A bicycle odometer (which measures distance trav*a t8ft1l55 kyj ujlrmt6gza+z 40oa7eled) is attached near the wheel hub and is designed for 27-inch wheels. What happens ifz+f56 uzlt*1 taj4aryjktl ag58 mo07 you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angulabmhjnd*+y9p hm 9 r39kr yv,)ir velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which casb 3phr*yk9+vi )mh my,n9r 9djes would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single vab4wxduks(vj *)s o*kf- n8 ;lilue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greatb a98o)d;y+t vs zw1gper 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 your3h(zob0 a 6r ;ggcnk4c hands behind your head than when your arms are stretche3gnocz ( c4;hb6kag0rd out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sproepy* 2d178ohtl k:u6rzr -jo v g75entfg)eh)ckets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprrryz-ed2t7pnf)o8*t lkue:)6 v1jh5geh7ogocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with flld,2 1tyejcx *m vi*w 3p5r4z;h7nambesh and muscle concentrated high, close to the body (Fig. 8–34). 2 hzimb*l*a ; exntj1 ,dpc7r3w 4vym5On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow be)clryw7 km)n:jwq6d1 v ci.nscx1w71am?

If the net force on a system is zero, isd7)2j lhun ad7 the net torque also zero? If the net torque on a syst27d)j 7lua ndhem is zero, is the net force zero?

Two inclines have the same height but make different angles witf5aox:fok/m tvl/.c;. fjf; ndfo;y *h the horizontal. The same steel ball is rolled down each incline. On which inclinet ;oofjl:*.ndx5f/ a.kof; m; vfy/cf will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start r,oefs gqu56 w sdjho/r(p99s)olling (from rest) down an incline. One sphere has twice th /5)pd(sesho6 gf9rq s j9w,oue radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radiu lf lta0h ah.wg4;2ue54i:dwsc6fv- u s and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which ctsha v0g44f de26huw 5ll:u;i.a f w-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 momentum are conserved. Yet most mx*g f xg1eb 3kpjpmy699l4oa9oving or rotating objects eventually slow 9*yk4o6 gfm3el p9jap b1gx9x down and stop. Explain.

If there were a great mig2 fuln-bp l8e97 :qebqration of people toward the Earth’s equator, how would this affect the lengu:p 27lee- qlbfbq 89nth of the day?

Can the diver of Fig. 8–29 do a somersault witni)h32pc.lcb5 kvkff9* v-x ohout having any initial rotation whvf-voifk *.5cnkc) 2 phb9x3 len she leaves the board?

The moment of inertia of a rotating solid disk abouz4t-u51d nrxut an axis through its center of mass iuzut n1xd4r5- s $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holdingb7;5tni g535 cmh g5fb/huuas a 2-kg mass in each outstretched hand. If you suddenly drop the masses, gfh b 57mh5su35tbn;ag5/uic will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have theyvahsa289 e ;n same mass. However, one is hollow and the other is ny29h8s a;ave solid. Describe an experiment to determine which is which.

The angular velocity o fkqiz *we7 ch5ze6e-6f a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular accelerationw ce5i z66ekh ze-f*7q 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 lat xfilgw0it8ig .: zd1.30au erge freely rotating turntable. What happens if you walk toward the cent z0 tx .a. f81d:ui3lgetw0igier?

A shortstop may leap into the air to catch a ball and throcwwuct4-+pn gj(gii 2ze 72h- w it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you wil+pwzi(uce tgj4h n7icw- g 22-l 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 momentum, discu8rop6(apv0i0w8 lur p9d d 5ugss why a helicopter must have more than one rotor (orur90rdiu(5wvl8 p6ad0p opg 8 propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 3nn0g w irr12 m:ekf*h90 $^{\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 be u i5z4kj1-bo,r1 tlxleg6mz. cause of an amazing coincidence. Calculate, using the information inside the Front Cover, thejg5ori m,1tl xz1.u b4ke6-lz angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 38ndn 9sf1;1 /p dgje/mc0,000 km from Earth. The beam diverges at an ang1/ 9fmg e;scnn pdj/d1le $\theta$ (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 ku(*zcwo/n5 ry5,hgx wcv46c3d: mgma rate of 6500 rpm. When the motor is turned off during operaw n45gu/ 5gc3ocv:kmxz 6cdyhr*,mw (tion, 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 thek1+ye rh5el .pzost51.jxtc/ ball makes 15.0 revolutions,jt+e e h1 rt/s.l1zyok5c5p.x what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolppgq q8w9kv(3 utions do the wheels m w(vg 8qpkq39pake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm.x19zt22xdb5 ngykc- g Calculate its angular 1 bxy9d5xzggk nc2 t-2velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round ma2srsm *1)c0gozzo os-kes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a ch-osr zo*gs2so0zm 1 )cild 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 Ems1w r 8 +npy*dg/ fu,4* xq*pfg*wubdarth (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 poi5lc59n3:c1q tjnc o kint
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0r2p fiv2k* ns0 cm from the axis of rotation is to experience an acceliksf2 n20rv*p eration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about itsnjwm5ylfkg3 -7dgp z98vye-: center from 130 rpm to 280 rpm in 4.0wpde8mfy3 - vgl59nzjg: k-7y 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 radiusu* qv6jrjetlko9b l0:q 693w k $R_1$ 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 thebvfii :1nf/8 hmselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from 8 vhbf:1fi/in 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 hosqr);ny --yi, sq1e elx;l+l l4b k3xwvvc 2)pw many revolutions did it turn in,e- k) 2syb- e14irv+qqlcl);;lls3v nxyxpw this time?    $rev$

  An automobile engine slows down from 4500 rp) -7l dna/iclo *krmf3m 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 oyfpnx7 s ;k a7j* 2b+eo4-oitkj5)ekff flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching e7 ;k-)y5a2fnjtk x4fo s7bi+epjok *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 diatxxis 8vkr4jmd:e 7*x737 pkpj5sdq 6meter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far willxm ixdks k*78d7 7s5qexpvpr:3j4j6t a 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 1569l.(w ,ihhtcv .gk u gp8gq-h00 revolutions before it c..p-k8 h(iqgtghuhg wcvl, 69omes 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 reduci*yzi0j t*o6jes its speed uniformly from 95km/h t tizi j6yo*j*0o 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 seh4or .yk3wg .jj9: nmbn l:kszf c22)peed uniformly from 95kyh )fjkwjz:rnokg2ne cb.9: 2.sl m43m/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

  A 55-kg person riding a bike puts all her weigl:g;qkz225 b (liteu)s4 gvdcht on each pedal when climbing a hill. The pedals rotate 2vs dbc)qi ;42lz(: l gtu5kegin 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 e; 5 xmc0n2nnwnnd of a door 74 cm wide. What is the magnitude of the torque if the force is exermn02x nwn;c5n ted
(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 wheey vlqp )wp;s* jq,n15gl shown in Fig. 8–39. Assume that a friction torque of 0q5v ynpsw)1gq,p*;j l .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 mas.p2mpavd/7ieo ma9w+sless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal posipp9wd a+/ai7 em.om2vtion and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of1,ur43v59yg+ :u dy0 ls xovzcmm4ejn an engine require tightening to a torque of 38 yx zmn1l43:+u o,905y4cde vsmvgu jr$m \cdot N$ 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 radi ht5nlj/ :2d dusoby r4o7v(g;us 0.648 m when the axis of rotation is thro/l54tn sdvj; g:o yb(7u h2rodugh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7rr7+arj :or 7l cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored 7lrr: rar j7o+(why?).    $kg \cdot m^2$

  A small 650-gram ball,zbpz3n h9bcywo,hy4(u 6m1c on the end of a thin, light rod is rotated in a horizontal cn69 3oubwzm1 y4bhh,pcczy ,(ircle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potteqo4k2p 5 (cndar’s wheel rotating at constant angular speed (Fig. 8–42). The friction ocd4ak (5nq2p force 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 z;d5w.cb f9aq sez*j8ge:5alarray of point objects shown in Fig. 8–43 :wlg*5bsc 9ee;qa5a. z8 zjdfabout
(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 ox e(:-s(eby gfox 04mlsygen 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 cw 7ae)9tksvqz 1o 3.gxylindrical 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 radius of 4.0 m, what is the required steady f3. stg a wkx9oe)1q7vzorce 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 r,z4nigca3m fy1/4 cirkh p0 9 ;olvr7uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. C m4zfi9na o4kg v1y,hr lr3/0pr;7 iccalculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it fro vhh *tpbwcq3b 6id1ie 3-27wvm 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 smalap*1.oc xj8qxl 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 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 torquecxaxj o81p *q. required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotatina-djrji8 dzb*1u rpluwi-*4+ db7 -qmg at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque ob8i4r lb*q ujujp7di1wm-- *rd daz+ -f 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 bk 9nf z;bvp942mh ms:gall 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 tjfu5)nkw c)/ thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ballnfck))u/jt w5 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 shown in qp ol(9z5k l8kFig. 8–4zpk9( 5l8lqk o6.
(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 consu,q ajsu(-gj7 ists 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 accelebwweo (91mz7 lzoi,3 krates the hammer from rest within four full turns (revolutions) and reo 91 lb,eo3k7izmwz( wleases 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 ccgle( 80+un) zh5 jzaa 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 devel.mh 6nzflh72wgw/ po4ops 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 and radius 9u7j*qoz0 , uct.0 cm rolls without slipping down a lane at *qzouc0tuj7 , 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun a:*o2l1s xpt gvq0,wqos the sum of twvwlqqp os t2*,ogx0 :1o terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 mej 6l* )sd t/vx+pae:y. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it i jse *a6p/ ye:)tv+lxds a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls wi,1 y,ros4ofkwel fa(*thout slipping down*y e(o4roa1sk f,f l,w a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. I9x + ws7qqg)n 8/oezsht starts to 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? [Hint: Use conservati)+woz9qes8 xshgn/ 7qon of energy.]    $m/s$

  What is the angular momw1+: n4tr -wo7cduhfaentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at ofcdun4wh7w-a+1t:r an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg upp(ln48q5jcb4qv 4ueniform cylindrical grinding wheel of radiusqjn 5lpbv u48 4c4(qep 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating atqgiz.gdi t*hzx f(gbh4)aix h o//)/: a rate of 1.3rev/s If he raises his arms to a horizontal position, Fgx .t/ g/qxh/ihi)h:f za()z*gobd i4ig. 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 Figgstwiuo2f s8k: z167a*o q1oq. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the2q:u ozo1s*sk76o fgti1aqw8 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 in9c)ccj5 l(vuo* fx.r acrease her spin rotation rate from an initial rate of 1.0 rev ev .r(ox 9cc)vauf5*lj cery 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 h j/z2kp b(n;r *tbf* 8gl,5ymtkooj9rotating 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 the rotating wheel. What is the frequency of the wheel 5 9plt2kbkhfzjt(; gn8oboy * *m j/r,after the clay sticks to it?    $rev/s$

  (a) What is the anguccsh63d u l7-u7x1ysmlar momentum of a 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 mo cwrz8g8u bz+: lvtba)+bc0n3mentum 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 cylindrx o,-tisg:) isical disk of moment of inertia I is dropped onto an identical disk rotating at angular spee:sigot- x)s i,d $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2o0stpg 3286lun4g jar.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 tq3 zqxjzq;n(um8y(0. 3jsvuzhe center of a rotating merry-go-round platform of radius 3.0zn z jx8mvyjqq3( .qsu( 30uz; 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-rounuuppy2fj rl(f564 2uh d is rotating freely with an angular velocity of 0.ul h2 p45ufyfup6j(2 r8 $rad/s$ 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 dwar0mx-rdbi w9qycwrx 6m)f,m6eyk,2 vk.hvb:8 f, losing about half its mass in the process, and winding up with a radius 1.0%m68, vf2:b)b iryr .,xk0 -qmcvyhewx96 kdw m of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (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 excwv/s,q8bxo3 0fh9 glw ess 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 ;yvjejix e6: 7nw7d a2$ 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 5q. 4mmh8vjxn(yd;- e kdf-pc rest, that can rotate freely without friction. The mome -v4; pjh8km -n(df cx.yd5eqmnt 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 +pvh yuu+ga+; diameter merry-go-round turntable that is mounted on fri +a h;p+uvy+ugctionless 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 groun1wxsx *upe2;zd 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 rolls behind the person without slipping. We wx;2u *xzp1shat 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 Earth. m;ezd h5446k fqvx*dbDetermine 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 orbitin5;6vx 4 z* dbh4edkqfmg the Earth.)    $\times10^{ -6}$

  A cyclist accelerates frbh th n+85,h3n )xqr gcm.adg/rgs(dwg uj1(*om 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 pba0yf8v3 5m5 n/cwpf uniform cylinder of radius 0.20 m acquires a rotawm 85 yn3avpcb0/f5fptional rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindricaf1b8 siamu3a6v4lzb gck.qw 7)y4jh5l disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.08a34u1.a z 4cmjh7 )sbi5wygkbql6 fv050 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 of 5cux,e f bhq2 bkfzbre1:-292v yjp)vthe 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 siz, bnd ly2c 1gdzbp8e59e of our Sun, but with 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 massb9b5dyc8,pn21e dlg z 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 energyz-g+ei: :/gvre khvy:0n)tu y stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.5): / e:r+:n-ugz yiye thvkg0v0 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrato*d:-05ryj n m vtvu2tes 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 s8y(gg lkd2ny 1peed 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 a:kg9tn7ja s3/ptsheu++) qwL can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Figss + 9g at3pnq/hk+wtju )7e:a. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the ezv:mm-rk:j;,ijsbnni1n2r( 0 7al ffloor, and we want to exert a horizontal force F at its axle so that it will climb av- rznlrbf : n:e7 n;sjii1a,2kmjm0 ( step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s ric8 t)a;ya: 2v *hggeon a flat road is making a turn with a radiuc * g avh8i):2yer;agts The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begjj1 m gca( /xnc6o:-t9rjfjk8ins whirling the rock in a nearly horizontal circle above his head, accelerating it from fn-ka cr6jj8c (/toxmj:g 1j 9rest 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 as thin rods, mtbhv rs(5 bi8 8a7qqj),d;bi.lwntlsp,z7m (aking reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with db rp;s8si )8 a7q.tjwmbnl5vz(lt,b q i7,(h arms held tightly against her body.   

  You are designing a clutch assembly which covl3smyq1u.r93v:j smh, /1 wry -fmyrnsists of two cylindrical plates, ofy.vmhr3ymyf :s1s-1 v , jrrq /3mwu9l 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 tr yl-)p i18t-xza0 fpntack of Fig. 8–58. What is the minimum value of the vertical height h thpif-ny- t0z)8a1x p ltat 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 do not assuqcyvld+v2)+p.e +v ed yvt.-gme $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unw) pinr)xvl0vr176fcv py3ku; 1p2 lbiformly from 90km/h to 60km/h The tires have a dia;vcv0ipl167 r lp3fw2r puvy1n)x bk)meter 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