A bicycle odometer (which measures distance traveled) i )aj2hn:28 klx6qz dggs attached near the wheel hub agkx 8njzh 2q6:gd a)l2nd is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rxxu 0,s corak,kbf 1rx;7. 9hiim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear 1brk0,x u.9xxcifa;7h, rsko acceleration change?

Could a nonrigid body bej2oy*xm ebr jm0s.9v/ described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain5gubdnj /3, vh.

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 ha)f r.6xbqdww 7nds behind your head than when your arms are strew 7.xr6wq)dfbtched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and t ou,+osbl9nn))yn,*w m xq m2qhree at the pedal cranks. In which gear is it harder xoq)*on)nwml2 u,myn qb9s+, to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run y4 2s l-csb lgzkqzy2ml(cx9, : q*s2mfast have slender lower legs with flesh and muscle concentrated high, cl*l( m:zs2qblm,x2kg42sc yq9-czys lose to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a longyp ,9mkqihh;va1ce: wy0 0u( c, narrow beam?

If the net force on a system is zero, is the net torque also zero,e mikbz; g9(1f.unlw ? If the net torque on a system is zero,(gf;ekl.w , nm9biz 1u is the net force zero?

Two inclines have the same height but make different angles with the horizontalzu 3tqhxo.:r vb*4nr3r fp+yqy,3k e6. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom q+zby, 3 rov reypq*k u:hr6nf.t34x3be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inclwg-ulm4zi 0 q7ine. One sphere has twice the radius and twice the mass of the other. Whi7z0 uiqm g-4wlch reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylindejjn0v hhh08v5* pdijg 67gv-sr have the same radius and the same mass. They start from rest advhj g 0v5hp vgj-07s8hj*ni6t the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most movinglimnn2-m;9eww l* c1r or rotating objects eventually slow down an2ie*m 9w -rl;m wc1nlnd stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wouldqix4 /rrk3wghe 2 9h1i thisrei g h/i1q3w2 rk4xh9 affect the length of the day?

Can the diver of Fig. 8–29 dj0l hl3 v sxxil5n5 bd49:x8fwo a somersault without having any initial rotation when sheinxf0:59lbsdhl xx4 l wv38 j5 leaves the board?

The moment of inertia of a rotating solid diqz,45 8aty v,6*a jmeemip zj8sk about an axis through its center of mass is , tqmmv5ap 84jyzj6e8az i ,e*$\frac{1}{2}WR^2$ (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-kkhc8 lhi0pm)6sf n*-ug mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decr)8fhi6np 0khs-*cumlease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and s; cpoyvx,z o9-,ihrmrb 479s the9 pcxv r,szy9-7m4hs, ;i broo other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. Inp;tx/ /0a3klj,aenf o what direction isx j0l;n3 a,f ktape//o the linear 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 rotatm7vu0ny8o jq:ing turntable. What happens if oj8uvy0q: mn7 you walk toward the center?

A shortstop may leap into the 1o ofyh323ymonef8 c(air to catch a ball and throw it quickly. As he thr2fc3 8 3ynoom(ohyfe1 ows 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.

On the basis of the 8edn7uwk: a3i4zr8) q rx7vxylaw of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operax7yv4:r x78uiw3 edqnz8)akrte to keep the helicopter stable.

  Express the following angles in radians: z kwc waq/m8jh4e2y dj9 qo692(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 becad6ol* 7rbyu-kbyc /pdda5 lnx mw91 *-4atq.iuse of an amazing coincidence. Calculate, using the information inside the Front r.a **btibny yal-qd- /d5u9co4lpmk6 1x7dwCover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 kmnpr5vbq 1whsi/ y 98 p4j)gvz, from Earth. The beam diverges awq5n,pjvg1hb/9py 4sz8i )rv t an 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 rotate at a rate of 6500 rpm. When the motor islp 46q)r7fguyuv bd. ( turned off during operation, thuqr6dfl) v(4gub .7 pye blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level flo/vgdz1:w vhv+q,c +c0x h) tacor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its,+vcxhdw):aq t/v0 z1+ vgcch diameter?    m

  A bicycle with tires 68 cm in diameter7o pynt.iure2k8nz 8 2awd9 /y travels 8.0 km. How many revolutions do tinwrzpytd y naou /292887 e.khe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 250h 9j4+ehz x*zh0 rpm. Calculate its angular velocity i*hxhej94+z zhn $rad/s$ $\omega$ =    $rad/sec$
(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,g52*s xq /sp*ytbmia –38). (a) What is the lineapbyx *,t/i sm sq5g*2ar speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its ort3-1 ds+0xwaj-b) ty -lsjzqxbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed r8b,hidz23d)ldey a0of 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 aea 8r2uhb,4v69ro6c/ fupnerdrdu mrl m8;0 8 centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,mde l9mf eo8hrr,ucr/r d na6 24v0r88;upb6u000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelmk.(f6mawwnng kg,xnw *i)**erates uniformly about its center from 130 rpm to 280 rpm , if.w*nkm* k*aww nn)xg(m6gin 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius nfmo2 7d6fs .v$R_1$ 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 abo/(h6*yw09y ziyiykuo ard the Apollo spacecraft put themselves into a slow rotation ti 0yyoiuyky*z/h w 96(o 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 uniformly from rest to 15,000 rpm in 2* qc0qz*4svov.r0n fo 20 s. Through how many revolutions did it turn vf*zors4*0qnq c.0vo in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in (v5kbuzgms)8gy p9r7 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 s djkxhv**1g do)zcnm l 2/f/d1tresses of flying highspeed jets in a whirling “human centrifuge,” which takes zn h dd* )1ldjmg*ofc/ kx/1v21.0 min to turn 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 uni* zw9b803wqio69 r t2 tlqhzdhformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have tr 3 89ih tlqdt0h2orwzqbz96w* aveled in this time?    m

  A cooling fan is turned o /7h0 xxptys f:+ coty/gwo59sff when it is running at 850rev/min It turns 1500 revolutions before it comes to a stopo/opsh:yc57x tty9swf+/0 x g.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the c k59nj/*x nbp oeq.jijzq3-1xar reduces its speed uniformly from 95km/h to 45km/h The tires have a. pox jqe/kzq3j 5-nn19jbi*x 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 reduces its speed uniform1fo mmj73o/ do0f e8nsly from 95km/h to 45km/h The tires have a diameter o0f1d7me j f83mo/s oonf 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal whk1u g-o0( :jtxk4piij7b2jz(/f w wtyen climbing a hill. i( jjtko 4ip(t ygb:j0uw/kz7x1- 2f wThe pedals 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 a)p ;-gm 7, m5woqpfqo74 cm wide. What is the magnitude of the torq5wp o 7fqgpaq,-om );mue if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the aw ly8gidei/agz6 5p:z4jqm 9 ;xle of the wheel shown in Fig. 8–39. Assume that a frictgdi;leyi9w 6z: 8aqpg5 /4jzm ion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of masa c.3w* jdjdt 9j*oaw;huz5s 5s m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and dij;jwodc3.ds5zwht 9 5a*ja u* rection of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening t06ew3i1m(bookqmba3 aym 8;uo a torque of 38 ;o83m m oki(u1w 0yba3b 6meaq$m \cdot N$ 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 sphec,xu26imc;cfu 0r52xq.ztb j re of radius 0.648 m when the axis ofc6f2 jcb;tx 5. ,uczr2x0 uqmi rotation is through its center.    $kg \cdot m^2$

  Calculate the momentjb tke 4g0hha5w-kg m 0,w1uq, of inertia of 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 kw,h b5gm, wqu410j- 0gtekahcan be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the */ oind w-nu -9jo*a/zuoom*b end of a thin, light rod is rotated in a horizontal circleoou*maw dj- *bo zin*/uo/-9n 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 rotating at constant angular speety.l wel4oa- lvy uo9n-.3: prd (Fig. 8–42). The friction force between her harw 9:e-y pyn vut-l a3o4o..llnds 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 point objecg 6ir,x3/sjv ots shown in Fig. 8–43 abox6/ ros3vijg ,ut
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whog9 m 1i, /xzbpk1 vslyye*5h3cse 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 *7uc c/.3ccuuj.qlfj spinning at the correct rate, engineers fire fof uccu .cq3uj/.clj7*ur tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with idur+ y ,- e:azdv.p(aa radius of 8.50 cm and a mass of 0.580 kg. Calcp-u vd(. yra,zea: +diulate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it frorg 0*l/ d +fdmkui*p/qm 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 a7 f2vjbk9n7q snd 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 o7f7nv9qj kb2 sf 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 torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,3skakmx9cx,; 5 00 rpm is shut off and is eventually brought uniformly to rest by a frict9 ;k,5 kmxasxcional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball at 7w.gy 0;zed 6/irf)i np $m/s^2$ 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 i uzyd+hm 1 gjn5gn* og:km0z)*s thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. Tgzmu *hnmd5jngzk+o 1g)*y 0 :he ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shownnufwqnou..5n) (+j lb in Fig. 8–46.wjnblnu..q5 o)u+( fn
(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 coe(lh3z9a vl8xu/ ty:dnsists 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 ree.km fhwn2l04st within four full turns (revolutions) and releask f4 m0enh.lw2es 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 halkd z 2kk)- kyh-m6(a5*nmfgw s 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 develom gsmp3*lf :1nps a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without sli:po 2j i2qm 0o5 *bi wskulb19gq0vee(pping down a lane av9u2 o5 bsmqij01*2ib0qo(ee w:kgpl t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect tvan8/2 6 eamf4spw (qfo the Sun as the sum of two terms,8vnm fs2 f64ep(q/waa
(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 maki6 ozi)y 46yiss of 1640 kg and a radius of 7.50 m. How much net work is required to accelerat4iyz6yki6) o ie it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.. e2iiluvtb)0gnjt y8 b9o/w se -88d8z/oxrj0 cm and mass 1.80 kg starts from rest and rolls without slippingsbl8jgvo//.i 92bedyuo8w ijt8 e xz8tnr) 0- 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 22lmz33lr hwss7w3o9f lk ax(balanced vertically on its tip. It starts to fall and its lower end does not slip. Whatlw(9 rmw o2z7 h23x safl3s3lk will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end/kho, o:u8q0y2f ,zng z*tryq of a thin string in a circlet:k8 oog fzz2yhnq,u yr,q*0/ of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a dm ;3fn z(njrcu- c:0k2.8-kg uniform cylindrical grinding wheel of radius 18 cm when m: f0 (;k3-nuc jrnzdcrotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform-p.hqgvow8n( ndr imwe467i /ofr2 9w that is rotating at a rate of 1.3rev/s If p -4i/h wn7 imwo fq26dw8vn. (groer9he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her mn6g dweeu+*cs :0. wptoment 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 speed pww6d .sge:n*u 0te+ c($rev/s$) when in the straight position?   $rev/s$

  A figure skater can n;xgz/ d2pnk1vq;n-dr5oim 0 increase her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a; 0ndzn p5-d1on;/ rgqi2xkv m 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 arounde fz-e k39n:pf a vertical axis through 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 theef93n-: k zfpe 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 of5idc1 y ktvjprd4 xb:lf1n335 a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum opy z2e38z/l;xx0ed pz gil8 1tf 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 ofr4os/.q3yp-ug-l* lda / im4at y1fmn moment of inertia I is dropped onto an identical disk ryua m4*tl lgf- .4/qnir3s -1om/pdyaotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atz7oo/2/bm84oxn r xdz; d(hgr8+n cdi 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 rotating merryf 0 wz x,idkwwg;4xlp ,q;c24m-go-round platform of radius wl;0 xcwp2z,wm4,g q;fdx4i k 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 angu e*fh+do z4qm;lar velocity of 0.ze4; om+*hfqd 8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half its mbukz) -.7io/qpcx 4 shass in the process, and winding up with a radius 1.qz7xb ocp4u/s -h.)ik0% 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 of 1j7p-mvrolv; )zt7 xa,20 $km/h$ 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 m/ie z5e hcg8x74mn)ek 5a2nk$ 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 ro wv7: pwagea)(vaov(*tate freely without friction. The moment owp)7avv a(w(*e ag:vof inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at7:qm.lp -3yx6 :gedu+wifp sv the edge of a 6.5-m diameter merry-go-round tur6f:gm eysd up.-:v3p 7+i xwqlntable that 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 thn7xp bw qflj5),l x- hfo,.c9re ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person wit)7 h 5ojxcpb ,rx.9wn -llf,fqhout slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Eart2yf* k ak+va;nh such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (abo kknf a*ya+2;vut 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 accelerateb;ydr3x ka ;9 69ryls8kxd7dps from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape orypcj5y3u ;j4+oa d2 af a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration.yo4au5+d3j2 p cy; jar Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of masgy yg/fgh 2,9 rrw,/bqs 0.050 kg and diameter /q,9/ryrwggg 2 f ,byh0.075 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, hg*pdz9 r2 cd g/yqg4l41dncb 2ow is the angular 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 ;+b/87dtlxh3y-qvus h* jnwg times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a-lv*dh yj;u 7hnswg8x t/3+bq uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollutiqk lzc7f6qt j2fsbbn92ub95(on automobile 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 cylindrical flywheel of diameter 1.50 m and mass 240 kg,uqbfk26q5 bn(97 2 z9tcb jslf 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 illustratj zsq-t d(eyiz7rh )g+ (f-8r3+amlzhes 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 7w+ z zob2u(psn3oi; vmkyjk:,/7a zva 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 lengtkm64p 2r1m b,c0,cxoi( ahcxth.)z m pg;)ts mh 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 released0,zcamxkpt 4b 1 .rc ); mpic6s,2htom )gmx(h. 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 verticaljm .m0:ee dwoqn1cg-(ly on the floor, and we want to exert a horizontal n.ecg1edwmmq( o:j-0force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a t +g ;gnk:unay0urn with a radius The forces acting on the cyclist :;ng ny+g0 kuaand 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-ko:p3 wtnskob;w58 dl2 g 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 the2nl wo wtbd8:3o k5;ps torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her ar7 k2kqa (.ypif (rod)pwul-;v ms 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 wi2-( kq. )a iru fw;pvy7dp(oklth arms held tightly against her body.   

  You are designing a clu-uw(.hg o40z/ ;qb .zsob tzg xd1-adetch assembly which consists of two cylindrical plates, of dw;x(4 .a-oz1b-dose /zbg.guhz0tq 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 and6+, sfdqa: b.0t/xik. :j.lwzwytten radius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest pol/a0s:+b..d:tqe k yt 6fjn. x, iwzwtint of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but mw(ntd ,zni5v 8h2b;ndo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revjnk az yk6)+w1olutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter kwk y+janz16 )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