A bicycle odometer (which measures distance travelc aro.,hcdv, 2*f(6as.bppwy ed) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels.cfw6.ap ,o y2s*r(ahbd ,cvp ?

Suppose a disk rotates at constant angular velocity. z x:+5vrp4dhvepxt*xo4bvaak 65a)0ci8 os*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 either05p6oa 5axve4c4pt8a:)z x+s x ib*hvkor*dv component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular 3 hua3*,s jg0 6ujcy)1hejd3h0vnfr lvelocity $\omega$ Explain.

Can a small force ever exert gd/vo n 9 4;tsdulc-t+a greater torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind your i 0arr26k26f+ kwxm sphead than when your arms are stretched 2x 2iprwafk6msk06+ rout in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at8zx :pn .o8v8 vo1: xyim)qndh the pedal cranks. In which gear is it harder to pedal, a small rear sprocko1in8o xh8:v.d v y:8pzq nmx)et 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 slender lowerkch)qq7,cbs, legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantag, b7s)ccqh, qkeous.

Why do tightrope walkers (Fig. 8–35) carry htk ky,(ek8/ ja long, narrow beam?

If the net force on a system is zero, is the net torque1* eds ty.uwssvdq.,; also zero? If the net torque on a system is zst. ds,.vd e*sqyu 1w;ero, is the net force zero?

Two inclines have the same height but make different angles withyf+nkc*tt0g/2citoe .2ophe 3m x opcf9 v30. the horizontal. Thep.he+ cxkpf f 2vt tcm0c*o/9nyoge2o .3t3i0 same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneoustre,o.j vd2 c :i2lce5ly start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline fild:r5.c j2,i c2otv eerst? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They st w2jlxr 9mkqdg*;ek: /art from rest at the top of an inclin2: g;klwkqm9xjrd*e/ e. 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 greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most mo)5qao b.o+glbh 7r*iu+zyob0; c m gj3ving or rotating object iblo uh57*jrm) zgobac y +o+q30gb;.s eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wrgwdd/ 31scd; jg:x,-zip x.num ;f 8oould this affect the leuog ;rsdgi- ;xwfp,j3 cz.md/d:1 x 8nngth of the day?

Can the diver of Fig. 8–29 do a somvo -gw;ig8z * cw9vb2x2opir;ersault without having any initial rotation when she leaves th ovrgg22cvw ;w ; zii8obxp*9-e board?

The moment of inertia of a rotating solid disk about an axis througl7nd4qu21f hh h its center 7d2ulhqnhf 14 of mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each chx4apl) 3 k:n : dtp7p386epzm;jov joutstretche cola dmn tvzhkep8 x:pp;j43):p67j 3d hand. 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 mass. However, g(p/io: y kj2zone is hollow and the other is o:pgy/2ik j z(solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points wesp5.iiqir y(z* e.9qh tut;s2h t. In what direction is the linear velocity of a point on the top of the wheel? Ifpqrtstizy q*ie.h;5(i9uh 2. 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 edge of a large freely rotating turzlj7n:;m;cks,6dzt0 9eso 8b0p brgxntable. What happens ifo68 g, z e;d xbt:0s pmjcrl90k7zb;ns you walk toward the center?

A shortstop may leap into the air to catch a ball and thrm d,xvj4w ce/15rj( pi/-/.csbsp ey tow it quickly. As he throws the ball, the upper pa bjtjed,5(- svp pr/ec4iwm1sc/ x./ yrt 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 conservation of angular mom,f3 or* 7s5fx xqui:vmc;/ hsaentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second if 7xo5v 3:u*m/,s;a rfq xhcspropeller can operate to keep the helicopter stable.

  Express the following angles in radinx u2:( ssassa ;yj3j-ans: (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. Ceo1l88hld+ielgf/7 a ad;v0st 2trf :alculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as se;de to:2d1a+ f8s e f8ag7rvlh 0til/len on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges aek r00nh81pw. -hvxj zt jv 80 peh0.r1xk wz-hnan angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rodcjy::x;cb g fw9 9y*utate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to resdg9byfccj:x9: ; u wy*t in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. qw3rg )m-f (atIf the ball makes 15.0 revoa t- 3w)rmgq(flutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How x,fy- nktrvt 32 tth3op3*2z xmany revolutions do the wheels makktt 3,x 3 f2prz-vy3o*xtt hn2e?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its 2np6jh3cyk:nfcv x 77 angular ve737p:vc jhn2fxy k6n clocity 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. 8–38). t 8*+xj/qgy297uauc-rf 6hpfas:f7yh4 z b af(a) What is the linear speed of a child seated 1.2 m fr+sy z92p ya /6f jgf u7 xq4ufarbcft7ah:h8*-om 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 around the Supo1 nev,oto d*xc7o; +n    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofv9+hul2c/ct0w j kaom/;/ fmt 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 parti15dnxb .7wmw-lj 8x( t; zlkgwcle 7.0 cm from the axis of rotation is to experience an acceleratilxd7zw w-tbn8w.;glm(k1 j5xon of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly a 8xd**qcgu:k fs;r n0wbout its center from 130 rpm to 280 rpm in 4.0 s. Detg* n0rwuxk :c s8;qd*fermine
(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 radiuf*2:pz ovf.2berz+l/(z v rwc 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 ;593v z;jt,m jpu(deo sgkmy/Moon, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1. 95mvtoy 3,(jmz;ejps uk;g/ d0 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 unifg o kl7i w.oq3.psi *ckd4,p,vormly from rest to 15,000 rpm in 220 s. Through how many kg qp7lkwi.v,poc o,3 d4is*. revolutions did it turn in this time?    $rev$

  An automobile engine slows jgg q,4x+-0:uxdatfhn(vl 0 d down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a wa4jerbd19yyj+ 9;z. zy 0tf xqhirling “human centrifuge,” which takes 1.0 min to tu a+ybj.jt4 1qzr0 y;eyxdfz9 9rn 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 accelerates uniformly from 240 rpm to 360 rpm in 6.58c6*o j; tabur s. How far will a point on the edge of the wheel have traveled in this time? cj6tbaou; r* 8    m

  A cooling fan is turned off when it is running at 850rev/mmz(q5 toegl5.,f,ovlmo ,4 dsb/x b0r 2q kvl2in It turns 1500 revolutions before it comes to a stop0,,(str f l/x vbledk., zgo4qq ml5bo2om2 v5.
(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 d8wl8gq.e-lus jsb/b5x,m9 o the car reduces its speed uniformly from 95km/h t- b mg/9l8elw.q so j,b58xusdo 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 mc d/0*bau(4 rs fwx2 r7eep-ynake 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h Thxy4 cwe 0e*u(/arfdn7sp2 rb- e 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 e1vo mx+g4y8gd ach pedal when climbing a hill. The peda+g1xgymdv 48 ols 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 door 74 cm wide. What is ths;+ : s-kk(mls :hg3c.ktckr:iy fn7 re magnitude of the torque if t yskg:nh;( : kkss.t mc f+:-rkilc3r7he 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 aboutoh6rgr/qy2nfy h8g, 6i e 3v1j the axle of the wheel shown in Fig. 8–39. Assume that a friction t o/f162hyn 8 hrgyjrg6,qe3 ivorque 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 which pivar 7 *ldk1kct xqr20,yots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and tkk2tyrd1x0 c, *lar7 qhen released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightenink;+ym h 8,slqk/q.vmi g to a torque of 38 ;k8mq,/hl.i yvm qks+ $m \cdot N$ 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 134 ,z nlhzwuz5dsv0,y0.8-kg sphere of radius 0.648 m when the axis of rotationl sy0,5dw3,zvu4 zhzn is through its center.    $kg \cdot m^2$

  Calculate the moment b---iwr:b7tt, h)vz jyedz; ,zehd2f of inertia of a bicycle wheel 66.7 cm in diameter. The rim and t hr, j,febvwh:bd-itdz- ze- 7y) t;z2ire 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 a hobznkf1 (e;7eo rizontal circle of radius 1zonbf ek e;7(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 zg(i;wbqp wuc5nsa4 6evx6, ;bowl on a potter’s wheel rotating at constant angular speed (Figx g a;;v6wziqsne54cw b6p,u( . 8–42). 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 p/i 6cvbvrx a*)mf ,f)ioint objects shown in Fig. 8–43 brv6c/ xffa)mii)v* , about
(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 o :gh)cunnr: v-jve1zix58 zz8biea47f 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 spinnihf ;xzjtb(9 1png at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is txthjp z b9f;(1o reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius ofu- ;sm-go 3qlx 8.50 cm and a mass of 0.580 kg-mg;slxo u q3-. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerlb.u3 5uln oguvs0g9l/7y w )pating 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 merry-go-roun:c r9dh4f bg 90nttx*od 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 o0xbd r *f9chtn:4o9gtf 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 ewe:vc)adzega x(c.8 5/ks3uis eventually brought uniformly to rest bye 5x8 ueacs .3vc(:dkz/ )gaew 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 aoy2og05 n 8 i(v1ewiuvccelerates 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 ba0/bg b*xf5g *xrf4rzull is thrown solely by the action of the forearm, which rotates about the elbow job 40fxx*/r5fbu gz rg*int under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered at 56tu.ob nrzf-c g6)w long thin rod, as shown in Fig. 8–46.6) .w-5gbzc n tou6rtf
(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 . 9 u-bpep i0rs,rwd8otwo 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 d.* blwlprc 3ord-e1e/,y s1v (revolutions) and releases it at a sp.dcvodl e3*b-lpe1w/y 1sr r, eed 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 of y : xsu() mjhzony63xj5i+og)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 ; ;5mdrih fu/qof 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 winru z8y j1 ff)ynrxuj;ir-ax6z -3/1a thout slipping down a lanx)in8-yf r a/; r6u1-z1xj ur yz3fjane at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum offh ui)lnlp: bl, +:.eg two tnelbll+hu):: g fp.i,erms,
(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 radij7 -ga7dqa-eu us of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 edaqa gj-7u7-revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest aii*v5 - 93ekx eeipd3ond rolls without slipping doevek3 3do*i9i i-5expwn 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 and its 8b:j6s5 pkmat:b(fcd lower end does not slip. What will be thek86djs m:cb bfa( p5t: 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 ok12)bs6ybqssz 6 wt --9y sysnn the end of a thin string intkwzysy --6 )bsy s19sbq26sn a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angularhl,eas:o; 2peo 1 fk8a momentum of a 2.8-kg uniform cylindrical grinding wheel of rads ah,f2 8poeolak; :1eius 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 atfk9ia rb xm f 9lc9*5k:n**kyh his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arl:999*y b a5*ffi khcrxm*kknms 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 hsq t ;:(h bd9g+ukwr(kexb* x9er moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speu:bx9;kbr+wht gsx *(d(ek q9ed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate fro,z vp+e8 i/t gviy0bf9m an initial rate of 1.0 rev every +/z0v vef8pg9tybii,2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis throughu ,trj6 bvcwdwy1+4a 3 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. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the centea 164 rwbwcjuv ,t3d+yr of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning ax*smy77j lj i7t 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 mozcq*s *cjchqxk4c ym1.+ i,)f mentum 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 disk of moment of ineq e8z4 vsdqhdr6 72q;xrtia I is dropped onto an identical dis;xdh 8 sze 62dqvqq47rk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a r 5739+xsq tfmpngr)ht 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rot 1zerkz)g4i;vpdy 6y2w/ /hn 2-adzriating merry-go-round platform of radius 3.0 m and moment of ;agwz)ikn yzzr64vd1y re //p 2 2-dhiinertia 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 i oqplpb3t -2mg x;y8ya2r-q2k s rotating freely with an angular velocity of 0.8xb3p-2;lay2rqtm po - 2gk8yq $rad/s$ 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 eventu yy6 ,jds6e/koally 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 the losyy6j de/ k6,ost 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 gnm9c. 677htts;syxgd sw;/w in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass 46 .nbjma- paz$ 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 fe( 6ikh*ofv 6s /:uuhhreely without friction. The moment of inertia of the person plus 6hh/e 6hfuo (:ikv*suthe 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 ofspmnyd.v 9njv 8 9n )oex:k9.c a 6.5-m diameter merry-go-round turntable that8p9njocxv.e)n nm9dk. :9yvs is 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 ground with the end of tlp 5 bk9l 6/kt7)icntie0w +ethe 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 bt9ik lbp)c7 te0+ l6wk/ti e5nehind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth sucfu*jpbnd+j)* dd 4j9yh that the same side always faces the Earth. Determine the ratio of the Moon’s spin anu9)jp +dy*f*jn 4jd bdgular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 b ,lpertmz- 23-kn p7sm/$s^2$ 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 sha1j4xwr *i a6qope of a uniform cylinder of radius 0.20 m acquires a rotational rate j6x4wai1ro q *of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each oz v,eg0f1wp)lf mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg anz) p1fevg,w0l d 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 z8d74tgj0u/gveu we2 u r,-g, qjil i6the rear 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 great, were /+mhrdiphuge6nh9o.,0 o 3ayrotating 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-r0aih.,yg6hop m9u3 hd/n+e oquarters 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 for it to use energy storzvv eooxr i 1q1 .:+a4:zvcka7ed 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 maaqv .er o1o14kxc z:iz7vv:+ass 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 illustra*xurz;.luz5ud 9q6hftes 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 (hoxo:z79(uzz c bop) 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 an,ny d2y7p/q3-mmx en ud 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 moment of release, determine (a) the angular acceleration of the rod, and (b) the2mnyepy dqmnu -,/37x 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 e(*ongn0zua; 6 y8 thj-u tv;fR. It is standing 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 ;fg;jvao e-(n8ut u6hynzt*0 (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s o)tc2 cvm ml4nkg9p/( ;rhdg/ on a flat road is making a turn with a radius The forces acting on the cyclist and cycle alr 9cp tnvg/2 )o/h;mg c4m(kdre 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 0.50l )vtg6)u bgl) .+g,pf6ekccw -kg rock into a sling of length 1.5 m and begins 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 achieve this feat, and where does the . bcltek,cv))g) 6luwg+6pf gtorque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her armjwn5b42p 5hrf s as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with25bp 4j5hrw nf outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylind.exy mjrpkfbc*2i2;-zy 4 jv+rical plates, of mazyejf2yx4-b.j*civmk; pr2 +ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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. 1dbfg1*d5gok- e;q px8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of theke 1*;1qbdg5go xf-p d loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, buts0frst.2oh3)rye1;j ;k u e gt do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as t s8z/zoisbrg 19muo,7he 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? o9z, go/ imur s1bsz78$\approx$    $rad/s^2$(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