(99v qyyg5+j*2d dro c ygxi/g xgya1-rmal 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 beg,dub: vqq,0l xins whirling the ro,q, 0dlqbvu: xck 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, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms ass8+nst)4j 5chv ;yjj/spzv jy o ,l;u- thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a s /p8j),4ns5 jy;lt s-o+hycvsjzu v;jpinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a j rmualm8f5/);)ezq0je1 cnyclutch assembly which consists of two cylindrical plates, of mass ) yflna5zrujecm/81) jmqe0; $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 roa-b:gqzv1q z9 cp-t ;yem;hrk4q: y-kugh track of Fig. 8–58. What is the minimum value of t bym4:vzp1ec:q; k-9y-ka z;tqq rhg-he vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but bi7:inw5m 79vbhewb (do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as thpjykv2v9;ff ( e car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of2fj;9v(ykfp v 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