A bicycle odometer (which measures distance traveled) is attached nease3 6iepxg guturi+( px,vao/(j7+ - vr the wheel hub and is designed for 27-+(iptvs+(-g7paveu,o3ji6g x/xr e uinch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angul/fdh4e e9wagj+gt r80p4 .yn: viul.tar velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformli p4dwrg .4v0ee+ yftj. 8tg:a 9unh/ly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a si )zl8kwk2* hsg oh8yr*ngle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? tf 04++qcs d/ oasuec1Explain.

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 he lw5auxmf:r f-i 17*ayad than when your arms are stretched out in front of you? A diagram may help you to an fl7 m5*a ix1fa:wyru-swer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pec 7jz4j99h wkhdal 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 jz9kc7 h49h wjfront sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast hav+t2b hre2c kz)e slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribut)k +2h rbc2tzeion of mass is advantageous.

Why do tightrope walkeg nru; 28r5dnmrs (Fig. 8–35) carry a long, narrow beam?

If the net force on aterg99o3 cwokr.i.3w ( ep pd4 system is zero, is the net torque also zero? If the net torque on a syste3eeig p9k.3 crtwod4.p w 9or(m is zero, is the net force zero?

Two inclines have the same height but make different angles dx,-w5ngkl0h with the horizontal. The same steel ball is rolled down ea0hgx 5dlk, n-wch incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start 7n,6l9iq:r* zjgwk rb 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 first? Which has the greater speed there? Which has the greater total kinetic ener grlz7 jnb6r9: i*kqw,gy at the bottom?

A sphere and a cylinder have the same radiup, 9bw1mc (qbus and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed a,cqum bbpw 9(1t 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 movejif q9jyp,rz9tst3y* )o*o i lc 5e3.ing or rotating objects eventually s*tc o.i)s9 3o9 3jeyt 5eyj prfi,qz*llow down and stop. Explain.

If there were a great migra/f 4hk1*nb)ovc5) larsql ss9c vwy/ :tion of people toward the Earth’s equator, how would this affect the length of the: y)9as* rl/qs/hv 1v wobcf) nscl54k day?

Can the diver of Fig. 8–29 do a somersault wak,cthy49 hs2iw)a9 y 6+r*n xm(vnfp ithout having any initial rotation when she le +6w2 )(4xr*cak 9nv,mphnfht9aysiy aves the board?

The moment of inertia of a rotatia; /*ycroeg: yng solid disk about an axis through its center of maea;ryoy/ :*g css 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 in24lbcp2fz8o mvqybn 7q ) +maf7*j( ma each outstretched hand. If you suddenly drop the yo8v) m*ncpl+2ffq ma2 q 4b(mzbja77 masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical andcqq/ 7gp13erlp3* o+e rcjfebg/)t( y have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine /c 31clgeqf+7j3pee grr*tq p)y /(obwhich is which.

The angular velocity of a wheel rvu1l dosbf +qj o;/r9+:d;h fwotating on a horizontal axle points west. In what direction is the linea j/dur+ qsf;wbd 1o+lhvo:9f ;r velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntqn3do-(b wy-t7v:b leable. What happens if you walk towartnbyd :l3q- o-bve7(wd the center?

A shortstop may leap into theofq2 y/p* xux mht56p. air 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 hipsmpo/x.qt2p xy*h5 uf6 and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservatiob - da z.g qcff*gx:xofa7q11*+y) edon of angular momentum, discuss why a helicopter must have more than one rotor (or propeller):fdqcg *fe* xyzaxd 1-1f ob.qo 7g)a+. Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angle; 4d dmxnw*pc:b43p lzwxr(-xzmr 7l8s in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calc)ccko5im6 e fqe2(rl5 ulate, using the information inside the Front Cover, the angular dcq 5emfor2lek i(6c 5)iameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon,j q 7qeo4m7+jf 380,000 km from Earth. The beam diverges at an anglemfq jeq4 j7o+7 $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turnetumio951vxx0x f8k wn +kfi: 8d off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down? m91x0fik fun8w5kx+8x: tovi   $rad/s^2$

  A child rolls a ball on a level floor 3.68zh0j : tru va3wwvn75 m to another child. If the ball makes 1nw jh 3t8w:ruz0 vv7a65.0 revolutions, what is its diameter?    m

  A bicycle with tires 6(3h0)l(yk)8zv hezmagw os4r8 cm in diameter travels 8.0 km. How many revolutions do tzk8z wash3 e)o(vhy(r4lm)g0he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calc t;xrd6,f9 ivxgsd2 e(ulate its angular veloc29 sg6(xdrd fivx ;t,eity 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 revols fk rmi7+5s9hution in 4.0 s (Fig. 8–38). (a) What is the linear spef k7h s95+msried 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 arol(zoiz6co: rmig-hhk-1 c( 2qund the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o:e t k-elivtx(/p)* 68zjulkmf a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm fr6n:sj3pix- xm om the axis of rotips36xx j:m-n ation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerate;okp-5 jm,umo jp:, nos uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Detj:ok,mp-5;om jupon , ermine
(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 radiusk 8 gzfmzj rmp 1r0t0hqp3d6 yvs-4(x8 $R_1$ 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 :cxravqv)t 02mw56i+vx l s(xthe Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft m 2qv tc0xv+)lrx asxv 5w:i(6can 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,r5 vi 33g0co6iv qyzg1000 rpm in 220 s. Through how many revolutions did it turn in thqvi6r50v33i gyz1ogc is time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculax /tcd1)/phycte
(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 flym kxgxe;+y3v. w*z5+d( dzqkping highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn throug*5ded(z3v gxpmzxy qk;kw+ .+h 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 r 6g*kn ok ,serhc48d)bpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this tno*c,eg6d hb)rk48ksime?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 rlv3he4 d )jb0 pfd-2hnevol4b2e3fvddp-hj )l n0h utions before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 rb-c6be1pt1y3;(g vkge j/bwpevolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires;tekb cygv6pb( jw3 /1peg-1b 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 6/xhm+ ll wmf9-5 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires hav+m /lm xl-9hfwe 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 aq 9h76j;t b9n*ttfeion each pedal when climbing a hill. The 6a t9 e*tq97 bt;inhfjpedals 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 do d zw-;:1* ad+qmtxpknor 74 cm wide. What is the magnitude of the torque if the1a*: kdmwqtd;z-xn+ p 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 shoge,lura: a0r ;wn in Fig. 8–39. Assume that a friction tora, l:gra0 ;eurque 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 3 hbv lu21*ymwqap83nug p3f: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 the net torque on thihpbu y3qpnwm1 *vu8 gaf:33l 2s system.

  The bolts on the cylinder head of an engine require tight44+ z dtj5x t1nvb7vmrening to a torque of 38 zmbv4nrt+vt7j 145dx $m \cdot N$ 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 radius 0.648 m whenqbd(84 (pzvq*8;bg/tj ojr s jde,j x5 the axis of rotation is through vjq 5jdtb(j zs e*,88rbg q/(pdx;o j4its center.    $kg \cdot m^2$

  Calculate the moment of inertia of(v l(+ghewsh9mb j x(1 a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignor 1(b+s l((ej9hmxhvwged (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizonta7 q8vdotph1 o*l circle of radius 1.2 m. Calqv 7o tho*p8d1culate
(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 e(k;as,c; zx6q eqi(ra potter’s wheel rotating at constant angular speed (Fig. 8–42) ks(rq6z(i xqc e;ea;,. 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 inedy;g1r1a,ao r rtia of the array of point objects shown in Fig. 8–431ra 1goyda ,r; 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 of two oxygen atoms whose tozffd n a77wt:2tal 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 spinning at the correct rat(r*;(atl tdkp e, 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 ia*( tk(rdl pt;s to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a ra0uz5p nf y;i0.k32vj8yweff 0bhvny 6 dius of 8.50 cm and a mass of 0. nkzv; .3j2vy yyef58pw00f u0nhif6 b580 kg. 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 ycb7v0bn5;n fec e7+gbat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and is wcwz+(*5nnr(i doe 2qable 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. Calculate theen z+w (w(d*2iorc nq5 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 eventuat3f502,ui, q,* ptxd w k3k*qjvgndqqlly brought uniforjqd*q tgf032x k*qt 5iq v,up w,3,kndmly 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 accelerhj*wy. y3yg.y ates 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 the foreanrni b(yvy2 ih 0x1zh/8( qgk2og ,kf)rm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45.q8(vxg y 1zr g2ho)h2in,kibf(0y/nk 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 considermnetc zk p64riua+9)9 ed a long thin rod, as shown in Fig. 8–46.) ti496m+kz9cpuarne
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two ,pi1a5g de-d pa4gu7lsdyisakp(s 9 n, 4vp):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 fo xa+ owb5c b*am,tde0c11 ifn9ur full turns (revolutions) and releases it at a sp *01dtoc5w9xbcb m a+nea i,f1eed 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 oerm.0z 7 ba 3vbufl0cd7c;qf2f 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 of 28c98cfpd5yc3l1,vvvbu 0r7muh( s8l k0 $m \cdot N$ 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 rbzt 7mm.l9k710 p ip zb+rary (4wzky*olls without slipping down a lane at 3..rp7iykt4* y +1wmz( 7zb0 akbm9lpzr3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Eartc/kjlnl3: o7 fd(b oj6h with respect to the Sun as the sum of tw:cd(jon6l7o fk/j l3b o 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 mapnut7c k*b tw6w-a7 3iss of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest a7-citwpn wt*u67b 3kto a rotation rate of 1.00 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 :/n6oo gix2bgb *gkw :and rolls without slippib: :i6wgxgbon okg2/*ng 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. It3h/ v z-et/ajfctgo 0) starts to fall and its lower end does nojzcave/g - t /3)f0toht 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 ofglq y;ya5x; 5dfa ;kpo.h7mn6+iys f8 a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.. of+qnayp;5;8ymx i;7 kf5a hd6glsy10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angu pnhm*-z+m7 thxrx;p,4 s2unwlar momentum of a 2.8-kg uniform cylindrical grinding wheel -nrxh4 xw*u 2s mmnth,z pp7+;of radius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform1nk(9t*ykaxe 7.l-fyfr*4d op) gscl that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal 7 pe*fcral1yl*xnsf(k 4k)y gd 9t.o-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 shof*v6)abdjas -k c- 3kywn in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck positionad 3-kyj)c fa-6skb v*. 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 i3cgl* yadvm dh3o,;)c3g,ongvv 3c1dncrease her spin rotation rate from an initial rate of 1.0 rev eve)cg coav n3dg1 v,3;ol*v33gcdh m,yd ry 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 ver/-b ld*npxple t9js 61rx4l/ itical axis through its center at a frequency of 1.5rev/s The wheel can be considered a tsn4- / 69peij 1blx*xp/d rlluniform 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 it?    $rev/s$

  (a) What is the angular momentum of a figure skatr yuj:i.i1c- her spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Earth+mwxt4 xg1 ok8
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an cf2rz8 ;pi7wfvo 3qi+ identical disk rotatficzr8qpw2 v7;f +oi3ing at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atl9j+g; j(1zjkfm j,+m vhxez) 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 at7ggx 9mkjg88c the center of a rotating merry-go-round platform of radius 3.0g89mcx gk jg78 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 rotatifmu y335t9tw zq3j s49 qw/tling freely with an angular velocity of 0.8ft339qyms/iu9t4 zjwqt53w l $rad/s$ 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 about half its m sh-rod3r .4j/g8gmjg0u;y d iass in the process, and winding up with a radius 1. rsmjgr3i/ y;dgj u8h.d04g -o0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess 5-9 ga :h:jzkc/2-ocngv ru 0fvu2texof 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 9a f;itc7(-n4 fy4sx d xhap)r$ 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 platf-lu:spfq bay29.k8 naorm, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform is qna y2f-9apu. k8b s:l$I_P$ 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-cu, /t8y jwfgod 9v6b7go-round turntable that is mounted on frictionless bearings and has a moment of inert t 9wcyo67f,dvg/u8j bia 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 grou /3 mvs2 h6wvswium-w:nd 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.w/mh- 3ws2vs:wu i6mv 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faceuu htfh-z5 f.e.hs+ t,s the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter caz.+f5 u-euhh,.httsf se, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates frompd99lv4n ck1 3nm1jip 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 8,lh( oyj2 qzrthe shape of a uniform cylinder of radius 0.20 m acquires a rotational ,2q jhryzl o(8rate 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 crf5wesj j:r btsq-.c7(;tcd+ylindrical disks, each of mass 0.050 kg and dia+-:rj5sft.c djtcq(w rb ;7s emeter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angk++6er w9qfz(1hli equs7jy :ular speed of the 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,v7c9(um7+ ne /5bluaogco vz9 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 oo9l uuv +9zbec5v7c(a7n /ogmf 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 automosn ju -bf -u4rcu,4+pvbile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cyn pju+4,rus- -buc v4flindrical flywheel of diameter 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 illustrate ;v;ye+3z s* 1azamy2a mmhdf4s an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3.3trtdz7z pvi00/1uhn 0 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frip:n wv6zfe6 8xf;6 pquction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of thfzq uw p6x6en ;v8:pf6e 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 vertical6i */efueer6 nht:fq 6ly on the floor, and we want to exert a horizontal force Fh66/eq nr6ufiet f: *e at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is ml0o6x42mwfupig 6d o;1c8 yrwaking a turn with a radius The forces acting on the cyclist and cycle are the noyp 64fic;r 18mu2wlog60dx wormal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins whirmz/,19ab(u s )y; ou98dxgzxd0nero uling 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 tondx ;ub)u8dm9a,o9u1 y0 groz szex/(rque come from?    $m \cdot N$

  Model a figure skater’s b hm85ge+qylcy( ;ts+wz6n:iv ody as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate (tszcygq +nev5yl+w8m 6i;h: the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consisg/+4vgvp52ct t qv1nw6 w +qegts of two cylindrical plates, of mass gq +v 5 +/qnvwgv 4pte12cw6tg$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radbv .*z ycq1.aol y3t/sius r rolls along the looped rough track of Fig. 8–58. What is the minimum vq3yzl o tv1a.*yc /b.salue of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but ibcf ge(*f,y5 do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutionszs*.,ocisqwz u0 y1r8 as the car reduces its speed uniformly from uy08, r.1qois*sczwz 90km/h to 60km/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