A bicycle odometer (which measures distance traveled) is attached :.savbyr4)pu/ i 8f fm :qv)wunear the wheel hub and is desvfusv/f. )pqab w:)u4i8 :m ryigned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocit3-s w7 ia+hjbx8 w*npxy. 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 cases would the mag83*w axs-j+ nb7wixp hnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angularc8vtphag2s3q/ygr sm7q1c - / velocity $\omega$ Explain.

Can a small force ever exert a greater torque t;d0o o(gh w:x)4yp.zej fuof4han 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 tm/0l2 fx0b ybho do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may l02/fhxb0 b ymhelp you to answer this.

A 21-speed bicycle has seven sprockets at9zh 5 nk(jsojh:g.8u q the rear wheel and three at the pedal cranks.(u jo jngzshqh8: .k59 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 pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have fb .(qj bv,ql1slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advlf bj.(q q1v,bantageous.

Why do tightrope walkers (Fig. 8z wtf (,ah9-oh s m1dw6,mz,,le)bnu reke1(l–35) carry a long, narrow beam?

If the net force on a system is zero, is the n;6dr39x tn hn.ydrq6 pet torque also zero? If the net torque on a system is zero, is the net forcr nt 69dxd6;qnhr.y 3pe zero?

Two inclines have the samezhbh yp.k1),jw.*/revzv r5 v height but make different angles with the horizontal. The sw5,e*1rvpvjv.z yh bzrh/.k)ame steel ball is rolled 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 imtcwm7 a0nc vov )(c:ze+km;8ncline. One sphere has twice the radius and twice the mass of the other. Which reaches tk;)cmvt8mv7 nm+w:cao c(ez 0 he 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 radius and the same masz t6oe 3bns)c2s. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greateo 6t)b e2cnzs3r rotational KE?

We claim that momentum and angular ry) vw-f. kih(momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Expikhw ry( v.)-flain.

If there were a great migration of people toward th 8wx7uy6ym8 3 rp2qtqst*sek2e Earth’s equator, how would this ysrkt8e2p *qw2 3ux qt8my67s affect the length of the day?

Can the diver of Fig. 8–29 do a somers p.tsm/9kz 9azault without having any initial rotation when a.kzzpm 99t/ sshe leaves the board?

The moment of inertia of a rotatpvh itiid63s 5ek .95hqo 6pk,ing solid disk about an axis through its center of mass ispi5q i .hk sk,ht9oe6536dpvi $\frac{1}{2}WR^2$ (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 ,lf+xuobq 6n0 a rotating stool holding a 2-kg mass in each outstretcul nq0bf,ox+6 hed hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow ands.k sv*9bae p* 7a-- imw(udbz the other is solid. Describe an experiment to determine whichs a*k-dm(pa * seuw9-v.b7z bi is which.

The angular velocity of a wheel rotating on a horizontal axle92u1(jv0u6uein f3 cnxafxk0 points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the un10iv jun ( xe93fa06xf 2cuktangential 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 largcb6 (/zh(igjqe freely rotating turntable. What happens if qizgj(b(ch/ 6you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he-c27c- log rma throws the ball, the uppgrc-7l 2-oacm er part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why g4,xd +;yz tx .+as7x jyyzn,ya helicopter must have more than one rotor (or propeller). Discuss on ynx xjydayz7t,x g;,4szy+.+e or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) xg75oe 7y)2tueq/ awp 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 ; +jw(str(ehacoincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as jw(; ae+sr t(hseen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380, sljn8y7.txc /000 km from Earth. The beam diverges at a7xsjytl. /n c8n angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When 5e4orw)n8 f cbthe motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down? e 4fn)c85b wor    $rad/s^2$

  A child rolls a ball on a level floor 3.b5 ct+h evqj655 m to another child. If the ball makes 15.0 revoluhv5q 6b e+jct5tions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revo(-gwxr0st vx0i3x z9*d sl2 grlutions do the wheels make?gri29lsvd (xrx003zts* wx-g    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its+c*r r f,qk*mjp 0w*og angular velocitwo0*+f qj*,cpr g*m rky in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one cow4-9rk5xjf jr oe0zc:0xc qh3mplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m fze 3f4jrkox:w05h 0cqrx 9-c jrom 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) in its orbit around the Sugy607tu qu s+dpyi4p0n    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pointp(; oipxs ntjdlx,) l;3b1f+f
(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 c up-igzc:*os3 entrifuge rotate if a particle 7.0 cm from the axis of rotati cz-pusio g*:3on is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about 8 3j78 z pmoi5ot fb+ngg;/jhwits center from 130 rpm to 280 rpm in 4.0 s. Detebj8op +fh3 mw;8tzn/5iogg7j rmine
(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 yt wtq+)++hcksuq)vx *bl3 9pf,lfh7rs9- k d$R_1$ 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 themsel2nck.u ;y7qefv1m 6 * 6cm79q ulckdljves into a slow rotation to distrib7mkuq;qe2mdv ckc 19jcu y6l nl76f *.ute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be 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 accelercb6uc*hw4euk7t y3tfg a( 56d ates uniformly from rest to 15,000 rpm in 220 s. Through how ufu5yc4tc6t 36adw (e7kgh*b many revolutions did it turn in this time?    $rev$

  An automobile engine slows down fr9peq hxq t-xqbir.z4 0t9+kz + bww6m;om 4500 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 fo i:y.6uad gmtx 1w)o 5e2: 9cx3nlgohnr the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0.6)xhmwxcl og oaund: 9t52:i13 engy 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 unif.sll2zpw:fnrf b0+ i(ormly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the :b+2.slr 0ilpwnfz( fedge 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 revoej7*asip)mj a7 y;t .siuqf x5-/rp6zlutions be 6ifa;sipx5ts/u7m-aejp)jq.y7*zr fore 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 reducepim/h*o,7 4ygnl5sqv2 1au ui s its speed uniformly from 95km/h to 45km/h The tires u*spii2m/oalug,vqy 7n1 54 hhave 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 uniformly frx(bg/pr68j usmqih7 0omg6 b7js(mh8/qx pru0 i 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all hh2 oljq84ohj 4er weight on each pedal when climbing a hill. The pedals rotate 42q o4jhl jho8in 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 cm wide. What is thelbra)h -v7)pw,tvsp,g.n jy; magnitude of the torque if the force isv,wj- pp;v7 rah.gtn)b,y s)l 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. 8–39yc) v.h m*fh6t. Assume that a frictih*f.mtyh6 vc)on torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of aekwtq4 t0ok am( 04s-v massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the hkvaq 4t4otw(00 kes m-orizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightfgw1 zq nmt;vuh3wvkm44;, r: ening to a torque of 38 kvmh;q wm:1tvwg4z;n,f u r 34$m \cdot N$ 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 the ape o c3qjq0g.5xis of rotation is eg.c 5 jqp3qo0through its center.    $kg \cdot m^2$

  Calculate the moment bx k4+ p5)v8gfhm oud-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 thok- +hbd5mv4)uf gxp 8e hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin1dh(85 xj:cul tsbl2n, light rod is rotated in a horizontal circle of r :b8ljnc1 sut2l(5xd hadius 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 potter’s wheel rotating at constas. 4wxty :oq36-ot-rtcu ar g:nt angular speed (Fig. 8–42). The frict: yu64:c-o -r wqrog.ats xtt3ion 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 theu8z8ktzucyp-aw 31roj -+vk 2 array of point objects shown in Fig. 8–43 abou3ur+ap-ko uz88ywk-vjt1 z2 c t
(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,js pm: 4bh(bekh4c6l 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 cylind2k gytz-p(u( 8i wi.8v7a wynwrical satellite spinning at the correct rate, engineers fire four tangential rwt2iag8-ukyw7. p(8 yvin(wz ockets 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 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm andvd(n -b-uwrwv2c3d jm+9*y bq a mass of 0.580 kg. Calculat-+myjr(bq 3-unc2wv bd d9w* ve
(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 v gd4ukq1ue4 +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 va20ozxdm7) fm s(w+30clj k ji3crtfr1x:)von a small 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-rf0do7mxtv(cf)kx1m3rj0w szc +iv: 2)3 jrl aound is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotc .d1um4ru6 nu-eog e0ating at 10,300 rpm is shut off and is eventually brought uniformly to rest by m .u4e-6ruengd0cuo1a 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 accey ldc088 g.ueblerates a 3.6-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.q(af11q hgl)3 nusjb ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. Thfh(qu. lns3bagq j1)1e 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 shown n e/6o*x 0o(kcffs1mt6z0bfc9*rq b e in Fig. 8–46b om06te1(x czfqroc6f k9* nefs0 /*b.
(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 two masse.f l2hlv3(un:ggsz z) k:t1ck s, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turns (re3, c/dowqqx rwc2w0 ,evolutions) and releases ito2cq/3cwq,rw,0ew dx 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 ha5untoz*-usac t d5-4gs a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque o0ug)w)0*yc/) cctfp txvg2 p p( ut0hof 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mas, oa:n-ll.kc: za6dot s 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.3 :6lzto a- cak.d:,nlo$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with gmt1g a6o*c zqkh (4r5 69lamdrespect to the Sun as the sum of two terms,da 4q1 5* thgmkoz(g9arl 6m6c
(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 r2frjh td c*1od,* m3oyx t7y9kg and a radius of 7.50 m. 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 is a solid cylinde t7oom39crd *r y* jy2,td1hfxr.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from resw2m65m1tog7l)91 fsg ef db lzt and rolls without slipping down a gmls 1 e597 lf21fdt boz)6gmw30.0 $^{\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 to fall and it 0y:g)zhoued9s lower end does not sliped90hu) yzg:o . What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg b6 w-c1m ks1/innkk o6,otj 8nxall rotating on the end of a thin string in a circle of radius 1.10 m at an angul16koi/ cnk1 n 8njok,-t6mxs war speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grindz73 wb7s ikkl0fwz v0/j5wb a*ing wheel ofwwz l7ivb 753asjzw* 00/bkfk radius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotate:*hx kj: iwn9ing at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of roeihw* 9x:jn:k tation 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 prbjc 2tst qj(c-b (67reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in(-s6tjq7pt2jcr cb (b 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 f2r dr48ozxl bq6w7- orate from an initial rate of 1.0 rev every 2.0 s to a final rate of 3 l x 6df 2zq7o8wrb-ro4$rev/s$ 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 x5-md,4fpz pd(ica/b 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 after the clay sticks -4bfdmpda5 , x (zcp/ito it?    $rev/s$

  (a) What is the angular moment y2gu)eoe53zu5 n fjy4um 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 momentum 1fa.wqdb0q aw 7alcb8eget8 0-9*xnet-ix; k 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 ocic-, tio2223;aysuy h;xlr bnto an identical dil aub;i s-r o,yhi23y2cxct ;2sk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at3t 9333d, e81nvqo qdymc6 iu2dnn vnty w(7rz 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round plogvz 73pah ;w* lw n69d:nkdo 7zj::blatform of radius 3.dpnzo6oh7 wkb3: ldj:;ln7 :va9g zw*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 ro zaum67tiwj) rnot( c) wn7;, i-pc-yytating freely with an angular velocity of 0.tiyu( taip7wz) wonn,myrc - j-6;c7)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 ( r6+:gdkcckxwq-b 6binto 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 rotatio(d wg-6 bq kkbx+ccr:6n 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 excess of 1l;t8y7+ h -jc/ )jlgyqnu aai*20 $km/h$ 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 ezl3.zuqi/ug 9kx4;g $ 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, izox-9f*u,x tk sok :,dnitially at rest, that can rotate freely without friction. The moment of inertia of the person pluxk :xus *ozk,9o-,tfds 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 ed. zjr*+f*od;xklji .g e)- js9d4cuoc ge of a 6.5-m diameter merry-go-round turntable *go rcjos).+kd9.l jxd ;*e zjc- iuf4that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls o+c1eu xjxx 4u9n the ground with the end of the rope lying on the top edge of the +9xx ju 4eux1cspool. 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. 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 Earth5u ym9s9 8 muwbx k+llge9j+4d3uwq/ e. 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 uqe 9ws3gyx+lmkbu45l+ u/98d je9 mwMoon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/n/r 6tnsd 9l, ):riexj :-mes2vya9et $s^2$ 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 avki ah8 .0uaiiuo2;*q uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque d ;0ivhq.i8o2u iku aa*elivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disulah6 er/fa .2ks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conser. ef2urah6l/avation 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 the:6zy d 2td*cfa 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,6t3 m cq)xu4di-wm(yfj je.f as great, were rotating at a speed of 1.0 c m( -jiyq4mjxe63)fduf tw,.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 momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is fdo5x -9q5-k 6ff6gvg p 5ebeb/6bzzbd or it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses pq9b5g dg/5kf6 -ex z e6bvbf5d-6obza uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates .hfdpp 5q; 8cq f8nxs.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 rolltzg0/xomun;6 ing 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 canhf3 z7h 6ymex9 pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. T yz36h9h7xmfe he 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 suhaal7u be; -60np0p standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step agap;70hbu s0e au6na-lpinst which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with spee3 0n9jczj.noprvxu6ku l3 3 )zd v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the jkrx pov0 9n zu) c33zl63un.jnormal 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 9/ 5 7m5rxbsiqa d.gvmrock into a sling of length 1.5 m and begins whirling the rock in a nemdra59vg b /is7x.5qm arly 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 as thin rods, mak ; xz5(uaxn:xlkv (7yging reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater w5( gaxzun:l ykx;( vx7ith outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consis(thpl75j hls,ts of two cylindrical plates, of ml,hlj s(h 7pt5ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 flyt 91 1mye7x;n)8pmk hvyqqy;: mk6track of Fig. 8–58. What is the minimum value of the vel hfk18exqymyymm;p 7 96: );yvqtkn1rtical 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 +izqxrvv/63 rg4 o6 sygue-q-do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions bp7d9b3jr3ltas the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 7d3t rj3bl p9b0.90 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