A bicycle odometer (whichk-o p )u,;qups4b(wwe measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. Whatp)so-b qwek( pw,4u;u happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim ha33jepc gh-/ 1 *mazv 6sr6tw*3zrhosave radial and/or tangential acc6 wa*jra36zhst*gv zp3 1 mheo-c/3rseleration? 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 descrkn)nw mencyo4nq7 ,76ibed by a single value of the angular velocity $\omega$ Explain.

Can a small force evb ;uq3*unxypv*9s:a ter exert a greater 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 h3ei-yb,nm ;sm9 c ng,8otkuyd:-)nfeands behind your head than when your arms are stretched out in front of you? A diagramn9uf8y;ny - mmedginoc),: 3 ets,k b- may help you to answer this.

A 21-speed bicycle has seven sprockets a,rwl7c*jj nknxd58 :ot 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 harder to ,cnxldjn8r *j75:k ow pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on b3v g*qg 3)ymzby4qzm0eing able to run fast have slender lower legs with flesh and muscle concentr3m4* yz)0qqgyzmgb v3ated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narm1 1y.bkedd9g row beam?

If the net force on a system is zero, is the net torquqz(-c d7erz 9fe also zero? If the net torque on a s-f r zc7(9ezqdystem is zero, is the net force zero?

Two inclines have the same heightqe)pz9g5j s3 d but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incg)p5e3 sz dj9qline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simulta ,omkh:quto1s3us 0q-neously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the great0q-m, k1 quout:ossh3er total kinetic energy at the bottom?

A sphere and a cylinder haocj ; 42friyv8ve the same radius 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 cv8fj2;i yr4o at the bottom? Which has the greater rotational KE?

We claim that momentum and 0kdpd9o350adggd dv4angular momentum are conserved. Yet most moving or rotating objects eventually 0dkgdv0o9d4gd5a3 d pslow down and stop. Explain.

If there were a great migration of people toward the Earth’s eq- u/ ebmv+.tytuator, how would this affu/tme- byvt+.ect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any in8nsaav c8h -h0itial rotation when she leaves the baas8-8 chh0 vnoard?

The moment of inertia of a rotating solid disk about an axis through its citn29u b) 82mtd l r;2 v*:slxsvwf5dqenter of *vs xdn9 qvdu)mf 5isw2t;r2t8b2: llmass 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 holdinv i1w1sexj m0r +*vx-+td ht(sg a 2-kg mass in each outstretched hand. If you +xhsx0w( +v jemds11 it-v*tr suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, 2 a y)q+,:jul ygw1gmolo vx,)one is hollow and the other is solid. Describe an mg) :,g2wol ),jvyyq+o1ax l uexperiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. jwg -6.h:cghdIn what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe tch: dgh-j6.g whe tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standi+zhzd76by-t74aux0kq8 ( z5vn fitvlng on the edge of a large freely rotating turntable. What happens if you walk toward the ce-k 80qtiyu+bz67znfa 7t vv h5xd(lz 4nter?

A shortstop may leap into the air to catch a ball and t.w6 mp20d tirrhrow it quickly. As he throws the ball, the upper part of his mr0tri6 p.wd2body 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 of cons3,tl6t ofy9o fervation 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 hey 69fo,tf ot3llicopter stable.

  Express the following angles 0g 3s spzdylsyqii ws4u)v,q(7e((o3in 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 coincide+-mdd5 6,mpuu wa76p dy cyvw*nce. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on dypm 6ycp7d m,wu6d-+ avu5* wEarth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,o++n t/o hz3ixzv*x8v2nw3qw000 km from Earth. The beam diverges at an ang/+zt i *h8w2vwzqxo3v3on+nx le $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender zw fdzck.ma.fs-e -.ak(zw 98rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the bwm 9azzk szc.(we.af- .dk8- flades slow down?    $rad/s^2$

  A child rolls a ball on a levelngebaq/4j8j(--qgl q floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diame g/jlqa(8e4nb-q q-gjter?    m

  A bicycle with tires 68 cm in diameter th oq8:lugn;:ocusdv/i 6 oy8; ravels 8.0 km. How many revolutions do thgoo6 lv iqh;8udny:u;s/c8o : e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 25);/lml bcjfd300 rpm. Calculate its angular velocity in 3;/j lbc fm)ld$rad/s$ $\omega$ =    $rad/sec$
(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 inn/2,l h+rbvv h;)hl8cdxt6h p 4.0 s (Fig. 8–38). (a) What b;vlhnc+p8l,26h) /xt hhv r dis 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 (a) ins*w , two5qfa4 vztw:t3va64m its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed /vmzlstey : vyf6i3*; 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 if a particl0aw -cusq 85*bku31c 7ywsf k+ f1wwzqe 7.0 cm from the axis of rotation is to exper+uc*a f z1w7ywwq 8f3sb-1wkc50 qusk ience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about i55v4k qu:ul- ;jwhttfts center from 130 rpm to 280 ruvf4 5uk5w - q:lhjtt;pm in 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 c u:nh,er+px;/dke92yofk 4 p$R_1$ 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 Apollo11 rhxosbws., spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of os.hxb 1,1 wsrtheir trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from *1cu6xz7dy x8*hherm p:fwd 7zx z doz:77tz2rest to 15,000 rpm in 220 s. Through how many revolutions did it turn iz:x7uedt7zr* d7x8dp zow: fz xc17 hhz6*m2yn this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcuhp:h26hk hr ,tlate
(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 fly: 9ge aow8qfznd: i;6bing highspeed jets in a whirling “human8of wz6b9ne ::qd ai;g centrifuge,” 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 from 240 rpm to 360 rpm in 6.5 sekkj 0ars ; e3anz5,+g. How far will a point o n rze, sk;3g5ake0ja+n the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850revaoz+3g-2l0kd fb o8aq/min It turns 1500 revolutions befo 8a aq-kb 0gzo3dl2of+re 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 m*i57z) d9n w-skp-v+vowhh eeake 65 revolutions as the car reduces its speed uniformly from 95km/h to 45kenp h -vi)5*od9 whkz +-v7sewm/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 red zy(lt(r se5:quces its speed uniformly from 95km/h e(srzy5q(lt: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

  A 55-kg person riding a bike puts all her weight on each pedal when climbintvz1j *;gv nk*4xauvd,+/xxa g a hill. tvvd4,x+; nk /** xagzvuj1 axThe pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the en) y+cn uyi,+u int:jr0d of a door 74 cm wide. What is the magnitude of the torque if the force :yu) 0yn ui+nc,+ jritis 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. Ascrz7 69px0+ ptm)o gd5eler:wsume tha )t6z wxl5e7p:rgrco9dem+0p t a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mt lt yo6*2m6ekkqk+ i5ass m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Cak y5kqt*l e+tk62 i6molculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of ) 1yudz9))tye tlz*x nan engine require tightening to a torque of 38 )nz)d)*1 tzylue9y xt $m \cdot N$ 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 t*6 l7/*iwad a -geghoof inertia of a 10.8-kg sphere of radius 0.648 m when the axis ofe wg7/*o*ath-ai 6gdl rotation is through its center.    $kg \cdot m^2$

  Calculate the momentnhrc t)tqt,o2 dm4a +gg -71pd of inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be+tto g)mpq ,74-2ddrn ac1h tg ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thiedx iuxj(g*dl-3(ldxf ;3 id8n, light rod is rotated in a horizontal circle of radi iu3dxx ;(ilf 8x(d-edj3 dl*gus 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a pottet s0q4wq5*o qqbz4w z(r’s wheel rotating at constant angular speed (Fig. 8–42). The friction forzb5q*0s 4q4z (qwtowq ce between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects show .esx)m4wi ou-n in Fig. 8–43 wm4.x i)oe-us 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 total mass i6,o3 ru.9pv mmd4da hrs $5.3 \times10^{ -26}$ 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 cylindr-h*o1n 673q mduertnfical satellite spinning at 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 rad q3 *d716u- thormnfenius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius o)ss dcrwvlw5/ +15qnif 8.50 cm and a mass of 0.l+n srw5 c)wq/vs 5d1i580 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 sw51yk qhxcg1 i)ings a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merryzg87yfjl1 u i:-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 diskujlg7z yf:i1 8 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 eip1+hp; v v sh(yq,)yhventually brought uni+,pqsp;hi) y1 hhv yv(formly 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 ujx k7rjts*v vfi)5da,+n j54-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thitq 8m*8(ez-o8jww birown solely by the action of the forearm, which rotates about the elbow joiom(qiw8b- ij*88tz w ent under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as showho h *(,qd.zg s5wveu+n in Fig. 8–46woq.*+ ghes5hu ,(z dv.
(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 con5a;s2 p)jvj:peqd 1 fnsists 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 acceleratesk9zfn/ mi,by. +e2 bnq the hammer from rest within four full turns (revolutions) and releases it at a speed of fiby+q2 . bnk/9,nmez28 $m/s$ 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 of inertia oftr3a2g*b/b;s7.dfw 6ic qgqt $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine developsooubm)06s fi/ 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.b 0k9yoskr*kp z:3x,l 3 kg and radius 9.0 cm rolls without slipping down a lane at9o*xl pyk 3 rsk0:,kzb 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to theznv mh.mnp0 kq6*l(qp:p;bc 1 Sun as the sum of two terpp(6np0 vb q.*mkmzlh1qc n;: ms,
(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 much net wou7y ep+hlf k9*ll*6dd rk is e hd **9 fuypl67l+dlkrequired 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 mass 1.80 kg starts from rest and rollx *7hm(q elyg,s without slipping down a 30.0 ,h7emqxg*ly( $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall q iy0/w 8bv( ylm,kkp,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? [Hint: Use conservation of e k 0i ,lkpb,8yym/v(qwnergy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a 3h8/twp *;kr.z mozc ethin string in a circle of radius 1.10 m at an angula8kt;*r hmz3o/ew.pc zr speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindr8luj yuf*7qi 0ical grinding wheel of radius 18 cm when rotating at 1 8u0 fl7jqui*y500 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 a platform that is rotatinqcmq.ux6dxzu,7ln da8q e:6 (g at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotatioxuqc8xqa6u .qd6m(z le7, :dn n 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 moment of inertia b ok pbl4/kof8pinb z(rvj0m*8.00j ory a factor ofp8n8jkr f mbvoklj0*roo/ z 0(4ipb0. 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 ink;f ry/kvp* ( :ttszh9crease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of yft/:prts;k* k9h vz( 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 tf+ ,c/qb 8vpp-qued+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 qd/b u +vp ,+q8cfep-tthe wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figurerva+uq *ge),o5u+ gon 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 Ea t7g*bp0u-at8 e5vel/ i;s+(irs osxxrth
(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 )af9itm7vs/x9 -ij wcinertia I is dropped onto an identical disk rotaif/acxw9 v -i7sm tj)9ting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 ya6f 4uj ffmih:18mm)lv,p4i $rev/s$ 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 q v.28u jpsrwem8lik;+/dq2 ythe center of a rotating merry-go-round platform of ruw2/i 8yv+emkrq.d lqs82 ; jpadius 3.0 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 3 kl/o hnc;i rwp20vc7with an angular velocity of 0.8 rich3 kvw2 /7con;0lp$rad/s$ 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 cohu)h zh4ad.tea 9zvo3 m( b.l4llapses into a white dwarf, losing about 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 nearest inte(4hhat .mhbezd lu.)o az3 4v9ger)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 12t6g*d)ugz.7 bk 2ylt* jc yoe,0 $km/h$ 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 : ch )x3w4)ewllzym3o $ 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 w;;;i ntgs/us1zojtxlktaj1xa,wiw r* e 7 ,;5ithout friction. The mo o1,xaa,tt tsl 7k z;wi1u;;xr wje;gs/n i*j5ment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m dia/i:t 0 f:c3euxke)6paiko; 1ql nfrk3meter merry-go-round turntable that is mounted on frictionless bearings and has ;f/lc fe qo ku:6r x:3n)ipi03aetk k1a 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 ac-c6kvs e h t5 3byd)w /wh;n1+vrpc,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 rolls behind the person without slipping. Whatpc361ynw5tv;rc+/ - b,hsd ekcvhaw ) 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 fa0 hqz5y(2ng ftces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbitin5qhz( 2gtf ny0g the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fromzknj( :ki* jb2o 1gk:i rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cyli)r zilg22 *y(h rbssd*nder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant anghilg* s2rr*bz2 s )(dyular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diskx(m zz7y 7xqs5b1s/ mus, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub o7x(bs qysmzxm 517 z/uf 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 th2)ap:w 6hqosu e angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with 7a5d6)iolrnh te t:p+ *b-mtc mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what 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 mome btdlrn-+5otm7 ih atp:*ce)6ntum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy s c0 mjv(c +yy:v(umm(kewd58 qtored 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 able to travel 350 km without needing a flywhee5mukqyd0+ wj( v8vmcmy ( c:(el “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 jfl b4a noe48jo0y wx(p:y3* z$H_2O$ 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 (:djbw-ze;j s g r mky/v35q0mon 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 can pivot freely (i.e., we ignore frictio z,op,fd6*zfdwrpc);kdr q) .n) 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, determf;r c )w qppz,o*dfd)dk,z. 6rine (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. +g +xxsf/xm85uqxzj radius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that i/mz+x+ xqg5jusx .f8x t will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling wifj 6jl2)ohkfy( j z.(cth speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle alyfjo (6.(jf hz k2jc)re the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0. p 7q87wu1btmo50-kg rock into a sling of length 1.5 m and begins whirling the ro 7um17 bqwt8pock 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 asfnl-1 pqvx) p9 a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly agfvp px 9)qn-l1ainst her body.   

  You are designing a clutch assembly whi p-j x0e626uwy1gtohpch consists of two cylindrical plates, of mas g0y2u6phjtoxew6-p1s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 track ogqx rfk wvbo 2i;x1:70f Fig. 8–58. Wrv2 k7qgow0 i ;bxx1:fhat 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 track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, b9ll g -/ iteo2b jg e(rzciic7w9q+,2dut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces it+k3 xwf4 feqv:s speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular4f: vw +e3fkqx 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