A bicycle odometer (which measur.b+hl1ncb/ 2/b0l +rdg n lgbdes distance traveled) is attached near the wheel hub and is designed+ /l/b 1dl2rbgh0c.bnl+dnbg for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doembl8x u- zdotbsj 57*i3sc8h2s a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increasjmcosdz5 73ibhsxlb82 u8t -*es 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 desch,:/5w:frgv xn ndto7ez1w7kribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain;.3 odsuz5lhdni f74a.

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 behindaih 0)g fu3jopjj kvvy +yx+6unw;9/7 your head than when your arms are aw; xj)gnyujokvp/9+i6jh f307+uv ystretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel andca9j7ct,o;b2.(6mq -4nbfknl d gkb x 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 7xk jf.d26kgq-t ;onc9 4mb,( bcbna la large front sprocket? Why?

Mammals that depend on being atz;b1mib/ ppahljr*vg+ 3 :sg/o*+mcble to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fim;jrmz+g*/ o pt cpilbasbg*1:3/h+v g. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrowx1waa+v+xjz 8gagec4 +v z29l beam?

If the net force on a system is zero, is the +- 7d q byaxa5l:*;siaxetzbl8l7 b,nkf:q9y net torque also zero? If the net torque on a systdqn ,qs5ll+abf:ka: ay7zy i xb*xt;9-leb 7 8em is zero, is the net force zero?

Two inclines have the same height but make differentqm*j0l v8vme 52hs a2x angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be grjq5v hlm 8a0m* 2xevs2eater? Explain.

Two solid spheres simultaneously start rollingyz +sp4 tgz cbvo5jxg *e1--;g (from rest) down an incline. One sphere has twice the radius and twics g4c-z+ tgb1 x-*yvozgj5;epe the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and rhni9 +7s:uxc vb(r 1hthe 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 energ (:i7nbh hvu+9c1rrs xy at the bottom? Which has the greater rotational KE?

We claim that momentum. 5i/ oouz:yko and angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Exizyk/: uoo o.5plain.

If there were a great migration of people toward the Earth’s equator, howz4 g7bq-g,id k would this affect the -7 igdk,gq bz4length of the day?

Can the diver of Fig. 8–29 do a soe5v e n6/ff.rlmersault without having any initial rotation when she leaves the board?56r/ lnevf .ef

The moment of inertia of a rotating solid disk about a)*84 d 7ius h9fp ca;kyesgcl7n axis through its center of masef yp; 48)sla7c schuig7k* d9s 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 holding a 2-kg mass 9l:f c x4mtfo5t n53pbin each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Exo5ctpm nf :5x tf4lb39plain.

Two spheres look identical and have the same mass. However, one is o7 th,4 +ki/t.lmdpw2uowmv/ hollow and the other is solid. Describe an experime. ohtd/7um/,o4+lp m wkt2wv int to determine which is which.

The angular velocity of a w n)1j+ifvb ,;*wbt)i6gvfmk z/.a ui xheel 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 accelerati/w)v ifjf1; . b tzv*ubgma)k6+nii,xon points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntabl eoy7qqov 8 h9+kj)h/xe. What )+ qh 7oj8q9yoke hx/vhappens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As3ezd4* 4fzmwn+.;f(yp13qcrup pme w he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (qep p4nf zf (zyweup+.md;mwc3*r43 1Fig. 8–36). Explain.

On the basis of the law of conservationqc iozmb shtdf5+b0ph- +6 80d of angular momentum, discuss why a helicopter mus hsb0ihpm+t8- 6ofb+dc 0zd5qt have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the followingz(y v2ow1xy,x angles 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 coincid ))iqb0a:w ou;h:y z7y/bjdidv4k+mh ence. Calculate, using the information inside the Front Cover, theki)+qddbhw;o v i z0 :yjb y7u:ma)4/h angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at tamcbc-7e/n 1jgof g6: pq* q9ehe Moon, 380,000 km from Earth. The beam diverges at an c mfe j*p:ogc/9e671bq ga n-qangle $\theta$ (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 is i c hqorh.ot0sm3* 7shwxefe3-2l.q3turned off during operation, the blades slow to rest in 3.0 sowroeh-eh0x.ti3* q7 mf32 sslh. c3q . 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 azmk 37eyz33r)wxh tqg(e/i8 snother child. If the ball makes 15.0 revolutions, what is its diamewz/) im7(r 3xqgstk h38ye3ezter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How *zs/ of3brh sp;6gv8. mwf8im many revolutions do the wheelsr s /.pmg8om8s*3hizwv; 6fbf make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotati fsaou o9p+64gh2v1s es at 2500 rpm. Calculate its angul6 hs4 1f2 suopav+9oigar velocity 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 (Fig. tu*ls* hosye9lj-4y x1 w4vd18–38). (a) What is the linear speed of a child seated 1.2 m from the center? *14*9 eyyjsswu4 tvldlh-o1x    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the 94z;vpwu3dfo o/6b libp*) hm3 xadu*Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of *t7h8cgs1ct1btujo / bf l(x;q *pgt8nk-6 wva 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 ofux: ,psn(/ emi rotation is to experience an acceleram(i:s x, penu/tion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its ceg udla(0ft emy1(wo4,nter from 130 rpm to 280 rpm in 4.0 s. D( ,4 d tylumfoweg(10aetermine
(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 radiuc5y9ku c p*8ysccu ; lel:oh;)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, astronautu o dnv5s)kcd 1 + srcjvma3t)x6r;g.0s aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of theirt3+6)dkc0a ndgvcv 1 xrmu;o 5s)r.j s trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates ux juhlg+*ite+,t6vl 8ytxn. 8niformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it tuht+6gt yt xuj8l 8.v*en ,+lxirn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcula n7mamw-8jt8 mte
(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 whirl/ihoy182+jw lnhd4d xing “human cdjn 8i ox 4yl/h12dhw+entrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerate8. hd:,vlhp0bpd8+km c38rccf vx +jes uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this time3md+vh .h8lcbvxpf8:+ ep c j80k,crd?    m

  A cooling fan is turn9w6sp/:w kk*ks u (fs1f9lqmned off when it is running at 850rev/min It turns 1500 revolutions before it ckf1*99 l usf knqkpsm/:w(6swomes 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 mak,nu w+zq((yige 65 revolutions as the car reduces its speed uniformly from 95w,q( un+i yzg(km/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

  The tires of a car make 65 revolutions as the car reduce- dgy8er uh)l;s its speed uniformly from 9 ;8yrl-ueh gd)5km/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 ridinm,w.+mt 9c9a,:xjc +djjvk,e -fyxi ng a bike puts all her weight on each pedal when climbing a hill.d+9nw,: ijj ,9ext m,x va-f kc.mycj+ 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 prmbgjq e,6x58ez3j o 8p:)d ydoor 74 cm wide. What is the magnitude of the torque if the force is eex)dep,:5j oqjr my883pb z6 gxerted
(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 wheebury4al,u ; u:l shown in Fig. 8–39. Assume that a friction tor4uy;, u:rabl uque 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 wn(*1h jtp7pizvf- q2 rhich pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate t tq1r 2jp nhfivzp*(7-he magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torqu;v3h.na bub:m5ihwzjh7 d . l(e of 38 l.ib;anjwdh3.5 ( 7 hm vzub:h$m \cdot N$ 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 momentjawtbsjdhn9 er:zx-l *-/ c ., of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its cen,lxjze/: - cr9bh w*. d-tanjster.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 c1xhu o:.lh;2c +d dnmj6 ouyb/a.:bohm in diameter. The rim and tire hn;/ob.b .1hd 2mu:6+h l yoj xcduoha:ave a 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 end of a thin, l mb25tqbg7 sewl).7e7x: 3c 2dayweh.v o k;zpight rod is rotated in a horizontal cbw3z .yha22 )w7ob7e q. emgckv7d;p s:lte5x ircle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at const r-y8o+r *eoypjk*;vb ant angular speed (Fig. 8–4e*yrk o brp+*yo v-;j82). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects raob9a/idx c l;arp: 6 a9rv4c:h/o4f shown in Fig. 8–43 ab;p9xvh6alro9d f/: roab/cc4aa r4: iout
(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 consiswzzh l n5h7z4;(yp;r hts 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 satellite spinninp ;+y 2g/caas9vk kox8g at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the sa k2yop9 cxk+a/s8agv; tellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8 y/ogm*5tk u(o.50 cm and a mass of 0.580 kg. Ca oyo*km(gu/5t lculate
(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,zhd + :8md2*rrvjh n*-exk fj+ accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a smal4 nyv)bji+g5adbd0atu rn ct6qx6 .*9l hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk vutgd.y+ b j6dr tca5a90 bq4i*)nn6xof 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 ig.5 9 fzn7xdcps eventually brought uniformly to rest by a frictional torque of 1z x59dp c.7gfn.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-kn6mn8qn-/tqv ;gajm - v,y(qd,ct j,l g 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 ba3 dg 0r1hgp*bull is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated un01g d3hbp r*ugiformly 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 inkfa*pba814pje 5 zw7vn,c(c(f6 mfwu Fig. 8–47ba8a1*(vfwzpm6f ckep(,j ucwn5f 46.
(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 massy u( a;nz9wn7ges, $m_1$ 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 fulllv+gn;l w ,cm.ocnmw n+2 /q:g turns (revolutions) and releases it at n,+nwg mmnc 2+o.vllwg ; qc:/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 moment ofn3x5 f2tznz,)bs1(qr unextiw60r/e h.etx ( 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 deve3h * )g)vgdhtjcprq70 g tw/:1kcdex 6lops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass ug5 +aw :ai4wg7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.5g:w giauw4 a+3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect 07iqw- d6 l8oxepzzi4 to the Sun as the sum of two8 -dx6ow7pi q elzi04z 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 rad d7(neg1i mc+jx5a vf6ius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.majgfvde (xi 751nc 6+    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts fq n mt0c1xxp(7rom rest and rolls without slipping do(cq tp1nm 70xxwn 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 i1x yenfn 95v+fs balanced vertically on its tip. It starts to fall and its lower end does notn9f+y5fe vx 1n slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thi)tkbha*qd *. jn string in a circle of radius 1.10 m at an angular speed of 10kq)d b*ah*tj ..4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform bugrp*rrl(q+x)to27 3q: fvitoy /a( cylindrical grinding wheel laq*yf r+qux) p(2oort:gvr7b3t/(i of radius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platfor + sle5cg*ug-jm that is rotating at a rate of 1.3rev/s If he rais*cs5j lguge +-es 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 ih ci/in9 (-/ccz ,d8lrvc o9cj*n. qbyn Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from thi iln 9cyc *z(8bovn-c,chq.9j / /cdre 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 initia* 4mwuwrb3d cuc1l 7b0l rate of 1.0 rev every 2.0 s to a final rate of u rlwd3m*cw1b4ub7 c03 $rev/s$ 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 p2x du; fda i1i3g: -skpfq)3pits center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat di2 3ppfk i d:1 di3x)fasp;ugq-sk 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 b+kt -ffl+z;vmomentum 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 *dzy h9;tk*e 9 o5;kwe7xhxydh
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto anmihi911 u 9t;qdiaq *j identical disk d ijm19*9aiu ;thq1qi 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.wi-5,f+p+zl; ,qhcto aa8 zeu4 $rev/s$ 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 rotatingt1/+ys,oq hy oz j4sz1 merry-go-round platform 1sotoz1s y/j4 z+,hqy of radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round3ktscvw6xfp+gjk *h*b8 e -r 2 is rotating freely with an angular velocity of 0vs3* wb eh2f*8-cptkxk +gr6j.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 nk55kc w:r3py into a white dwarf, losing about half its mass in thwy3k5n rc:k5p e process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new 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 excess ofgev t7 a0 lqo60( qhzm0v49ptr 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 qr h;(sd8 fw,y$ 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 roa7mboa+b wg u(eq-1(r5u4h n jtate freely without friction. The moment of inertia of the person plus the platform i(qr1(bea 75w4hauo+ m- b ujngs $I_P$ 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-go-round t :sk1vtv- gk6furntable that iv 6tvk1:-gkfss mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the gol1jm j(0ex.:ichsp 9(s 188 xlle 6vx g5mesrround with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the s1s8 xjslr0 elv x69l1em85 .x(hcm e(pj :siogpool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same si3y iz.zvxjv6 9zi +2-oylq7vtde 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 thel2vzvyoj3 6-iv. 9+yizq ztx7 Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rayvzo4;d7yt7 c i1rv4c te 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 kn uh)q9;jk;m of3o liy,69k6jpxo ,qradius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constanp qf;k ;,ih6mlk 9) nqx y3o,jojouk69t 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 0.050 kg and du bhrpgcs14nm d*swq2j50 3, liameter 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 releasqcu32 s0 h ,g4rl5pbs*nd mwj1ed 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 whmqp0ws7)qg 1 hm;n,ds eel ($\omega_R$) 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 si;uac3tuz,hw sz-t +5.s vij:xze of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revo vhstz,; -uu:is x+3 twz5ca.jlution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automot vy.7-hl:k qg.t 3trabile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 kqrt-t3akvlh y. g t.7:m 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 illustratesf ;kx6zj-3he bw 0xy2j 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 kyy:gni g+b2 e( j mon-+ie4r5-z,zhv rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ig; jixvovm+n)bij3- (jnore 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 m oi+)bvv nxj3;jj( -ithat the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standingis gk+l iruuf +q; da06*tx0i7 vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). Thqfut+ ;al0i+6 g*i k0rdui7sxe step has height h, where h

  A bicyclist traveling with speed v=4.2 axc6bfp.s/ ypyq-7z14fo um 5m/s on a flat road is making a turn with a radius The forces acti5zpq1ba y -y/o.pxfs4m76fc ung 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 ,y f;0mntbh- .hp-ge dlength 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from hfpdt g.nh-0my;, be-rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her armj gez::gze y3ht.v): es as thin rods, making reasonable estima.::) yve3gzg eehtz:jtes for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two c80zmc6 g-h)nz3qw udqylindrical plates, of mq h8dgz m w-3zq)u6c0nass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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.:icwl a*wu qzli07e,- 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 of the loop without lea,clilw-a:7 uzeq0*i wving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not a*ddgc eu- (4cj6hp**al 5uia4gk k,xa ssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions aswyh9z6v vs;.b4um3k aw7m;r j the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? 67uksvh;4 y3z9wmj;v.bar mw$\approx$    $rad/s^2$(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