A bicycle odometer (which measures distance traveled) i.b, k0 1rrmo4eacj0ej;* lm/1t acwpf s attached near the wheel hub and is designed for 27-iw kf/;a ce .01l, 4obtajmpe*1r r0cmjnch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at co98;0z hbr w1yt++g +irsyd tuinstant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitudew+br1z9i0hsi t8gu t++ry ;yd of either component of linear acceleration change?

Could a nonrigid body be desc 6p4m qjt3xjox5ar q;n7 4ch(sribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a gry-9bp2(n jtz6wf o( pdeater 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 yourw4njl 0se1cj rs/n3h*c2x 8l jvb k8n8 head than when your arms are stretched out in front of you? A diagram may helpvexr8jc3lw 80n ljsns*h /14bc k j82n you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thr6t) h4 wtqw8yxtmw6/nt a -0niee at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Wm)qw/ 4t 6 8x6th tynw0nat-iwhy? 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 fl)xilesy jjouq,8 )67q of 7p+ast have slender lower legs with flesh and muscle concentpue6xq 8y,)+f7i) oj l 7olsqjrated 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. 0ij;zhb;qc7pou4sxz yro,-rjt- t) ,8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque a8 f s747y wf;aoj,or5l y ad-gma0o2zolso zero? If the net torque on a system is zero, iaym7 ao2wf7agjfos;lo5dz yr8o 0 -4,s the net force zero?

Two inclines have the same height but make different angles with the hori9c 9a fbqrb (nv1,ss3azontal. The same steel ball is rollav39ac (q 9 fbrs1,bsned down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneouepjg9 uqp1gq 7(uk85yeno86 ysly start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Whick8q1ojyq96uye p(egu n pg587h reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They sta f/1 fd01ji i cqcn-p4-akxxvzk1w52 hrt from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bkxa /n -k-i fw5vdi 2p hxc0f4j111zqcottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Ye.r a- h2z;a 6jaq*m9 xleggy6ct most moving or rotating objects eventually slaal6-hgmj62ez ry g.c; xa9* qow down and stop. Explain.

If there were a great migration of people towi ago.79 m:dye d0fpn9ard the Earth’s equator, how would this affect 9omenafp0:.y 9d7g i dthe length of the day?

Can the diver of Fig. 8–29 do a somersault withouhj 9ud1sdk.5a t having any initial rotation whahjdsk91d u.5 en she leaves the board?

The moment of inertia of a r-othbdi 2.4y:y j ,dphotating solid disk about an axis through its center of mass4yh:o bt2hj. py,-ddi 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 hol poh.f(e m1b)jding a 2-kg mass in each outstretched hand. If you sudm jh1o. e(b)pfdenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identksowv9cvma7o d,:;6z6 vl cy2 ;e(j tyical and have the same mass. However, one is hollow and the other is solid. Describe an experim:wv6;,7(cvcm 9o jkdy2v6y ;szaoet lent to determine which is which.

The angular velocity of a wheel rotating on0.y,jsrmz zvas1.q3 -e+np dd 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, describe the tangential linear acceleration of this point at the top o0+za3,ysnpm esvj- d. . qd1rzf the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge 1 82lmjb7nd d;17v enz 4uyz0v-t txhpof a large freely rotating turntable. What happe48n0;tl- 7h1 1vu7ymzeddjbxv n tp2zns if you walk toward the center?

A shortstop may leap into the air to catch a bcf a1 )mxh na0k9m.*8nmc* gji9gsjo*all and throw it quickly. As he throws the ball, the upper part of hij *c o9 8a*mn)mi*sfg9 cmg. ax1jnhk0s 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, di -.vnlp+p1 cgi -tlzd:scuss why a helicopter must have more than one rotor (or propeller). Discuss one or -zd:+tlgi. n - 1lcvppmore ways the second propeller can operate to keep the helicopter stable.

  Express the following a zo9 ktajm0abrov 1h,7a/u 13angles in 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. e ;; njsmxug75m hd j,m)qd+j247urea Calculate, using the informa ,me4 5ajjjh rgd+uem7 q;msu72dn );xtion inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. Th2 eocl 8jb)s/sa;y tk/o+r vt m:os-+re beam diverges at an 2sect+r/vl8oosam;ry)b:+to k/js- 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 gp,)pn 3ud9nu rpm. When the motor is turned off duringug d 3n,p)up9n 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 e( -9pnglbrvnj :xg-7 a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diameter?(-pl b7- r9jegngn xv:    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do/5u1ql+7fdq *r+h ujdf8ae :hmgsf*e the whee h hfgd*erf+7j8leuf:1dm *qua5 s+/qls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcudij 6 1rd q+idx,7rp7w)v8scglate its angular velocityxd+r6siq87dgwj71c pv ,r d i) 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 complzci f0v2pk/ k+l 3klo3ete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the /l30f o 3 z2pkickl+kvcenter?    $m/s$
(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 d9btyb5sgd+.2v wfr 2s.k0 w*2hnhnsthe Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spee 0jq.0. * x6tbk+ncd 2zucsjrad of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axis ts5uufahav6cltp j-s(- /ge0j*y 8 6lof rotation is to experien*cvt 8ah u(a-jlus6 yp6 e-/sj5tg0f lce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleratgr3-7+paky 2bbf jp 6zes uniformly about its center from 130 rpm to 280 rpm2pfzbar6b3g pj-+y k7 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 radiuzz l -i7d0ochnu9z. 1ks $R_1$ 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 tc1czp41 xz+ j+mcgs:fhe Apollo spacecraft 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 intp:xfm+1 jz4gc z 1+ccserval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Thr,cvx-gy (n7 laough how many revoluti a-v,lc(gxn y7ons did it turn in this time?    $rev$

  An automobile engine slftbkj3 i:*bc n;s dzp *::smn9ows 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 flying highspeed jets ipf;m ,5:vsyzzzko+0 u n a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete reu5 ,zk: +z;pszoyf0mvvolutions 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 diameti noa xy-2ydk7q0e8 wjc-b02ter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge oiq8o 2--b2w jyx0td0e nc7yakf the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turnhomw.o+ bbr-6 s 1500 revolutions be+hw.b-mr o6 bofore 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:xff(w j2d u es49l3kq 65 revolutions as the car reduces its speed uniformly from 95km/h tk(djlfu: 43ew2x qf9s o 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolf +lxc5-dwuh fz.j0 2wutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires hhw0c5dj z f -wu2fl.+xave a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbingjh(o4e.ccojo s5;).rs9i jhg a hill. The pedals rotatj.;5.hsoreg ojc (9)i o c4sjhe 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 fixf ;+u25y; rg+s -gy eg 3dzduz5.huthe end of a door 74 cm wide. What is the magnitudeygufuygs2.ud di-r;fxh5zz 3g5 ;++e 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 o)x5 aw1 0w o8h7xadaiif the wheel shown in Fig. 8–39. Assume that a friction torq )wiaox7 8daiwa51hx 0ue 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 mashu cvpsr (c 3a y,.0d;sctd(g8sless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system. hscg.8psd3c t,au(cd; r(yv0

  The bolts on the cylinder head of an engine requts ; 5ya06l2rvr a/ny6hp9qobire tightening to a torque of 38lya052b/raq yt 6 s6op;v9hn r $m \cdot N$ 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 1sre 0- z9ih4nrn0k s(i0.8-kg sphere of radius 0.648 m when the axis of rotation is through it( 9si-0irsrnkh40en zs center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm u/nn;8u(q3r o2nft z6bzq2nx/xm xc6 in diameter. The rim and tire have a n28n/tx(z;z 6onrfmb 6qq2x n3ucux/ combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the n/de/ 1n,8.digv(juq 9bc 8j fsb:bdm end of a thin, light rod is rotated in a horizontal circle of radiu:bv(j8 g m/ ij 1b.9s/ e8ndbd,udfcnqs 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 bowl0 xk3c5* au8utluh.tg on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the cl.5c ah *ux gu8ltt03ukay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point ob1jz. lr)s8pz4aq b/p9eff)yt 2lmgl 6jects shown in Fig. 8–43 ablpllt 6erjz9)ypz4s .b qf 2 /am)81fgout
(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 cona:nbzqu5l s u2sa))3z sists 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 cylindk f69we6dgv) nyh.o;a 47cgymmi f:+urical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each roa d y;nci :y ouk vm9f+mweh.6gf)4g76cket 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 radiu,m7 gc6g7gzrvj1j mwhf jq/.6s of 8.50 cm and a mass of 0.580 kg. Calculwfm1j 7 r qv cgmz,6.hj6gj7g/ate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it -f,)hrs(tt6ib7l rcwfrom 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 tan w5( )km9ju9fmmt tpva2ou 4e:gentially on a small hand-driven merry-go-round and is able to accelerate it from: tjpau52okmt4emmw (v)9 f9u 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 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 eventb(:v2f ,r s)p ddr97 z:nhqepz9 qru/0yi gg+pually brought uniformly to rest by (p 9rv2qdpgfgpn :diy+e0z)/7b,zsh 9rqru: a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-ktx 8hx:tmf ,*gg 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 iswq oloeq1) kyi55*mu lp+ 5npdd d4/b+ thrown solely by the action of the forearm, which rotates about the elbow joo5dkyio)n pm55 wdl*qu4e dl+qpb1/+int 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,jpdng 0:ip4, a as shown in Fig. 8–46. apj 0n:i ,dpg4
(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 a (ow76r3blcvs, $m_1$ 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 t d),3zxh)l(.b:xeptj4 cfpa .umxm )aurns (revolutions) and releases it at au4: p ) .axmxfxez lmd.tb,pa )c3j)(h 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 aj-dd;j-l -1kk6rn8qditn rdow ul *8sp3( u2z 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 of8ky )dyc8 0c 8r9hkmvc 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cfqu- (gnbrp /5m rolls without slipping down a lane at 3. npg b5u(fq-r/3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to sx;qw2g7x)x4l7j fq cthe Sun as the sum of two term4qsx7w7x l )qc2gf ;xjs,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net wa)k mo f+ *:fo;xf3gko8xh0rx ork is required to accelerate it from rest to a rotation rate 8)o3xx;0g:rm+* akfkofx hof of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.2w+x iubnku78 wv k :0wsj5yk-0 cm and mass 1.80 kg starts from rest and rolls without slipping down u0-:bkkw w2y kn jvxiu8 75s+wa 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fa,dmt epacmi5 5/m0d.p ll and its lower end does not sle5atppd/md .m,c0m i 5ip. 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 03dwdv6t :, qbmbpj. 7o.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at at o q pd3d.v67j,wmb:bn angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg u * 8;aflgwr0dslu 2e- br.mwu.niform cylindrical grinding wheel of radius 18 cm whe.w; df8ermwlagubu.l *-2 r 0sn 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 sideks26abqg 9z4r , on a platform that is rotating at a rate of 1.3rev/s b24 zqa ks6gr9If he raises his arms to a horizontal position, 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 modcp4 u41 jnf9m-pkz :yment of inertia by a factor of about 3.5 when chap 9cdjyz- kp um4f:4n1nging 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 rate c0 q);8 jfmxo3+qnnumof 1.0 rev ev8mu0cmn+ qfjq)oxn;3ery 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 aroundxru1 aextl ke ;:k r35)iurx 5ewi5(u* a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and dre5xe k1(;5:*)utwux ruli iear k3x5iameter 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 spinniy )ex*,.y +tsxv5vq usng 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 w4n efgih6;6p 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 m/l.k y(i -dvn8zyev; dropped onto an identical disk;8zd/i y k.l(evny -mv rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 :a pc4 2sk 7xgowx2b.t$rev/s$ 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 vt7. njfv-2yda tc)v 4*2dhjbcenter of a rotating merry-go-round platform of radius 3.0 f7-b 2djvya4vhj n2c d)t*tv.m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with a u1m-.yj-n1 ktqaz+f bn angular velocity of 0j1z-afy+ mqu.kt-n b1.8 $rad/s$ 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 into h9tt m.vt2s-58bn olka 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,8bt tsvt-m 2. nlk5o9h what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve windkq(9cpc h0llbcx :d25s in excess 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 ica-y sx*bko8k-dl:(x1 : i a(oicj6 c$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely ndfi p7rq5yu+1/rq j2 5 ,3ketdq+jiv without frictionq+ 57 15die3krqup n jyt/,2iqfvr+jd . The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands atq:ao8p28 ds8tbcnd8uy, -b* a xcum3q the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearinqoqps2 mb:xu*ctad8- ,cbua8d8ny38 gs 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 groundlyan/-4)0dk x ph4 q6rhgade 7 with the end of the rope lying on the top edge of the spoxradkn4y lp04dhge/7 hq- a)6ol. 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 ale ca93ooi 4m9yways faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital ang oi9e3ymca 94oular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a p*hzcf39ey3row1 *( hfrqy(grate 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 cylinder of radius 0.20 mypf 8/vees q4bte (;3o acquires a rotational rate of from rest over a 6.0-e8q f4b3s /yv;tpe(eos interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass c+vrog;r7n*+v,2gj el sc3sc0.050 kg and diameter+2vs3 ccn l;r+cjogv ,s*7erg 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 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 rear wheem gojw:0uuwd:c59kb2 v;9 verl ($\omega_R$) 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 sizlhst1/ rh)-;svqsjs ; ;h0i gqe 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? Assutqs hj-;qh/vr ;sl ;his0s)g1 me that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use enerzyhr.acs/ lt: /80c(m;wvzmpgy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.mmycp(vstz r8 z:cw/l. ah0;/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 anaw9 7pg-j nbu6x ou8j0 $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3. q0 (+1ranakvb3 $m/s$ 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 frez)k :f *,wd*errr-hp7 4gowc7zqxj8bely (i.e., 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, drx*hj4 cz *o7bq- 8kdwz,fwgpr e:)7retermine (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 hazn3;d mw:9v zts radius R. It is standing vertically on the floor, and we want to exert wvzn :d9;3m tza 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 mh5 r7koih s*yn.9l qv:aking a turn with a radius The forces actr*voyh9 .7h 5ik:slq ning on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins wh1w )x*,y5m7x1zxfdyqlim j )kirling the rock in a nearly horizontal circle above his7lw)y)iz5q dmx, *mx1k1 j fxy 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 hem*o8+n7 (tocb(lmc6 io 5 npypr arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds fpp*85oi7c(6tol(mmob yc +nnor a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly wf5: x91pv p4jhznbwp9hich consists of two cylindrical plates, of 1:jbpxw4pv9 h9zf5npmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fig. 8(pfwt:29k r3i0zcx mf –58. What is the minimum value of the vertical heig : (0mkrt2icw xz93fpfht 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 d ery )qnvt:8)io not assume $r\ll R$ h =    (R-r)

  The tires of a car mak j83x7*9p b/mnhg j4ly ne8eege 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0ggn3*l9p8xjj8ee ey hmn7/b 4 .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