A bicycle odometer (which measures distance traveled) is attachei99ya u1mkqiy e+kon-ax s+:; aww1b: d near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?ya:9ko :+1anx 1ea+y ku;9wmqiwbi- s

Suppose a disk rotates at co nz(* vw2xp u3a,dpoz;nstant angular velocity. Does a point on the rim have u,3p *xp2o;ndwa z v(zradial 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 the angularfu)m1u s xd9e9 velocity $\omega$ Explain.

Can a small force evb4cdo )psi( )g**cnoeer exert a greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your heak8 puzo 1pef++d than when your arms are stretched out in front of you? A die+ f+18oz ppukagram may help you to answer this.

A 21-speed bicycle has seven yxw4+ ;,zopdoyb s5m3 lo.j46 ad+ yyrsprockets at the rear wheel and three at the pedal 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 front +d 6o,wyayys. 4b3jp myxlo5+;odzr4 sprocket or a large front sprocket? Why?

Mammals that depend on bmynv83ts9n0q.nt 7p3si v.he eing able to run fast have slender lower legs with flesh and muscle concyv 7shtte.v.n93mn i 80snq3pentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam4na m5q2f4k n ho2uyt,?

If the net force on a sy9slmzz0 ;4u iostem is zero, is the net torque also zero? If the net torque on a so;0uz 9lmi4sz ystem is zero, is the net force zero?

Two inclines have the same height but 5:b tds+fp vso-.- emfn 6bf-rmake different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Expla-s fbr dtnfop.-bv+-m :5s e6fin.

Two solid spheres simultaneously start rolling (from rest) do 2e6h.ss93niqa mt5; prnx8 chwn 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 energ sr.3emht5p69xscn 8hq; n2ai y at the bottom?

A sphere and a cylinder have the sj w/ gotw o-ya/,t :cps;u/fw.ame radius and the same mass. They start from rest at the top of .foy -:/ /apwtuws,jw/o ;tcgan 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 mfn z6 sdkgxwvw9)eq5 ;9 dg4s*oving or rotating sx6g q 5 snzfvkg4w9 d);we*9dobjects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’srcmnw99-an hbxychc+ :at x3o**h,j9 equator, how would this a-cyhn a9c w*,:*o ajh39rh x9mn+btxcffect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any i6 q5h.imuuohmcsi ;3/nitial rotation when she li m;oucuq3 /h 5sh6.imeaves the board?

The moment of inertia of a rotating solid disk about an axis throughjt 2e n9,d8my(d-q ujt its center o(m-tetq d2d9 8y,jnuj f mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in en4nwuy8 9 +nf/zpxv9v ach outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same?9npnf 9y4v wn/vz+ux8 Explain.

Two spheres look identical and have the same mass.phku47)v kha ywu1 .qi( j0d9y However, one is hollow and the other isqujudvhy9h0i.)y71(k 4 wa p k solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotatinks*m:ql.e*ri+x 4qd-p5tm d wg on a horizontal axle points west. In what direction is 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 * +dmx k*ieq4ldms q. tp5rw:-at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a largeb eyu zph.g49-/w x8 fe/l:qos freely rotating turntable. What happens if you walk4/ x-epheo:8bf/q sl9wz.y u g toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As b tp aq3; of.(8dn/otd 8:tqimhe 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 di q:d8 bt.(t8m; pq oniat3dof/rection (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, dx+w w imy1q6-riscuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways th i1wqw6yx mr+-e second propeller can operate to keep the helicopter stable.

  Express the following angles ieukp w3tr g01j8.g0fnn 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 amazk3rmzfu x h2bt 04)/b i)tec76f-kzq oing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radix z 746kh)/iek)30fbof-2mucq rzb ttans) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 3j+ q iktz,e3m:80,000 km from Earth. The beam diverges at anezmqt,ji +3k : 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 is((rbs s 8an w5tzvc;4-nd zk qo.c21xp turned off during operation, tt2ac 4qpcs nsz-v85k1 dxrwn;(.z b o(he 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 floor 3.5 m to anoc10. :b y5.yy0w mp vcuum:jrn0br0lf ther child. If the ball makes 1. c0j:0. myyn ypc:u10 0brrvwu5 fblm5.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diametero*2q; :l gnbu l y/.o*wjie84mcm fmqbv .21hr travels 8.0 km. How many revolutions do the wheels ma2nmqvo o :2 y1bc ;hl/mrmle*bju4.*.i fq8gwke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcu3 wv0g9 eybm sapwj *se,;8a(ulate its angular velov3 gwsy0ua(;b9 e*m, p8wsaejcity 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 co :t2oy:-mbqid nh)4u jmplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a)obnm: uid-: q2th4y j 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 aroundu7;:pf 0klzf hg3n1lf the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of k;noiu6nqg+g 6g.lfy( om5dwv,7fk 1 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 rota c.sd ;e-0s*kitn 7tdyte if a particle 7.0 cm from the axis of rotation is to experience an accelera*i t0kssyd - e;.tdn7ction of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center 3/sto9qp)de(rb.)f hm b+r affrom 130 rpm to 280 rfr pqa(r3)) b/ oh+d.9befts mpm in 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 radiu9lhwnnuy.; +d.vt*i r-wc*f b s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronautzp-,l40w udrn s aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s ene0,u wrlp4ndz-rgy 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 fri5 v1hw x4btx0om rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this t x4x01h5iv bwtime?    $rev$

  An automobile engine slows down from 4500 rpm to +f60rmwwpj ;z 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 iej aq e81(lxz96xj)mw6t+ g qmn a whirling “human centrifuge,” which takes 1.0 min to turn through 20 co ejzqe86x9 m(xtwlamgj61+ q)mplete 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 acceler 65qsbozk73xm ates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge ko 7m6zxqsb35 of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running ag)6jqlhq,ef3nc5fe 9t 850rev/min It turns 1500 revolutions before it comes to ,jc5fq9 ef)q e6nlh3g 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 revolutk3v/gsi bus 7a3- b, h 9mrj,*eb arcyz.unc*9ions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have u9ckhzr*,.an3bcesrmib ,sgu v 9/ by3-a7j* 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 tktro; n4ne a j(z*w1l.he car reduces its speed uniformly from 9*4k nwn (azl; tjr.o1e5km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal wh+ hjt ix1qpsdqz * ad;yq-h-82en climbing a hill. The pedals rotate in a circlqhd 1s *2qa8+;pdiy--tjzqxhe 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 endhd ar(gh t) i).o4ark) of a door 74 cm wide. What is the magnitudog)) dt.ri(4 hrakh)ae of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel s0qw)x/ bus ;jvk34 xyuhown in Fig. 8–39. Assume that a friction tobsuxu/ )0x;k43w qvy jrque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached3 .zr gp dkhy62t5zrf5 to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of th hrrz5 635z k2yp.gdfte net torque on this system.

  The bolts on the cylinder head4; hne.f183zkqqvp wb of an engine require tightening to a torque of 38 ;zkv.wbhq8f4qn 3e p1$m \cdot N$ 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 when thwq5i6/dxscd4a br0i 2 e axis of rotation is through its ces6a/ br ixicd 405dwq2nter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle w2bt/udeq;8v )*xi d vtheel 66.7 cm in diameter. The rim and tire have a combined m 8xt2 ivd u;ve/*dtb)qass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light egiquad 6;z*s hv 3uf c8py26)rod is rotated in a horizontal circle of radius 1.2 m. Calcula;sa zg6qp8du3)ci yhvfu 2* 6ete
(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 consths7te z + yx1ba,imbn),7t/ntant angular speed (Fig. 8–42 ty,ebnb7s/nh zti17a) + m,tx). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig. 8 b 9srha a. ki.np1vmu lp56x*t)5xa*b–43 a auv9 k. )p6n a1b*ra mixx5bs.pt5*hlbout
(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 whosevrqx;/ qo:b aj. .jp+j 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 cylind8j fnalv*5jw8 rical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0*jl8j8wnafv 5 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinde e:/vmd ya39vxr with a radius of 8.50 cm and a mass of 0.5vea x9v:y/3dm80 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 bat, accelerating it from0ut l.mdt) 8 v0 zlpvuups69+b 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 isctb) ;ay o:;xd 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 ofbxd;;)y t co:a 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off aqwbu (n: h+ y*qh6;ejn/k6xb( dn0eie nd is eventually brought uniformly to rest by a frictionweq;(nj u60be +dhn(bhxq 6k/i*e: ynal 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 balll2)gr aj/l8sz 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 t n k5j q,9xc;dtvk.ss7he forearm, which rotates about the elbow j;j sv cdk97k x.q,tns5oint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shoui,2 e5/mn ayhwn in Fig. 8–46a2yi 5e,u mhn/.
(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 coelk4 tx1ot : zwzj ro,hax43b 4si,w5-nsists 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 fromfz r 9y i3ajumhj*ibu: 12yj8/ rest within four full turns (revolutions) and releamj zaj*3j /iih:8b f1y92yuurses it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment nm glzemwe (p skj/*8e1t6+t+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 develops a tv3liw *6mein+ms bw 5xhna67 /orque 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 9.0yli *p8gjrc5/(rrxj m. 8 ob9w cm rolls without slipping down a lane at 3.3 r/(95oigyr.wrxc ljp8 mb8j* $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wio6/gqr do(6mw+pq,f mth respect to the Sun as the sum of two ter/p+,6m6q odgq(rw mofms,
(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 hasu/rgplzqw95 aos*fp4 )9p i c* a mass of 1640 kg and 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 8.00 s? Assuwap5 l9p/ *gciz qous*9)r 4pfme it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg st-ycvs.ycfw w3p, s: o5arts from rest and rolls without slpfycw3y:w -co,5. svsipping 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 ihbd d ylj88agv)(f*0ss balanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hintlhgd( )8by as vd0j8*f: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rc wh8.4 p vpla,g:sulrsb08kb3 +0g ciotating on the end of a thin string in a circle of radius 1.10 m at an angga u8lw,b:k+crb0sgi8 30l c 4h.p spvular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2dso j)nc: e66gv hrk;/.8-kg uniform cylindrical grinding wheel of radius 18 cm when rot6hoj);k 6sgd n:/vcer ating 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 that is rotating a(tr+q1k )e:k xjc 8o cl+ewx-zt a rate of 1.3rev/zck xw8)r oj:eexltk- (1++qc s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of in7ohy+.ameizuae) o - :lc3 ak:ertia 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 ioza -3:mlaa)e.+ :heuoc 7ykwhen in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate;corrh6p 7vt: of 1.0 r7p:r h;6orv tcev every 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 ardgw zuo 5)p7gs.d.i. vound a vertical axis through its center at a frequency of 1.5rev/s The wheel can be consideg .og.wds7p)zvi 5 .udred a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skc5 b :pln(aaxtmy6*p5ater 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 momen/- 3x9n4 -yqgi2dpx 9rrv hwyx 71xsentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped ontm(,r/ me;of ;lka1 xbho an identical o l ,( ;/ea;hmfmk1bxrdisk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns *ij if 9 +5srd bplg.4diq6k+xat 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 merry-glfxc1 c bn8zbov0m/1(5kw 8eqo-round platform 1 z80 (olcq1fbvkemnb/x85wcof radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely withy2 ep c0l d0z6u*zwr8youu07b an angular velocity of 0870uuyb02zzw 0 6oldr*pe cuy.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 intojcrjva+;88 rp a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotaprj8;vaj+8 rc tion 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 .fuo6m, k;wviexcess 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 )k i,cx-e4y muvo6fj2$ 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 aqvdlad0 i 5 4*aye tdk3zx55;z platform, initially at rest, that can rotate freely without friction. The moment of inertia of the person p*35;d0 t xi4z5yaqvezld a 5dklus 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 at the edge of a 6.5-m diameter merxrfxd ifr7 02+y3 eb2 8 :btp:oioni9iry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1700 ir38:y0odx:f7exip+ ft 2oibb2r n9 i $kg \cdot m^2$ 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 r64ir 24cim8fs +w:qofjfv wb5 ope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center64w4i rfobmq8j f+: iwv 5cf2s of mass move?

  The Moon orbits the Earthnql q)p8o+sa8(aj 9r c such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Mool(j c)98n +aaoqsr p8qn as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of *d4 ei0d4zwl q1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinpd)jkn 3ff9 zif,;kf 5der of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at p dkfz5ffnfik9 )j,;3constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made offz ay(t*25g (jfx b6og two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculat fa5t(yz*j bo(fg6g x2e 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 isj .coezev m6 9ikg17 ltd*ca*( 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 8.2mn7tf6 :we md*bwtm ;0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the prwtb mw ;t7*nm2m6ef: docess, 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 itx 5ass- xb*+72n7kdv uqgic0 u to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 i* 0nu7+scudb - xqvx5kgs7 a2kg, uses a uniform cylindrical 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 illustrat:oi((0ru2-be x66wm d i yuwnaes 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/xujvq9q58t zd xaol*8oq+)b 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.ytxs :s(s1-jn , we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horij-ns xssy(: t1zontally 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 verticalpnm-ue/eza 4z yv04ln u7 +p4fly on the floor, and we want to exert a horizontnalf07 p /p 4y+zuzeeu44mnv-al force 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 spsa6evyhzdje8af r6r jt, : -,)vr (nt,eed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the normerr-8 6jvevdfy)(hr: 6n,aa s tz,jt,al 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 l;)i kv 6y0xsnluam4hj)g i3(dfg..a1.5 m and begins whirling the rock in a nearly horizontal cir .mh;dl.fv axa0(i4kng3 iy6s )j lug)cle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, mr; ,hl5cmow1 aaking reasonab;hrowma15l c,le estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch axd 7 aho5 xki49 7qfqm 1;p/troolf+v9wta+;qssembly which consists of two cylindrical plates, of ma;qo ho+q rf+d;tf 49wkoa5lxi1t779xq pvm/ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track +g)sh:d+ygxq.5;ljif)n v uoof 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 point of the loop without leaving thegn;odv +)qjf y).hsgu:x+5li track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bqpw13 c68pmpiv81 qj jut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed xj;btq3*f2a9 h w0ysuz ho;8s uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration o9;w03q*hjzs yxt2 su ;afbh 8of 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