A bicycle odometer (which measures distance traveled) is at*81a b)kc)sblp)/eg jng 9t*1 ypl ixhtached near the wheel hub and is designed for 27-inch whj al1 t*esgygixkpb*)n 9 )) l1h/p8bceels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angul znrnap ,-za60emi c,8ar velocity. Does a point on the rim have radiz,r-i ,zm6a0epn nac8 al and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be des.pzxvpg2eo/ :cribed by a single value of the angular velocity $\omega$ Explain.

Can a small force eve cb d6s58fo7w;gx8tsdst2nrwc 1-bc hd6,-mcr 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 hands behind youri .nk4skvp/t 0 head than when your arms are stretched out in front of you? A diagram may help you tokpvi/sntk .04 answer this.

A 21-speed bicycle has seven sprockets at the rear wheel a 75fnhf3c3 w48*n unahfyle6 dnd 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 pewl n8 *y5d3h7fh6fcn une3fa4dal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fa5(sj7) i/z6vu:ggjri pv. m gbst have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basi7zg :g 6u ribg.)jv(impjsv5/s of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow bea vknl4ky5 ,kj/p(9yalm?

If the net force on ; zm1c/qf+ zp)cq6wdo a system is zero, is the net torque also zero? If the net torqm/ +c1fzz qwcd)6;poque on a system is zero, is the net force zero?

Two inclines have the same height but make different anvl flj3ze(* 1qi;myh7 gles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bott 3 ;lj 1(y7m*qvziflehom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down hd8jv a 6c:1+u gfmcb7skw3og+i m a9*an incline. One sphere has twice the radius and twice the mass of the other. Whm39mcgjbdaosc87vi*6:+ u+hg 1 kw faich 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 sameoa 9y2yrsis+* 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 at the bottom? Which has the greater rotationalsiy29osy r+* a KE?

We claim that momentum and angular momentum are cp r:rsgyj.5n 0yzl5- konserved. Yet most moving or rotating obsl5 nypgz5rr- 0j:y .kjects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’sxk0mzm 2 o:tb( equator, how would this affect thetm (mzo0xb2:k length of the day?

Can the diver of Fig. *+n 6wlio:: zovsq -gs8–29 do a somersault without having any initial rotation when she leaves the bz:gqsowl-s *n6 v:o+i oard?

The moment of inertia of a rotating solid disk about an axis throug qp/),c8sk/ kqne7y laooy itv*vb59 rv,8/ gth its center cp,i ,k78t)/r q*at5 / vv ngoosqye/ kv89lybof mass 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 r;,q(b)1ziof 6f dhlrlotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the safid ll),b ;q6f1(ro zhme? Explain.

Two spheres look idento 3af6 nn:jj5e3h2ew aical and have the same mass. However, one is hollow and the other is solid. Descri25n:waa6oef3 j3h ejn be an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle poqewu; +z; z7/ lk 5fwf*trak/0q(h owpints 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 tangential linear acceleration of thhpqt+r 7zw0( feaf/kw zl/koq; u5w;*is point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large vmead )lk2g(hxod. vv2x4 ;(xfreely rotating turntable. What happens if you walk toward h(e xldx.v (k d4vax;2m2v go)the center?

A shortstop may leap into thetbbq4j,wn761 e43ft iskmw 7t air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you lowwn isk6m7e,tbq ft4jb3 71t4ok 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 conservati-vhlnekf6gb5f :n/)e on of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways thhn6ebf) /nke-5f : lgve second propeller can operate to keep the helicopter stable.

  Express the following angles in *dds :rhqg8ssvxq66+ radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculatel (h+ dmmclsc5)) yki:f0l2zy, using the information inside the Front Cover, the angular diameters (in rakzs) +ycmfl5 dc:ylm 2lhi0 )(dians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,00 4l0a 7gng:te:t gvgb9zn am870 km from Earth. The beam diverges at azmgn9bg 0egtt:a778 :nl4vgan 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 the motor ) 9nhjjvlcm9m8/i i 5ris turned off during operation, the blades slow to rest in 3.0 s. What im iir9 8jvhl95j)/ncms 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 chkrv2ii5w (-t -jv3uwa ild. If the ball makes 15.0 revolutions, w vv wi-u 2taw(kjri-35hat is its diameter?    m

  A bicycle with tires 68 cm in diametjw51osed,ve0 er travels 8.0 km. How many revolutions do the wheelwo ,sed10 5jevs make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angu enjajk+/a8a 6*,mn erlar velnk*/8a jr + aanm6jee,ocity 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 complete revolution in 4.0 s (Figd0/nbi lei withv -524. 8–38). (a) What is the linear speed/v4id2 hnii 0lebwt -5 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 ovzodvjv6- o1 + +ixez:60rdzb0; dspr f the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed6,r(vn np r fh3qbd.m9 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 rozo.6ksgwd( /9 ahre4holu)e/ tate if a particle 7.0 cm from the axis of rotation isuadl)9zsr oeg6 h/kho4.e w(/ to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates+q7tk/ccozwoa*qem 0f 7g +1f uniformly about its center from 130 rpm to 280 rpm 0c a7oqe+/k+ zo17tcffm wqg*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 radiu8y+7jn*9il 9gz h hsdns $R_1$ 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 themselves /epy mrpn.s*f)ee ywz03i5cq,-pd: q/ ry4ic into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thoughtsyp 3f c/ ci*5,4 p0n)em rd/qiepyrq.ze-w:y 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 rpmadtgrv b6(e- 5 in 220 s. Through how many revolutions redgv5-at (6bdid it turn in this time?    $rev$

  An automobile engine slows i. isuouzhb2,4w6io* down from 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 for the stresses of flyingcxc4l5,kfwr0 highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reakcxwfc l,40 r5ching 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 rpmc* eke qy91 ems77fgox 2-p)pg to 360 rpm in 6.5 s. How far will a point on the edge of exe7g g2omqys 7 fck9e*-)p1p the wheel have traveled in this time?    m

  A cooling fan is turned off when it is runniw n5k)05l6g *qw9cs jnqmzyx (ng at 850rev/min It turns 1500 revolutions befqq)l 5 nx (kjwc9m5zg6nwy0*s ore it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its spee/nrlej 7cbb i igdg6:6bh-od yk 9,4*sd uniformly from 95km/h to 45km/-gne :i*rbs ,hc6k97gbol ijbd/6y4dh 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 t-jg-r80e4 ofrx)cw8 vc;nzsg9 4lm aqs)e2 q speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.8 4xan290z foe-mqvj-t qlercc)8; s4 )srggw8 0 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 climbing 8mf mygo:vswv /q76f pf6u,v8 a hf /67 v,f8fsmmovp 8q:ygvwu6ill. The 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 end of a door 74 cm wide. What isf .whgt c/r r+ +i mw:dwq+*rac393mpg the magnitude of the torque if the forcwahrcm:q dg ig3+w*3rt/+rw 9+f .cmpe is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in t-neyx htldh ( )zd29/Fig. 8–39. Assume that a h/dtxe-t)z dyh l2(n9friction 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 a massledbkv1x t(y:4 qss rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the ma x(q 14d:ybvtkgnitude and direction of the net torque on this system.

  The bolts on the cyli i.bvm qhbb5 edf*:wmbm ou9.-:2gh.inder head of an engine require tightening to a torque of 38dbhm-om 9: q .5ubi:fim.evb.w 2bgh* $m \cdot N$ 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 inev((:iip k3kcsa 5rs,crtia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its ccki pk3is (v,5c(: asrenter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diametcvx80(jvkear6 s4 z e.er. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can ( 60xkece 4sav. zvjr8be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the enj xyoekjv5d03h*u5g 6 d of a thin, light rod is rotated in a horizontal cir jxgdj50ok63hyeu*5v cle 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 ss id ah/5ty5 wwu/bkdh(1),son a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her handsbk,/ds1 h5 witu 5d/hs )(yswa 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 ce22vx3jk70 (puyh 6k lu7hsz array of point objects shown in Fig. 8–43 absh0 c7zv2xu6e l3yu(h 27k pjkout
(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 oxyge xe/l5/fv9vs-pyil k3n 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 cylindritnk3 h) g4*egdcp 4c4fcal 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 steg34cgd4kfnh cp4* t)eady 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 8nw e4r+y0 )swhwe sxux0,m:s9z,x ao; .50 cm and a mass of 0.580 kg. Calcuywwx,h)nx +s ;x0u4wo s m0,e:9ea srzlate
(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, acc2a 91q/z*yc zq6m2rnkbi7ye gelerating 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 hw*x;tgbmv t : 23t-zuy;; dz.cnohtj/and-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 torque required to produce ty3 x;noz tjtd;z;.t*w g:ct2mhvb -u/he 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 eventuau,6mc tj:nh9 un.t i:4jvxow6lly brought uniformly to rest by a frictional torque of 6ih ,ux j:u:n4w to9t6cj.nvm 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. 8–4twsiuxq8c 07 25 accelerates 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 ball is thrown solely by ths;a (oj34r+jf ii8leme action of the forearm, which rotatei3+soaj8jim (r4 fle; s about the elbow joint 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 shown in Fig. aqfh3l-i- j3c7s2kb. tnz ukiz900u4bah*ei 8–4hilhj90t. i-k 2 z40-*azubbnk u3fcsqa ei736.
(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 massesc ntls)0)s.-uoh-fsol*s s z :, $m_1$ 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 withinq dkbf15q.un ( four full turns (revolutions) and releases it at1nu(fdk .5q qb a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment6f*gn)0ic jvrhlof98 of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 28z s-n7x6 5wcxkzsd9.od ik62g0 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg angrm:wq/g 1j2 ad radius 9.0 cm rolls without slipping down a lane at 3.3g r:/m2gq1waj $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to tc v)yk0..g om2ys/mwcl9jx+4:n i qj jhe Sun as the sum of t4l+0n9y vcs w.mjcqkix g)j/ :2oymj.wo 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 massgn- z6 +cpi/glmbb +: u4v:ss gvmt;6n of 1640 kg 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/vlg4cug zvin6g +tmn6sm -: b:b; +sp? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm qe4 v 6f h,l+ep*u c7:obwew4land mass 1.80 kg starts from rest and rolls without sliplfc*b 7 w64,uee+q4eh:wl ovp ping down 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 i u.ocw00 2kjtmte:zs)ts tip. It 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: Uts2 e00cwm j:z tou).kse conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of k.1 jlv7zi 9xx(fnif-a thin string in a circle of . jfl 7xv-f(nii1k9zxradius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum ofp3j*7 9mgz:3s,0hr ov ov aijxo7 q(kr a 2.8-kg uniform cylindrical grinding wheel of radius 18 c( jjrm3hpvr,9 gv37ooox zkiqas 0:7*m 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 d)u4emxd -n:krr/o5y a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, d:yemkx)/dn ur5-o4r Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of in+ 2 u7mn p9om /s8s*daqldtlo(ertia by a factor of about 3.5 when changing from tp7 talo+ 9*dms 2 u(mol/8qdnshe 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 increase her spinb)ie) :;lkd sz rotation rate from an initial rate of 1.0 rev ev:)ed;s kz)i blery 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 rotating around a vertical axis through its center at a f v j8829xo)k23hl bk4 n5s pdlz5dwdlarequency 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 8 xsd5lozd)l2njk9 35p24bhklwd8avthe frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angu;9(66uai ahg lutqm1 jlar 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 momentum of the Eart-5 dji3wlmm5ch
(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 o9 v ov. 9qy owu s2z;ftl-ar7lt3fqz17f moment of inertia I is dropped onto an identical disk rotating at anlof3w z2 vttyoqvrf ;1.u77-9a9 z sqlgular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2bh rhz3e96 jyjn-w*ylu waz:( rf.j;7.4 $rev/s$ 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 tphyzv;o1da5mn 69elbub(o+ . yx:0eqhe center of a rotating merry-go-round platform of radius 3.0; yb 9ezduov6h .y5(o0bl npq:+mexa 1 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 freely4ly0d4 d hfbqw4lp0 6y with an angular velocity of 0.qylh0 4blf6dd04yw p48 $rad/s$ 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 6r() mwtbrd5xd /mqz0into 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 w rmdmb/6tr0w5 xz)q(d ould 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 excegcx1,rnny(n ie:;og9 ss 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 o k1vbh 6s(re:q.4ugf$ 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 platuv n0kf.gak *d1t/gco38it 7c sxd5v5 form, initially at rest, that can rotate freely without friction. The moment of inerso/v 0td*fi 1t7. a8k5k5vn 3ucggcxdtia 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 diameter merrye /slr,uu z,sia/ k3nke3 :aa.-go-round turntable that is mounted on frictionless bearings aaea3:i/s,rznk s l 3u/k ,eu.and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with *9xwdmm d : frtly9;(ethe 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. What length of rope unwindsd(m*;l ym9d wrt:9xfe from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the 5 0f1 g3hnfrv xrq5iwkb(cwj n4ge(0. same side always faces 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 orbiting the Ea f3q4w ben( 0rnh( f5gkgwjir.vc0 51xrth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a r))w9u4 urnci:p4 :gq1my reahuuvc +*ate 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 cylindi hogielb1c1 ,6j- rj.er of radius 0.20 m acquires a rotational rate of fromrjlg o. ibh,1icj-e16 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 soli6axu -wp:u 4(q lb*pql52iubw*lvfx7d cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energ* 7uqfbllux5w2:xai4w* bvpl upq6( -y 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 i*nzp5*vk(g94p fwu tdmrd-/ o ,keeo.s the 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:k f6:lpfxxwf eh+d ; ,,7ss.fr,klyg-c )rzc 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 mass in the process, what would its rotation speed be? Assume that the star:;l:rk+,,6es-7,ffc)hfygz wsxrl xp. kdc f 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 e*k;f +os 81evntb *ig32h laklow-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a tot2+loisg;n 3kee8 a*h vfb tk1*al 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 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 illustratea p*car i3va.+hn8kh/g1xr q )s 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 horizbt)(wbks ,v uc z0./x, j(wulmontal 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 fs5x2b s-kk:ks reely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod isk2-: xkks5sbs 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. Its-+q pj-x; /dxpps/mr is standing vertically on the floor, and we want to expjs q;d/xr/pp+- -x smert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with sh2sanm y7 .9wupeed v=4.2m/s on a flat road is making a turn with a radius The forces actin7 .s9nw2yh umag 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 lengthsnk *xfn vn72d-;wje6uqa0c.,av(qz 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does(.;0zqfwkunnn7- xvad ,vsc 2a je q*6 the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as r f- j3xc,sxa7thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for-7xj s fxca,3r a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assemr s9t o1kx5ad+bly which consists of two cylindrical plates, of x+ o5dkra 1s9tmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 of 6q+m:v;b ru*d xg7 8q(gd cpxoFig. 8–58. What is the minimum value of tdx+;q:76v* b (gp qxgrcu 8modhe 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, but7x k.t mk7ll,b do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unif9+mpnbj,dw y*lsqb 1. ormly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular a.byjnp 1,wb*l+ q msd9cceleration 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