A bicycle odometer (which measures distance traveled) is atta mb)d2sj+wg2(5zydn dched near the wheel hub and is designed for 27-inch wheels. What happens if z2(d)+yg d5 nmbdwsj2 you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point ont (j 7z5yprqq/ the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformtyrjzp(q /57 qly, 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 described by a single value of the angular velocity f k+zld.u*jl5shm) 4g $\omega$ Explain.

Can a small force ever exert a greater torque thani6d-9 :0lmnz(mm cfv n 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 hart:ff e7p(yrq ev179 ands behind your head than when your arms are stretched out in front of you? A diagram may help you to answeraf:17p(v rrey eq9 7tf this.

A 21-speed bicycle has seven sprockets at thex *- mfh4 0refk0oz3jk rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large froohe* f03kxjz0 k4fr -mnt sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs*d58mhc8qof*j ha2 ki with flesh and * c imhqa f*5kd8o8jh2muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carx6 q1oht)kfxy :)) kge1na: icry a long, narrow beam?

If the net force on a system is zero, is the net torque also ze,;; xb0woprut ro? If the net torque onp x 0obr;t;uw, a system is zero, is the net force zero?

Two inclines have the same height but make different anglesd,5l )2hv o (xeexak -mnu12tj with the horizontal. The same ste1l hjetx 5kv2o2( -),admxneuel 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 resknac fg ldu3/xu1h70n1 hn/6v 0mzd7t t) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic enem1u31axnzu07v6c 0t 7ngn/d d hk hlf/rgy at the bottom?

A sphere and a cylinder have the same radius andqj6msp ho+ zb/6l y(r; 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? Whichh +6 qm6s;o yzrljb(p/ has the greater rotational KE?

We claim that momentum and angular momentum are conseesmc3:9 o) k4,dkkw:katrl d .rved. Yet most moving or rotating objects eventually slow down and stop. Explasekm.ok)ktl4 dc:kaw d3,: r9 in.

If there were a great migration of people towardpzftq.zu6(e raw0 q+4 the Earth’s equator, how would this affeczr +q e40wzpu.at6 (fqt the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation epbpyd m v ;22.4x1zy6fa3o hhwhen she le 2mxp2f;h1dao36v hy 4 py.ebzaves the board?

The moment of inertia of a rotating solid disk about an axis through its2(o-zc4xv e8sn hij5p center of xnsj -ep4hi(z 2c58vo 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 rotating stool holdikxk;rig.1 zqr7u 6v r96k7zl yng a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, orlxri197zg6 6v.u 7kkq;zk rr y stay the same? Explain.

Two spheres look identicaav6yq0o.ox5 w8u1dficv,-g;kzjuh.b hdx + /l and have the same mass. However, one is hollow and the other is soluya8cjhi/;f,w6hz o1ub0k .vd+ qg-.oxv 5 dxid. Describe an experiment to determine which is which.

The angular velocity of a ad+f-ar:0qxi 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 acceleration points east,xq-a+ ida:r0f describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotatw6 o9eltpatlbiik b(4+5.:fao*b -xh ing turntable. What happens if you walk toward thetbiohf(* .6baiol l+:p-4btw5 xkea9 center?

A shortstop may leap into the air to catch a biu8uxi j 6cn43 (bst3xall and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips anduisbx63nc x( 3 tij4u8 legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, c tz1vrx *cmc 2dyv6q:8q2ze3discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second p2 xe6:y8z cv2d3ctmq*zv 1rc qropeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a)qv hvs+e0jfb( /+vxl62fb )dh 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 coincidevc/ e8 4o7cnt;co utsek.w,0ance. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun ankvnc/0 7ou8. e;ac twst,4oec d the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Ea7j7vi8htqa om9wri : )rth. The beam diverges at an anglair7ij8m:9othq7v )we $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rowa9z109 gp 4 rn 4gcz fwl(d(jz3zaof5tate at a rate of 6500 rpm. When the motor is turned off 95zw1(az (f 40 gnwp9 zr4ga3docjzfl during operation, 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 anothy.v*lq, kn/tser child. If the ball makes 15.0 revolutions, what is its diametnqsyk. v l*/t,er?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. fv d;3xjkoh 1.yd-bf: How many revolutions do the wheels b3;dfvyjhfodx k .:-1make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rprn z:),ce4jlwn5b,bm 9 bo:gkm. Calculate its angular ve erl49 nz,:m bbnw,bogkj):c5locity 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 jrs a u8tfz-;/one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of sr-tafju ;z8/a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the -/k*hxr-hy ( fcpe pp7eo,g8gEarth (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 ao*v5ni oc7i w4.+ bv0f)f kvmj 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 ce+h-xj8 gi*brn:19nucr hwuy 7ntrifuge rotate if a particle 7.0 cm from the axis of rotation nc*hw-x +:rgbj7n h8uryui19 is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates 55 ipooxw(+kluniformly about its center from 130 rpm to 280 rpm in 5 5+x( wkoiopl4.0 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 radiu6jn l*: dgeunj+xh0 myhsj(03 v c-bi,s $R_1$ 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 spacec n0t5lnd ,+fdkae6d5zm wa /g.raft put themselves 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 c +f.,al d5 mean6dkwztn/g0d5an 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 accelerawk2mew4y 44hkx3hiw kw5 0m1bws*m*stes uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions die4*40x3hssk kk* i5y4mw w1wmb2 hww md it turn in this time?    $rev$

  An automobile engine slows down from 45w2vow .3)tjcg00 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 flying highspeed jets in a whirling u d +lit cspz*k,v ma0rkda553p1-zh 2“human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachic,udamd1arz l5sk pz30+5* kh 2i-vpt ng 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 rpml2p )ktr2/vxn jb*r(o to 360 rpm in 6.5 s. How far will a po kvl*22/x(b)ortnjr pint on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when ) h+ewun9- v1eica6jk it is running at 850rev/min It turns 1500 revolutions before kejc wnave1 +hu69)-i 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 rgax9h(s8 ff9 o keu c7pow,,t c/lj,w3educes its speed uniformly from 95km/h to 45km/h The tires have a diameter of fsc, kl79 ,a9 hwt,3w8pxgo j /eco(fu0.80 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 revoluww6;t hj9+ yqytions as the car reduces its speed uniformly fromyh6w qwj9;+ yt 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 allxgik26 1x4tkv 7tnq ut:t1tz , her weight on each pedal when climbing a hill. The pedals rotazt x2,1ntx4t7 kkti6 tqgu1 :vte 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 1s+4neqylna pl12momr4m /l+a door 74 cm wide. What is the magnitum s p+qm 42 mllaol1ny4n+re/1de of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle otdkpel el e8)50rd.b4g:tf ucc )78uof the wheel shown in Fig. 8–39. Assume thablg5k 08cedctlr8) pe:)7.defuuo t4t a friction 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 massless rod whr5q)g.z9 qd bdich pivots as shown in Fig. 8–40. Initiallb d.59)dqzqg ry the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torqu/ i9b g0ovv,.*tm-eq zy*(vk+1ql rz1mtfomx e of 38 z*otm/f,(+v -t gi0zk moqvy.r ml 11xev*qb9$m \cdot N$ 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 swpdzh+ry 69g3pu t,1bphere of radius 0.648 m when the axis of rotation is through its center. z hp,d63gtypuwb +r 91   $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Theb y1d7 un:)kok rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why)o1b 7dykkun:?).    $kg \cdot m^2$

  A small 650-gram ball on the ena r u6/xu3;wg/h rl qq7-)0g-nruxr pdd of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calcu/3uruawr)/ l7d 0n-;rhx q rg6q-gupxlate
(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 c zh7/suhcbl( gr*va0(onstant angular speed (Fig. 8–42). The friction force between (rbh70ghu* c v (zsal/her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of w6s82czwmozk: z 59gh point objects shown in Fig. 8–43 abouthzg w68omzz2 :5kswc 9
(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 consis9 w;qgs rwoc38ts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical sateq-kw6pil(4qb2 8npm(e..gl/ bh gp hbllite 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 anq.mqh pg(l w4kbe hlb ngb 6p-8i./2p(d 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 and a mass of 0.ywbbvl7fo:38rka3 y ,u6wk8 n580 ykf r8w6k ynl owa83u,7v3 :bbkg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, acip2lalw6p080my gxq 5 eaw+,p celerating 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 tangentiallyz ps0e9q6y /u.g-jqhc on 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-round is a uniform/zsejy-0cu96hqg . qp 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 rotating at 10,300 rpm is shut off and is eve(4 :hpf+bdvau qt 5lbq1 k-d8antually brought uniformly to rest by a frictional to- uqbl+5:d1fd a8pvh4(qt a bkrque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-k9zfmnv)qm95(2ozmb4 8r j wudg 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 the action of the forea58ekyu y8cw q,rm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from 5wuk8y8,ec yqrest 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 lo5xlk6eyv bu5hru e5x(1m5w / vng thin rod, as shown in Fig. 8–4hwee 56vy b(um5vx5kxul/1r 56.
(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 7n ph n2,dgvqw5oy 6glwk3;7y masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest withi5 ciw.sx /ilk8n four full turns (revolutions) and releases it at a speed of 285 s/ icil8kw.x $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia 8 fzw6k3vz+vsof $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of i qnd9ok:h4vp+ gyib-(q*m (/tpz(a x280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls+yw- wnxhvf a6pq-* )m without slipping down a lane at 3v *yx)f +hapw-mw-6nq.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energaf :bu0 4pnyazkki6ja9(.q8w y of the Earth with respect to the Sun as the sum of twop0uwzn96.k f qi:b8 ky aj(a4a 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 m.9k ymt +tjd6; c(l dl/iz3-7tdj (kdfe How much net work is required to accelerate it f+dj(lm d kzj t9 ft-6(i;ycddl/7ke3t rom rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 qs24+ppe ek0x kg starts from rest and rolls without slipping down a ek 4q2exspp +030.0 $^{\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 f sykw6vsx 90m2v:b9emall and its lower end does not slip. What will be v0s9xwmvmy296:bsk e 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 mo g -k04cnm2tlmmentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.1042kcmm-0lgn t m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of, y-mi/m ,sceu8- jxmr h0afw( a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 (j w-uxrh-cs8ame /,m,my 0f irpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his sin /xikvb+h-c- wyek( /de, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal p/ b -vix kk-/+cn(ehwyosition, 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 reduc;xg da,akm/kwk9l5. we her moment of inertia by a factor of a axd la9;5g.w/,mkwkkbout 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial ratrz4c-sd )/ 53 dvfoip: k.pnoue of 1.0 rev every 2.0 s to a final rate /co5):upn vrf 3d.pdi4zk-s o 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 iui;eu+ /vvx8amh6b6mkiq /x;h 4ok(rotating around a vertical axis through its center at a frequency of 1.5rev/s The wheel can bex(o q 4mv/6m x/b vhi6 ;kiuiak8eu+h; 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 to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3.kemu mvgyo 6f k4,lp+wd 4g.735 $rev/s$ 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 mome8wat6o)b .a*kgheb.f i)4gk untum 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 di*z; mfoxx,81a saj zt(sk of moment of inertia I is dropped onto an identical disk rotating f8a;x tmaxj,o1(zs*z at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns db(y jff9 m3 .tl dj-:p/z8uktat 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 rotatiwju 9 kv/h/o9erqh6 z )vb-)emng merry-go-round platform of radius 3.0 m anh- 9)be h9 j k6w)oz/ureqv/vmd moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angulc+j+ pl+ gwd*r) des+nar velocity of 0.8 pd*r+s+ g+ ln )ewj+cd$rad/s$ 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 dwazlldy88:wz 2d rf, 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 t88ldlw zzd2:y he 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 exc0 p;1k /rmqjbito 1an+zv5j x0ess 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 80 v/ainidd6m hylf +5i)5vn 1v1wabm$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotat irs4tco.fwsmm p;.5g/57 n19t fpkmj e freely without friction. The moment of inertia of the person plus the platformft7sgmpmo/j 4m9 rwkc.p5s;.1fi5 tn 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 merry-goj1 njllr8gc/;b zp -9; g.rqzhpeud3 --round turntable that is mounted on frictionless bearings and has a moment of inertia og.3;bglrpdj - przhj1 c; uqn/-9z8le f 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 rz250dnf a: n5 vgb3, mkljs6pvolls on the ground with the end of the rope lying on the top eg6m b5f z:vpk2ns, lv503nj addge 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 unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwae( uxd-kvf86s ys faces the Earth. Determine the ratio of the Moon’s skd 6u8-ef(xsv pin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1)j(ktcki nk yls 1w6/5 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 cylinder of x153h /3yyw, nmqc jggradius 0.20 m acquires a rotation gc/h15qy3ngw, x3ymjal 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 cyh6g ,w qq1lmp;;ca p;vlindrical 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 energy to calculate the linear speed of the yo-yo when it reaches the en6a m1;pqwvqpc ,lh;g;d 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 r qf( )6lyyzp3mc8a:vvear 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, bu,xo4rpxz;o8qf w4zf(( l(hprt 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 wo4qwpxfpzf(ohor 4(;z8 (rx ,l uld 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 au2ezm;rl nd i*5tomobile is for it to use energy stored in a heavy rot2 z ilr;dne*m5ating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a 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 illustrat jl sxd19 *(3wpafmrt.es 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 horizon )4hzzho vw gvweo6;.5tal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e. ;kgs 2zwch23d, we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, 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 ahg ks3;wdc 22zt the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standin,gz.qn6z +ghm3qx8h44 ffk a gg vertically on the floor, and we want ta 6.hm8z gkx3,fgzq+f4qg4n ho exert 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 speed v=4.2m/s on a flat road is making a turnpa0xs3d m;9bc with a radius The forces acting on the c m930 apxcd;sbyclist 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 5; ;z bmd3lxkm7,gqbbphi. p8begins 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. Wha8k;,57hm3bp ;xlgb.z q dbmi pt 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 zam75sm6a rj 1cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with ou61 j57sma marztstretched arms, and with arms held tightly against her body.   

  You are designing a clutch0)8ok iozf+xa3 g: si.*lfme f assembly which consists of two cylindrical plates, of mass s ik:+fimo0.l3e8 )ox gzaff*$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rou5 51zrdz -/bx ch6ohsj3xg .kxgh track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point cd/h5x.x6so bh-1rk 5zgjzx3 of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, butp)p+k(4a ll)c,*b jcntcv)c5i)c um p do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the c16 cue)b7 jrugar reduces its speed uniformly from 90km/h toj bge)u6r1 7cu 60km/h The tires have a diameter 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