A bicycle odometer (which measures distance traveled) is attached near the wheel 6u+ r6jk2-pkqpb+ apc7aruv)5 t dr2f hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-incpurj f+6k-2 v+ 5urp qc72kaa)rtd6pb h wheels?

Suppose a disk rotates at coe) pzizspauk ),g/e6- nstant angular velocity. Does a point on the rim have)e-i)6uageppkzs, z/ radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of req4 rsa vl/k);p)dr8 the angular velocity $\omega$ Explain.

Can a small force ever exert (8qzgagu); w ga 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 f* -vl.e5rpep hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer thisflp.5erp -e*v.

A 21-speed bicycle has seven sprockets at the rear wheel and three at th;dz jb bu2-i*ox* 9v ai(qurk*e pedal cranks. In which gear is it harder tob* ;d*u2( ua*q9bz-oiri jxkv pedal, a small rear 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 ablj;iy*9u 8u mije to run fast have slender lower legs with flesh and muscle conciu9 y ujimj8*;entrated 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. 8–35z5cu zc i71wiz(p(c xrefa6) 6) carry a long, narrow beam?

If the net force on a system is zerv-wb *vecsr3 ap++:0 kq nnjv,o, is the net torque also zero? If the net torque on a system is zero, is the net force zero?,+rk-j3e snwpv+qna vvbc *0:

Two inclines have the same height but make different angles with tw1c(tp4rmhp0 tqkb s**9mor ; he horizontal. The same steel ball is rolled down each incline. On whwtk**(sp19r 4pq; t0ommb crhich incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inscshlc 9- wt( mzdr;49ie63vjcline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the grrt(3 s4 melc vj-ci ds699z;wheater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder*ccfb:fe *x 6m have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottm 6ffexcb *c*:om 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 m96b vhq2-a9)n rwl2xcmfm0o q oving or rotating objects eventualnqwaqhf o 6cml2-9v2 rb )xm09ly slow down and stop. Explain.

If there were a great migration of people ttx3lh( osd+3z oward the Earth’s equator, how would this aff hs dx+lo3(z3tect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation r3r-t 3fxk 7 m)dm i4fh)*ne mb3:rsumwhen she leaves the boarf 4333rdrmm7mke f*r)): tu-nxibms hd?

The moment of inertia of a rotatigp3xyg 1w q2o6fyok)8 ng solid disk about an axis through its center of ma362gpg 8fw1oq kx o)yyss 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 yto-q,cd06u wp1iob+ in each outstretched hand. tc6wo +oqui- 0b 1ydp,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 mas8 shz; 4rc-ztfh ht(5zxa:qj2s. However, one is hollow and the other is solid. Describe an experiment to determine which is h-:qzzcj8th ftz5( ;hsr x4 a2which.

The angular velocity of a wheel rota cw7/qx5 jqpbr.fkb9b3: i c-eting on a horizontal axle points west. In what direction is the linear velocity of a point on the top of t3qw:bpb- 5b i.7k9cf/cqejxr he 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 zw2,-riwk f05+w i hlsof a large freely rotating turntable. What happens if sh,+-l5w wfwr k0zi2 iyou walk toward the center?

A shortstop may leap into the air to omi 6rq2nmp5au9) ruo88iz, a catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.io,api95m q 8ou n2) arm8zr6u

On the basis of the law of con/uzvoa+:qzm0:bb j l0 servation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller cabl +/m0zu:jabv :0zq on operate to keep the helicopter stable.

  Express the following angles in radians: sm6(6gxh4k 7ovs wc y;vqqjk ,:qu6n)(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 becg/ ah f+/h0szbause of an amazing coincidence. Calculate, using the inforb+/ az hfgsh/0mation inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380ogn+v48u /:o+bdwq du(3xxn n,000 km from Earth. The beam diverges at an angl8o+:/ b3 xvu gdxq+dun(o4nn we $\theta$ (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 q+ kgg18 zue/qnyvvm7qa0( k3 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceler8/n gkyev +0zv31qkgau 7 qmq(ation as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 mrda; ytbqr9 c*o r;)f( to another child. If the ball makes 15.0 revolutions, what i;ydr 9 );ortrcba*q f(s its diameter?    m

  A bicycle with tires 68 cm in diameter ; 0wry1mrk8os travels 8.0 km. How many revolutions do the wheels makkm1ow; ry sr80e?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. C;w6ej38h di eoalculate its angulareho 8; idw6j3e velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one compl 6o )xdk*q+ jl+cej1dbete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speeddecjjd+bq1o 6+*x )l k 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 Eart+ mfot( tck1v8h (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speede*,4tlcq9 cm v of 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 from the ah3a.y fz* rgmsrk*q1( mz)it,xis of rotation is to experience an accel *i(tzfyr)3 skg* ,rah1q.mmz eration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformog xxsr2ai)/ *3 4mmnrly about its center from 130 rpm to 280 rpm in 4.0 s. Detemrxr2)mgo* snia4/ x3rmine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius 2q/v 4w//8oer dyqxx jukeq5 ($R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apoxuj0*vlh i/g9x7b:e v llo spacecraft put themselves into xh:0euvx gbvi97l/j* 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 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 un5 tvxx ),0p1+wdnwr fuiformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this tiuw wx0 ,5xv trdpf1+)nme?    $rev$

  An automobile engine slows down from 4zoz fcs.zm:7(b , mm sh-l+v9d500 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 whir r2*/nkdy h0wqm )l(ec wp-mb7vro4t.ling “human centrifuge,” which takes 1.0 min to turn thb/ r4tq-vkny(m2l o7d*mw.0 p) crw herough 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 tsmk rk7 (a+5m; -3ebj.vrsjl lo 360 rpm in 6.5 s. How far (k.ej l 7mbss;mlr-3k+ja5vrwill 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 07owrkfa: o:e 1500 revolutions before it comes o7 :refw0 a:koto 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 my7 terom;qu5 .ake 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h 7oyeru5;mq. t The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reducesrdqko,v6j9 :imjn5*h its speed uniformly from 95km/h to 45km/ d,5q6 :or*jjkmhi9nv 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 wesl)d;ue c5 llfw(5 c)iight on each pedal when climbing a hill. The p c5l()delf5lu;)cw siedals 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*ci a0gzyt-(p a door 74 cm wide. What is the magnitude of the torque if the force is exertt y *cipag-z0(ed
(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 ine (xwtra:.s2cxk13gp Fig. 8–39. Assume that 2 gr(a 1x.cxe:tk 3wspa friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massleii5s ki) woe)3ss rod which pivots as shown in Fig. 8–40. Initially the rod is helo5sk3i)wie )id in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightezm j3s-7 1w nie5t+x zrl1h/yzning to a torque of 38 3 lnzs r t1i1z-wmj+/7y5ex zh$m \cdot N$ 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 y8 ,vse 82xnkurz5sa1 radius 0.648 m when the axis of rotation is through its cy5ers8v z 2xn8us1,kaenter.    $kg \cdot m^2$

  Calculate the moment of inertia of a ux +q4, ffdi9b jwf0 1n9jp,hkbicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass, jjq+u9fknfwph 1if0 4d,b x9 of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on th:cogzh h f9/rys;cg-b50v6 f bv ie2i;e end of a thin, light rod is rotated in a horizontal civ csiyfg/o z- ibg0 9; r;eb2:h6vfc5hrcle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rdah -od-o f0* s(be1tiotating at constant angular speed (Fig. 8–42). The friction force between her a *o te0- odf(di1-hsbhands 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 inertiaic )a6( n8wopwzpimsvi6hjz v( 1(ng3 *77os z of the array of point objects shown in Fig. 8–43 apncswjo7p(az vi6z(i m(ng*6s vz7oi 8 3h1w)bout
(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 atc(2xsz.p4,off p96 u3 ui6n ingtb su*oms 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 cylindricgxp/jd;cd c o+ -+)vaoal satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If thexcvdoo;g/p a)+ j-+cd satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm qrh1c )cjv81r and a mass of 0.580 1q)cjc18 hrr vkg. 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 baf2(31c4dxx llx5xkca hbk23 rt, 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 i* o0afrh(ea,(g/pvde s 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. Calculate the torque required to produce the acceleration, neglecting frictional torqu ,(h/0(ve fpr*egadoae. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotatingvku-.rl*xd( v5kp j -3 4geyp w2bdv ,1l4fzau at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of uv b xpl*4(kug5lef3j,v-dkazry1pw 4-.v 2 d1.2 $m \cdot N$ 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 accwr/s0h. ffk*eelerates 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 thesbg(p,qo,z(s-g 4.it0 *zdxa ie/kkf action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball s(z0qia4ep sf/b (,t* kog-i,zd.kgxis 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 ca 8n 6dcm;;qbmtet9 hh(n be considered a long thin rod, as shown in Fig. 8–4q ;m6h t9 he;tmnd(b8c6.
(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 consistsvu2 uc++w6:hlcww qa8 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 (rev8 l0b erl,.d;03bhg r pe9bs5t7mbpdqolutions) ;7r bdsd3g,rt0bepmqlhl9e8 p.05bband 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 has8 a(r cw /pumqnfqd,.rt(.): .lks jtl 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 develop(sn-l *6ez4b+v0p glqw:a az8b2fg nzs 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 rpvpj9f gh3o(r),-n*w ol i ac*olls without slipping down a lane w-(i o*npac*fv9o3plr, h)gj 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 p3un85jtd y0w, b 9ct;,lzsmrof two te jtzb3 8m,t,0dpn9w yscr5ul;rms,
(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 a.i./;mbdgle)jn-w4y n(f q mund a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8nl; m /-uiy(gwnqmj .d .efb)4.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from resgm 62hjx l1whql7hn79 t and rolls without slipping down a 30.0mw9gl6nhq 1 72 lhxjh7 $^{\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. l5tv,mqyz g g/c.-4 ihIt starts to fall and its lower end does not slip. What will be the speed of the upt5g/ic z.g ql -y4mvh,per end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on thes/j.mxy6ri4d l3:h bb end of a thin string in a circl3.b r4ishy l/ bmj6xd:e of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular moaj n(+:y,stxv mentum of a 2.8-kg uniform cylindrical grinding wheel of t+(xa,yjns: vradius 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 ah9 bee;,cq kxl+.d5rrt his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotationl+e.,;k9rcdq xe hr5 b decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the onep2)o (3y5mtarvn ru j/ shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when chanmuro2 n 3y/p) v(ar5tjging 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 increase her s/u,iw4 s)du c96an pbrpin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate rpb)/n9uscu dwai 64,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 through i ye1 (eba:rnk/ts 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 center of the r/: e(knbyae1rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figur czs ;kc8 )o/t3(qc)i3gftgn xe skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Eartzzow ro/e*5(e h
(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 ob l2u6q/ymo .)x el)iqnto an identical disk rotating at)l2x bluei)oq .yq6/ m angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns 5y* hzp b+qg uqsdu9al8w149iat 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 y a, lh(jp0;ehis8b*ya rotating merry-go-round platform of ra hspej, h; yl(0*ybia8dius 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 angular velocity of mjzj4 zs7j 6v,0.6s4m z jzj,7jv8 $rad/s$ 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, losin 4ppu2peg,px7*,d 7ga y7xl7o-vwz hzg about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new roo 7 gypavehp7p4*x-72z u,7 gpd wl,zxtation 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 h-0* skjnuseqk/es 9dunc3.oyu/e) 3 winds 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 gynm:q(q ktl4+sf ;)+ lza0ts$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely :lw czd8rjb( + 72s41+6ygdvlxji z khwithout friction. The moment of inertia of the person plus the platformj48x (cd6ir 1+ :ds z2yzklhv+l7gbwj is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m dib / d56/qu(yo gj34bifpgx f2qameter merry-go-round turntable that is mounted on frictiqxo y( 2 bpq 3/5g/j6ufgidbf4onless 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 the rope lygl:6tevu3-orl mp2:y hib :8b ing on the top edge of the spool. A person grabsp6lyr2: io8gl: mhb-:b e3vut 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 spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such7j9e5iub-n*jgbbh (fef2.w m 7 ab*sp that the same side always faces the Earth. Determine the ratio of the Moon’s spin angula( .5suej2bfjbai * 7f- 7*w9nbemb ghpr 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 m/inqhe bmg0,.x3,y85f we b v:e$s^2$ 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+u i o t*;e,ehn +d;ujrf6yx8j of a uniform cylinder of radius 0.20 m acquires a rotational ratx if6+e;e* ,uj8u r +;tjdyhone 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, eas0/np (j dfc 2.3vtf(g. .eupk-xhmscch 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 tot . d ( .( vn2sec0fkpfh/-xsugpm3.jc 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 p,1 8;p3w5xob0 zmh 6ihpdlis 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, but with mass 8.0 times as gre-:::q f 8zcmt)xd f mpp)ey+fb,by zn)at, were rotating )cnz: m)dx: yy8,fq-f m: bpf)t peb+zat 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, 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 cc )7h(zw ove9energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travv o)7hc wcz(9eel 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 uoxlc/ku**0zrol 5q ooc3 ;t(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),konf/xrr .brlx ixrv+/u1 1usv57m - 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 length L can pivot freely (i.e., w te 1gkd)hmby+o,dh- 6e ignore friction) about a hinge attached to a wall, as in Fig. 8–5 o6dtkm,1-hyg d+be)h4. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume 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 t*m*+s :6a ;jebywu,z3 xsljiv he floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which y se ij*m*;, axvj63l:+szwu bit 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 making a turn with 8w6;pk1 m +.pojssu p ,b8uzib0dwn3pa radius The forces aci+bbp3zuo0w. np spp wjudsm,;6818k ting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 3ou) am e, g kn4.clpcdfg,z+l1zf+a /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 torque comenlgczfa /kugp,z m)ld. co+1,3af +4e from?    $m \cdot N$

  Model a figure skater’s body v- bm ;-k j2,e6gokyrzas a solid cylinder 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 tightlyek jb6vk2yg ,-o;rz m- against her body.   

  You are designing a clutch assembly which consists of two cylindrical plt-s1n*d)w eg kates, ofg*d s-1)etnwk mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radiuso,ake - aw++yztxg114p7-( lsrtgoh;*c ymu q r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the r7ey1x;g+u -*acw1 k(zly og q ts,-h 4otmap+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 do notjn*;fglm cpc 6:b6c h,vbj4,l assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reda6ubye9m3 f+ euces 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 eac 3ae+ 9meyf6ubh 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