A bicycle odometer (which mmz rn9z*x9bx 5 5gsc1weasures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels9*m bx55zsc1wn9 grzx. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constantjb; *bhf0 6sk* f/fust 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 magnitude of either component of linear acceleratih*fkb fs06sf;*j /u bton change?

Could a nonrigid body be described by a single value of the angulao br/.e+eet5dus:nf;ukimv*u 6l 1 9hr velocity $\omega$ Explain.

Can a small force ever exert a gkqe2 0tu qj0i-3tagb5)rkq5rreater 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 svem9hh6 6 ct8r6wq d9 p,-ipziit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer 6vihm8z-9,wi t qd e66rp p9hcthis.

A 21-speed bicycle has seven spros + gz rdn6)f9gul56ckj6isu 2ckets at the 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 gj sgu 6ks cd n9fi)6lr2uz65+a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower nqt(vs4(3,jol eqbib 4fn ./3x;x u2tow4tcdlegs with flesh and muscle concentrated high, close to the body (Fig.x3 v3c;s(4 4xubtdtb o(. ej4i,lqn w/ntqo2f 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, uqg2h/ruj.obd+vm56 narrow beam?

If the net force on a system is zero, is th,iib8d d;v2dy. l: proe net torque also zero? If the net torque on a system is zero, is the net forcei dbvo:8;iyr2.dd,l p zero?

Two inclines have the same height but make diffe t -5bl;u)gbszrent angles with the horizontal. The same steel b ;)z- 5lgstubball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneoungh1avt y(r4)fz c 4;osly start rolling (from rest) down an incline. One sphere tv nf4 4hcyoga;r) 1z(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 energy at the bottom?

A sphere and a cylinder have the same radius and the same m0l.eak xq-p vkxy;m(/ qzjcn/z6u.)g ass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greatep0(6j qkag/lzz; .xq c)-n/ .mxekuyvr rotational KE?

We claim that momentum and angular momentum arlk6.jiu;e ub(t1 (e:xcmz- f algql4i4jb+a( e conserved. Yet most moving or rotating objects eve l4ecija.j;+ fq- uil1(x z4u (e:lg(a6t kbmbntually slow down and stop. Explain.

If there were a great migration of peopge .sk4l (1ywble toward the Earth’s equator, how would this affect the length of the day? (ye.swkblg1 4

Can the diver of Fig. 8–29 do ao2( ,jy. ccidx somersault without having any initial rotation when she leaves t.,x c2ciojd( yhe board?

The moment of inertia of a rotating solid disk about an axis tzubcm no-ku 2 , +ffs)rz*r8h.x.en5through its center of mass is rf n.b*.k8,xnu2fu 5)s ezrzocm- h t+$\frac{1}{2}WR^2$ (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 (on9ofl54.haal:s xi ws2ta0ci9lj.4 vhhn.stool holding a 2-kg mass in each outstretched hnvj.2n fla.x09hl it ah o:4s.4si9l c(o wha5and. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same m ajbplx)lh,v ;7b t23yass. However, one is hollow and the other is solid3tyb72 x; ,jp ab)hllv. Describe an experiment to determine which is which.

The angular velocity of a wh:fa9 gie-n3leib*9e heel rotating on a horizontal axle points west. In what direction is the linear velocity of a p*nhi g3e 9eb l9ei:fa-oint on the top of the wheel? If the angular acceleration 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 ths 0u 1mamt8*pue edge of a large freely rotating turntable. What hapsut8mu p1*0a mpens if you walk toward the center?

A shortstop may leap into the ai59 nyr)m+ee/izvo-d 3gr2bsgr to catch a ball 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 and legs rotate in they2 smzg/+e)9ird o35rgn bev- opposite direction (Fig. 8–36). Explain.

On the basis of the lawx28o;n t gkqdq45u2k u of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicoput28q ;k5on qg42 kdxuter stable.

  Express the following angles in rfwfh+6 )bfa8ysusgr1e s 38 3vadians: (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. Calculphv5;si2(-l +oi zdnvate, using the information inside the Front Col+5;v(s ipz2di- vhon ver, 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. The beam divey( g77we tefqql*dh. 0rges at an adq. ( legywt0e7*7h qfngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender remx wt2fxktp.1-(s e bq8)oa;otate at a rate of 6500 rpm. When the motor is turned off during operation, the blad)2.8 fqeakot-m(bxx tp ew ;s1es 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 3.2plrndg)d 0 yw+p1t ak: ;f(fwo2 jf 5k4q3exa5 m to another child. If the ball makes 15.0 revolutions, +ot2pkfj lra:e)wn2wd1d 450fq3 k;x pyf( gawhat is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions doppzu ng 2)4y tu:)7apo tonpg:a)yp u2 tz p7u4)he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in tfhww-rx36 bysqp r3e14 ) a2sdiameter rotates at 2500 rpm. Calculate its angular velocity inta13p rx erw-y whb4)3ssfq62 $rad/s$ $\omega$ =    $rad/sec$
(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 ay(r3.j (j0 , oljhykos (Fig. 8–38). (a) What is the linear speed of(jj yakrohy.j o3(l0, 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 Earth (a) in its orbit aroun il*mha hvet72-i;j 97.e3twvpqypp+d the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed u avwcq: 6t+g2of 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 parti8 4 .14buw g;stp rmgmdm.lr3zcle 7.0 cm from the axis of rotation is to experience an 4 s lmrgmdm g8z.r4u;.3 pt1bwacceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about-e obtfuun);r7 edfae4.(un - 3 c99f5nqkogn its center from 130 rpm to 280 rgf -ue-uf4e9 fda7)b5;onrotn n9n.e3( ckuqpm 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 radius w s;az7h/v2ssess/y + $R_1$ 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 siiu)k )k,/locpacecraft 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 )u oil,k /cki)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 uniformly from rest to 15,000 rpm in 220 5 5o2c5slyd7)tb1vl oo ls +ois. Through how many revolutions did it turv5s)752l 1o ooy+lbs5 ldto cin in this time?    $rev$

  An automobile engine slows down from v1bn9r wlzh p0j ib+ft*/; ph9v- p5p :;pecuj4500 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 ofwe9s uim4f9uk * s8qj/ flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.e /q9u9s ukimsf4j8w *
(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 diame5bb61o*efw8 au:.rzxmj iy)eter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on thmoe i) b6je85yw * ua1bzxf.:re edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It t hf3y4,tuybf ,d7xw/f s dxy(/urns 1500 revolutions before itbw4,7/ts hxyd/fy (,3yuf dxf 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 res fmy);yt, ;:.arkqszj4*pq xduces its speed uniformly from 95km/h to 45km/h The tires have a diamesyy*,4; krq :qxs.j;)matpf zter 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 askqb 7;;ogzai u 1rm-b1 the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter o;mqaorz g1 bi;u -b7k1f 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 climbin6 +r+gmdfm xh rl02mac. 2:beyg a hill. The pedals rotate in bmc6lrf :r22.+ mxdmag+0hy ea 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 5k oy.xs9grn;9*nartf t5 /ad25 N on the end of a door 74 cm wide. What is the magnitude of9tr*5n9k; x/ st ag 2aond.yrf 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 torquehhqb qz c 1m4jk,fb)lx(*x 5l2 about the axle of the wheel shown in Fig. 8–39. Assume 4xhb* (bjxc1 m fh)lz,qqkl25that 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 at:z i+wwm*; (dh4okzqpk lyd zai 04:165e ofehtached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizo4ya: *wi;m lo4z dihd:zp6he 1 zfe+(0k5wk oqntal 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 tighten)i,rocm sm 78ling to a torque of 38 rm)lsc,mi8o7 $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radiuzl. 4sa(: btuus 0.648 m when the axis of rotation isz(sat:4l . uub through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyclr*mnq jhv .m4vs-,8tcsv)5sw7v f )b oe wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hvtvc wo4 j))fv ,5 *smnq.b-sm7rv8 shub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizon3v + c9akd7asv9k-/ lvzsqnq.tal circle of radiuzq99kvnsqc+akd3 / - slvv7 .as 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’qs77mxa a)a:t//-z :cpsp.tsl7 dom ks wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N o maml7)s q sd7pkpxsa.7ta:- //c:tz 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 sh/ylau kfw c-qiv5,( p j0a0u6nown in Fig. 8–43 abo qa5k(ni0 vu6ajwp-fu/lc ,y0ut
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass z6 pa+yukkej vj7/9q9 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 spinning at the correctl*5d.9korbw vu 2 uxe( rate, engineers fire four tangential rockets as shown in Fig. 8–44. If5ru9x. *v(uke 2w bodl the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder wit6fgay:+qx ; 9qquw9 qm)l;ydv1t9isvh a radius of 8.50 cm and a mass of 0.580 kg. C9vv qt1u xsliy w6 :;gqdqmq+; 9y)9afalculate
(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, aii3,g0 +ughodnze8 ht; )u9pp ccelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven m p9579we zd s6bo bpb0(onxu8 umql/3berry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass oo9m7bbxno50 bp dz(6/wul83qs epu9bf 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 eventually brought f ,wtsvr lm2j03xz+u 1uniformly to rest by a frictional torque of 1.2uj 0w 3 rzlm2st,xfv1+ $m \cdot N$ 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 aobq6hc5 :qwn uvx)b9 xb* *c,qccelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of vpa8k0 g)uoi;c w3r4ythe forearm, which rotates about tyoik48wg ;vap3 )0ucr he elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thir5984 ueah +z hsubb9kn rod, as shown in Fig. 8–46.bheu5s 4 h 8zuk+ab99r
(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 oflfzq+fan0cb3bh18 q0 ee+c2k8hf7u c two 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 kv:lc 7 hniim)lx59 f7v2 g9jv from rest within four full turns (revolutions) and re7ch9 lxn:fij2lvgv vk)im97 5leases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor hxz0, hrj*qi1;p a adj7as a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque +gecc yyta a9z : ;n*ud3c4o;nof 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 cm rol*s1r3qz- / pf r9tzvyyls without slipping down a lane at 3sqzzy v*1y pr3 t9f/r-.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earthn+a;ni6h5vedeu dv)/ with respect to the Sun as the sum of tvne;udhvn/)a5i6d +e wo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has ;enca4a0vb 97zpg x8 ra mass of 1640 kg and a radius of 7.50 m. How much net work is requirecr0 ;apa n8ez4v 7b9xgd to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 204 l2x4 c+):n sszyyoiu.0 cm and mass 1.80 kg starts from rest and rolls without slip+ :2xl4zn ci4s)uyosyping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and5e2 w(ck w/mhw its lower w/whw(5km c2e end does not 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.mt6 pxv4k .cc a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at akpxc 4.t 6cv.mn angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular moment;*r+d1j2ud pm+9sfo.o mz gj nq3cr d1um of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotatingcr9qus* dgm+pdo +m 2r.j1o;z31j fdn at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform thatm1oqb*j-m 9z39 arziy4dnm1 a is rotating at a rate of 1.3rev/s If he raises his9-oyin1d*ab a 4qmm31jm9 zz r 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 mome-pnagj sp3jiq3u 17:vnt of inertia by a factor of about 3.5 when changing from the straight position a7in u-vgqpjs3p 1j3 :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 eh+xnw2 j sljajnd+ t./i2 i8;her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final;+t2h/i nd+j2lwinjx8js.ea rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis lh(g7wfj ): xdan(ea ls;t3t0through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. Tht(slael)afj3g 7; hn(xt0w:d e 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 fot)6zw1,gu c 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 s3*ki7x 3i7o:snaj1 iywa f2 zmomentum 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; (8zik :t1t5oiqm0y vk/8q8 lydnp e/qjb (wl disk of moment of inertia I is dropped onto an identicjq w 5tdlqzl; kp8eqm(0k/tyni8 81: yobi/v(al disk 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 2.4uizi-f2 /uq;i 2)up of $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating mer:m/( cz5 a: er8wrbhktry-go-round platform of radius 3.0 m and moment of inertia :/r: wetr8bm5 h( zack920 $kg \cdot m^2$ 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 ns0kqoyai+.4an angular velocity of 0.aqy4+nsk0 o.i8 $rad/s$ 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 dwarf, losing ab9i*jf4rwhde2fv,ll +p svb / 7out half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s 9h/evffpl,bwdj+7v slr2 4* inew 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 of 120 atqoyoip d)f-h0/-dy *la q7/$km/h$ 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 ho; r:cj*nheq 4st3r4 $ 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 w *f 5uj2u6fjlhs2ab u:k8zzb/y- * yzwithout friction. The moment of usuh wujf62b- * 8 zlf5 zk*/ya2bjz:yinertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-rournp8jlm0 -osh fka2 ,,s4uug4nd turntable that i80ofnkl g4 ss p2j,aumu4-r,hs 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 g:t.he* 1vmd msround 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–hs.v1 mdet*m: 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 Ea w9y;/dn is*9)hjyvdmlo: g)frth such that the same side always faces the Earth. Determine ;y n)9h sfd 9vjog)i l:/m*dywthe ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at fe 9a:h+l:gzg a5hm*co; k9nta rate 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 unifv+d-nyk)y5 1wfvv. gb orm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant an g.k+fbvy ) ndw5-vy1vgular 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 xdx5 8svc7:ho diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0xdhosvxc 578: .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 re 81l6p c1 4pbzwkx1ymaar wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as gag *wbbul z7zta2.089 va) fmhreat, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational amb)g2ut9hw aflb .a80zz7v*collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is f vq8ayas8oa of4/tyfbk-2q ,f *h1kp2 or 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 diapsfyv-a4k 2yoaqtb,f* 18 /a8kh2f oqmeter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an 2 lwsuz;tm1-a6zv no3+t, ke j$H_2O$ 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 gjc e2c p+l+-n,1x1h(vbf puehorizontal 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 bnf1dwb91 p/+ aofa b6and length L can pivot freely (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 bwbo/p+ af b6af 1n19dmoment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. i3hu : i:dhf2mIt is standing vertically on the floor, and we want to exert a h:d miuif2h 3h:orizontal 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=4.2m0+ 9e8ddh+vo)pkz o qay8ot 1n/s on a flat road is making a turn with a radius Th oo 0qayn 9v+kztp+d eh18o)d8e forces acting 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 ,swe t 2kipnn3g nio5ke9/m,m 2byn) 8begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achi8embwn)os 2k,yt n kg2mpn,i9/5n 3eieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, makiglqke hae0f( uxm-o4*g-j u )7 ,.bbrpng reasonable estimates for the dimensions. Then calculate the ratio of tqre) lb70*u 4gg.ah- f xo(kb pj-em,uhe angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch ass; x.ox-(e*md :lpsswx embly which consists of two cylindrical plates, of mass wpme-l.;d xx(sx *:os$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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. 8q k+z-3w c6+c0xyi nwh–58. What is the minimum value of the vertical height h that the mawcn6+i+0 3c qwhxkzy-rble 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 do nowub+ rqd.3 8la5;tfvu(cc w7ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as tqtr.gj..tk0 khe car reduces its speed uniformly from 90km/h to 60kmkrjq.k.gt. 0t/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