A bicycle odometer (which mea8vumdtrte f96 y;g-03 s*fxfdb)aj;g sures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bvx8 m9;gga3fs-df*br fudt y0j) ; 6eticycle with 24-inch wheels?

Suppose a disk rotates at cons5jny dsvu* c1,an; hyv63np6 otant 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 hav61 j ynu c5o*s,6np3 dvhyn;ave radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular vezecj- l b6)uk y3 .xe3kdrdlz*) hyn):locity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? ) 45;o*y4toov -nammpc. vof tExplain.

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 yourl.in,r m:3x .r)o*9cbad:ssk3:o ng 0hcv mqt hands behind your head than when your arms are stretched out in front of you? A diagram may help you to ans, :o9)kqmrdbish t:nmcsg3.*lcr 30n.va o :xwer this.

A 21-speed bicycle hjhal/+) e nqs-as seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rea)/es-qlj+ ahn r sprocket or a large rear sprocket? Why? 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 fast have - xm59q0-zk 547nwcw -ekeqkzfos0ea slender lower legs with flesh and muscle c5eq04sq -o 7kkxkez-59w -wnfame0 zconcentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow 3mtr0wwcvv-u a4xak,uv 9+e 4beam?

If the net force on a system is z)a3zqxixc n6a0: bb7yk catvz1 j4g:-ero, is the net torque also zero? If the net torque on a system is zero, is the bagn cc::zaa-zqbi46x 7) 3xytv0j 1knet force zero?

Two inclines have the same height but make dif uez;20,dy l7a/;qg 6p dtqgcmferent angles with the horizontal. The same steel ball is rolled down each incline. On w6t 2 lc;0;gmd ,qzyp dqe/7aughich incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an *v(y:hacjp k 1. 9.lttejp,ciincline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater tota9l cec(,t vypjaikh*tp.. :j1l kinetic energy at the bottom?

A sphere and a cylinder have thqsll+.v ph* :qe same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which +.l qhs:q*lpvhas the greater rotational KE?

We claim that momentum and angular momentum are conseriv,7g yu:/lfcczjl2 ( ved. Yet most moving or rotatin 7jv:zlg f/i2 yu,(lccg objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Ek-;i liw( :xpc hnrd3)arth’s equator, how would this affect the length of the x;hi)npkcl id-( 3rw:day?

Can the diver of Fig. 8–29 do a somersault without having any initial ra (/t h-m/gbwnotation when she lenag(m -wbht// aves the board?

The moment of inertia of a rotating solid disk about an axis s qs-kvy*5jo* jjz99o through its center of mass is 9 kvj*9oo sy5sj jzq-*$\frac{1}{2}WR^2$ (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 in v3ygb q+ zcw6hfb4*0m 0zdk77alc(xd each outstretched hand. If you suddenly drop the masses, wilfbcmz 4c*bh0dwlq0ga x3 6 z+vk 7y7d(l your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identicalj-/ wfk2f)j aqdg4 pl gql+i/7 and have the same mass. However, one is hollow and the other is solid. Descrp-kl w f)fqigqdga 74j/j+l /2ibe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal jy t4tx1jm9(7zgzge vr3g t,6k0i jv2axle points west. In what direction is the linear velocity of a pointet 7 gzzm j0t6vit 3g,gj1y9r(4j vx2k 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 the ez/*b,nfh(rb04ay p uidge of a large freely rotating turntable. What happens if/f04u,ri*(hbzpany b you walk toward the center?

A shortstop may leap into the air to catch a ball and thrgchr87g-z z:gnh+5 nn ow it quickly. As he throws the ballgz5hc :nnnrg8 h7g z+-, the upper part of his 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 ch :b -etdr:v*tac awi1*7*qk zonservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). ct wbkhi zv1a r7*qd:-:eat**Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in rdfvajafe x:2. )/ kr2eadians: (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. Calc, vo8)t; ;stf2 jcoanoulate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, aova2os8c j;fn ,t;)o ts seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam dxd)k5-su + 3 hcytug+8ax lq;jiverges at an ang g3l +uc5xa-tjkx;8yhqd+ u)sle $\theta$ (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 mokpsujfvl8)a. 2wn3el( 4vj0stor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slo(n0 upesl)w3fls8jakj24.v v w down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the bal8vivklxx,r 8(l makes 1lv88,r( xkixv 5.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter tr40knr. -diluyey5 h0y avels 8.0 km. How many revolutions do the wheels myrynludi0h0 5y.k e-4ake?    $rev$

  (a) A grinding wheel 0.35 m i ,ym8qbm). nhcoyy2 6xoct 19rn diameter rotates at 2500 rpm. Calculate its angular velocity in cb2rm yc.6n,1o hoyt8 9qm)xy$rad/s$ $\omega$ =    $rad/sec$
(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 inb(/q 6hw/qg 3zzetkj3axew6wz / 6j:c: tgw2e 4.0 s (Fig. 8–38). (a) What is the lineww eg/ z q6t3 kzbe/(cze/wjx62:3gqa :6 jwthar speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit aroune7j:b*f gza e+d the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a (vq6s )-qicyspoint
(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 i :exf601/xon1 :t cfw buebl)zf a particle 7.0 cm from the axis of rotation is to experience an acceffx1w t)n 6bec0u:exb z:l/1oleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centep 64 5,/keynrynlno(b r from 130 rpm to 280 rpm in 4.0 s. lo,keypnn6y/ n 5rb(4Determine
(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 radiu+ *v1ykyswxjljls+da8b09f c i8u4b x.f ml;6 s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts abqz;xw66q g /ecl0r )aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accb)qgz6a ; x wr0qc6l/eelerated 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 uniformly from rest to 15,000 rpm in 220 s. Through howo;yo0 h23mod1:+5mz s mx,dgibn/gfs 3do oo:yigbz,5/x n;md02fg+hs 1 m smmany revolutions did it turn in this time?    $rev$

  An automobile engine slows down fx+ nhy.lo5) 4xhhkma9 t,pgw 4rom 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 fly(ma (k,k h,hvding highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachingmakh ,(k,(dv h its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in y * 0nd8q)ytsn- zj0kw6.5 s. How far wil-8t0j nsq0*nykzy d)w l a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 , mfxc, h-sgyun)gc( +revolutions before it comes to a stopfny sg+(hgcc) ux-, ,m.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed unifoa s8ag6bn-9wle6ol( w f2c 0yhgu 85ur;iwg)srmly from 95km/h to 45km/h The ti6b-e cu8 gusg6o8 w;0af wnw5shia(l)g92 lry res 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 reducp22cb k :ostbp m)zw) tqihe --r/h(24d4luyz es its speed uniformly from 95km/h to 45kph/lq -u)ck2zz) tiowy-r 4phd2 :s4(eb mtb 2m/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on e c0lv(wp0lstf// j g1hach pedal when climbing a hill. The pedals rotate in a circle of radius fwt0p/0jhs(llgc/v 1 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. Whf.*or kbew4cs+7bhse*z (rst h5 f7i8 at is the magnitude of b 8tr (cs5fe 7*erz7hsho kb*.+4ifs wthe 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 of the wheel shown in Fig. 8–39. Assu2dsxw/ rvvm1 8me that a friction torque of 0.4 m r28 xsd1/wvv$m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, aresra;: n gzb bok2kd0:7 attached to the ends of a massless 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 t;k7gr :ob nkb2dz:a0 she net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque 5i, pvqay;vd:gg5r3 g r)(cl.1 x:sbuiu.t yl of 3855iyvpu,urcb(vst;x .1: g gad3yl i)qrl.: g $m \cdot N$ 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 racpfh 8)r8 +jsidius 0.648 m when the axis of rotation is through its cente8 )ji+hrp8 cfsr.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Th4cce mlvk dp0)sm:j 2;e rim jp km;sc 24d0evlm:c)and tire have 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, light rod is rotated in ,0 yv 9b 2rjqad0:ctn6i8eoyt u,zu3xa horizontal circle of radius 1.2 m. Cal3tq6b, d20uyvyxc,iae 8z9 n ju0 tr:oculate
(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 constant angulmtsa ko9k0uul6 + 3j 08fw;vhnar speed (Fig. 8–42). The friction force between 9l8 3skfu 00 awnm tjkhu6+v;oher 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 cgp4y/q qisf )l2-(qz point objects shown in Fig. 8–43 abo)qc-f(p q/sygl4z qi2 ut
(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 s njp* *kv71ev0zy*q8j: dkpcis $5.3 \times10^{ -26}$ 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 correct rate, eqd82biilsf430cma t.ngineers 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 rocket if the satellite is to reach 32 rpm in 5.0 min md.at3i48c 2qibfs0l? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a rad6rbil kns p2()ius of 8.50 cm and a mass of 0.580 kg. Calp6r)n(2 bk lsiculate
(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 from rest toni d9 1uyr vuvv:)q ,v5) n9:qeq;htht 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-dr mcxo-,rf+u i;iven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Ca,;mifuxr+c -o lculate 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 brou9nxm2 d* e:nright uniformlym nixd29 ne:r* to rest by 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-kg ball at l z2tx2r6)z ehy/ 0u(s2 btlc rl(u3ci7 $m/s^2$ 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 b an iq8.;dwm0mz(bw-vall 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 ( nvmmw0-8zaiq wb.d; 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 blhv y v(vf(v0o:bld.5 iv.sf/iade can be considered a long thin rod, as shown in Fig. 8–46hf(:dovv.v. 0i ifby vs/5( lv.
(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 consiymqc gb gew7fvya h59z -)5h,l0gd;j1sts of 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 from rest within four full turns(7 up zduyai:j51/noi (revoluti o(di/zjnu: 5aui71pyons) and releases 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 has qgj)lq x(njwt,(*pu,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 developbwxl7)w wy1d 4s 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 7.3 kg and radius 9.0 cm rolls without slipping down a rqmc0m 6sjj-)lane at 3.c-srj qj6 0mm)3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Ea;qt 9 7-cdepw3g5aa ycbg8-ecrth with respect to the Sun as the sum of taca5p3q be8--; 7y wgtc gde9cwo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius ofrnj 6s+- apqvma/qj. 7ff+sh) 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? Aaqr+an p-/7j)sjqm6fs. + vhfssume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and roll-,9 rxgcziwffx :e13/zat hcy. nz.k /s without slipping down a zt yxcer f -z.,x9hcn/f3ak1zwg/i .: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 tndrd.y0*d5(u g,rfseq 88ti+v-vn wyv ac7;fip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Ugnerq(dsvud5- wai.; , r*v f+y08ydn8f t7vcse conservation of energy.]    $m/s$

  What is the angular momentum *g.d rtqj9r+tiief- eohk 25 ;of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an an95i*+ io qe df-;ejrkg .hr2ttgular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angul; jam2npct32 tuka:(iar momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 1c; (2aik up32: tntjam8 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 platform that6f fn9qpr1zr)k yq:8 f2l;vju is rotating at a rate of 1.3rev/s Ifvy q: u;rqnfjlkp1r26fz89 )f 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 onej)umt)0o nrsj-4 bzi m/kn)i 7 shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 k4iz)rmm7 j 0obn-i)u/)njts when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increhvl c)5kqbt: 6ase her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate ofcqh6v:)t 5bl k 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a4 r8 *9 y7exl(dfbx+ fcr4u6zre4xpr s 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 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 ibr* lp (c4yzf97fxxrr de+48re6x 4ust?    $rev/s$

  (a) What is the angularch iar8qu ;-ib*n6s.p momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of t:xn4l0zn8v hjc2ce0 s he 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 dzyoxd(iy ;8hv6pn .u 3isk of moment of inertia I is dropped onto an identical disk rotating at angula 8i6oyynzh;pv(u3x .dr speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 e-f; /gyb+dc .m)nsuh4 bzw1h$rev/s$ 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 7v2yw yv6ju+j stands at the center of a rotating merry-go-round platform7+y2vyjj6v u w 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-round is rotating freely with an aw7hlo3 ag+ 4ytr*0 bn9k:npze);xtyj f cq80wngular velocity of 0;qnth)+y0zkbranfo9c0gpjw 4 y*l x3ew :78t.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 eventualj7bqd;)k ts1fidb.6t x 4 d ci,sn)2nqly collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming thbqd j42)cnx1t n7sdtsdf,)bk 6 ;q.iie 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 inzjn s 54oa8 am-(/ )p:+ b;reezdsioyi 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 w ,uj)a0npfq.30 tgyn$ 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*8nw 5*z(psz duqyxmqjs f 5:/, that can rotate freely without friction. f* np/5z:s8wzxq y(d*jqs5 umThe 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 :uzi2oqeifb .3m3x ;vat the edge of a 6.5-m diameter merry-go-round turntable that qf3 2mox; . vb:iui3ezis 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 gr7j +re.kzx u9td -7w/ vcyxn;iound 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 spoole9 -/zd x yurkw;tjvci7.x+n 7? How far does the spool’s center of mass move?

  The Moon orbits the Earth suce.6). exet j5anz ruc8h that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its .j5t )ece.e 8nru6axz own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist acceleratesms9(ay6s cmn 3qgdc ) 0zs4etys 46aw; from rest at a 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 unip 20criw+fa. dform cylinder of radius 0.20 m acquires a rotational rate of from rci f rd+2a0.pwest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid zyt p dea;052/zcx.xs6qt ry) cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the lin rxcd6s0yxp;)ytt eq.z z5a /2ear 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 whe/ftw;o /yka2tw81 w;(aoregh37 kfbv el ($\omega_R$) 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 c3ogcvi 5t;+kof our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times,o+ k i5c;gc3tv 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 energy stor52aj /4;u /ezo zipmxced in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diamet /xiz4oje a 25c/mpuz;er 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 v9h.bnok( drvep./e 0$H_2O$ 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 (hoo7 i+n py925j 3eaxrjtip) is 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 lens*wz wu49yb( q)rmgn;eltdo+c pjoq66u*89 mgth 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 moment of release, determine (a) the angular acceleration of the rod, and (b) the li*9qup9;s)qj d*n 6wuomr +4 zbl(yc68wmtgeonear 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. It is standing vertically on the flote 3f 2n. *n,pnp;fux5tn1 is9 gobb0uor, 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). The s tp 91nn*etf5s0;bigf u3ux2pn bo,n.tep has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is zd:tw.,a c/)hi 4zc mlmaking a turn with a radius The forces acting on the cyclis.ch c/azmdl i :w)zt4,t 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 hegkvg0-y8 1vand begins whirling the rvk8g- h 1yv0egock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid c.*ws9u c3 hjxgylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.s9wh.ux j*g c3   

  You are designing a clutchhv3s o2gr4c*yr2 hq*t assembly which consists of two cylindrical plates, of mas*3cq thh rvor2y* 42gss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 looped6ja8q6ku4* q.yt(mbkjlq22d 6krqn b rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must jq(by4qqb6k ra*qnd6 ku jt .l2 6m8k2drop 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 notun y8cxdz4be5+p+7ue assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the c-grn7f+cy 2w car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter ofr2+w7 cfc gny- 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