A bicycle odometer (which measurz)k af0l;pd (k2p1quwes distance traveled) is attached near the wheel hub k) w; k2dp(ap u1qzf0land 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 angula:k js:v-izn,y m 0cdvqhi,n5qu ,7i6mr velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear accele,q:ii-s q,nmcvk u vj7zih06mdy n,5:ration change?

Could a nonrigid body be dlbe/yz2ng i:) escribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever ma9,uac7m 9nv 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 he*sx cq(rca f hyo 9c( izs.4o,q2f,g.nywos33 ad than when your arms are stretched out in front of yor3ca ,ociq2gf.9 .zch q4foxsws*no( ys(3y,u? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear i4c zqb0 ms1k+n +vpq -hep)7zwheel 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 sprocket or a1eiz4 v 7b0s+nzhqcpm-p+ kq ) large front sprocket? Why?

Mammals that depend on being able to run fast have slen9e4s9i7l +b zkf fj2cxder lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of7lffe9 xcb z94 +skij2 rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35p4*x9 v3abdnnsmli,k 8d*zk5 kuo-t7) carry a long, narrow beam?

If the net force on a sy6 0 0k c,d5enttou8w,zquaw)istem is zero, is the net torque also zero? If the net torque on a systemzw6ut),8a e k w otqiuc050,nd is zero, is the net force zero?

Two inclines have the same height but make differekc mdkz9s6r ::- uy7ksnt 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? E cuk9-m:z 6ssd k7k:yrxplain.

Two solid spheres simultaneously start rolling (from rest) dem6f p37f uzy2own an incline. One sphere has twice the radius and twice the mass of the other. Wf67y2fzuem 3p hich 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 cylinder have thetw;lw:( x *khpv- 0aeuvzc5w 7 same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first v*wvwwak5 ;h:e7x lcz p0-ut(? 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 anguli1y0 n13w0mct / *1waqhglas nar momentum are conserved. Yet most moving or rotating objects eventually slow 0wmich1 3 tn1qwy*lgaa 0/ sn1down and stop. Explain.

If there were a great migratio 1kf5*8hw;kh7slfh :l zc86drqx6wpi+ j-d skn of people toward the Earth’s equator, how would this affect the le5 7 hrqz+1xk:pl8fj kh d66kwwh-;d8f*lcs is ngth of the day?

Can the diver of Fig. 8–29 do a some) r lyw7 8/ rl-n5+ykeaj*zimorsault without having any initial rotation when she leaves te-mll ya7+8n *o/kyijr5rwz )he board?

The moment of inertia of a rotating solid disk 6 fio,e d2obupn gbj/c9q33i2about an axis through its center of 9,fbce23pdi i 3oqu 2bn/g6ojmass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstrefps,c0.o/kp( kh-/8e +auge pvvk 5 pxtched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explaine 5f skg o0ae+cpxvv(8-//u ,k.kpphp.

Two spheres look identical and have the same mass. Howevevr dr i(se619yr, one is hollow and the other is solid. Describe an experiment to determine which is r9yi d vsre16(which.

The angular velocity of vz,r 8o n-thi*a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on n*to-vzr, h8i the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating xa)wn2aa g omu:*.hi. turntable. What happens if you walk toward the center?wgoaahumi2.. * ):xan

A shortstop may leap into the air to catch a ball and thropnp 3hqc0)go /w it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will noticn0/3g )oh qppce that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservatio3p u l(kl2iy0gw w(7rqn of angular momentum, discuss why a helicopter must hu lp0 2k(qgl3yw 7(iwrave more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the followinghubj/a 8isf yzs(n51 9 angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because o2m 780 b6zemie p0/*;bykrpcj)llraj q j3wf8f an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of002mqp ckjrfl8m)pjy b a; w3 ibje*8 6elr/7z the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Eartb wmog,77 cez6m8t,ei n9 -(u8haqnqsh. The beam diverges at -e,e8 (mm9,u78n ia6tbncqh s wgo7qzan 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 turned o gae(6z),woso8(vtcvb)4 xo)j og,sm ff during operation, the blades slow to rest in 3.0 sa)osvzxo,g) o w v8m(jo4),sce6bt(g. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If wa8p/4pt--a, irsdzdps ( k v)the ball makes 15.0 revolutions, what dv s-(-rpw/ ,tiszppa4kd a8) is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolut;o5*z:* u;z4q;m but ciwmx ceions do the wheels make?;z4cbei *x w*: tz;u;m qcu5om    $rev$

  (a) A grinding wheel 0.35 aabx6- gmn3;ih ,p.pp m in diameter rotates at 2500 rpm. Calculate its angular v,hxpb3p.maa6ng p; - ielocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s:nyi -ktpf-/caf n gaki c/;/9 (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from thg;a9:ti/ cykfn- /apnifk -/c e 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 ong) ,tx+lga6its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of v/zpa q y9;/ 7:orersea 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 4l1no j i1(:hhph)apj axis of rotation is to experience an accelerati1hnj ho4 a)p:ilp( 1hjon of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly1hawl 3ul2z8+;jm wfk about its center from 130 rpm to 280 rpm in 4.0 s. Defm;8 +3a jw uh1wlk2lztermine
(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+c- dn 3;duo.tgcimf, $R_1$ 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 6q*c- zxti+p gthe Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution xqi pz6tc+*-gevery 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 220 s. Throughgh0 u-jl1mi5 k how many revolutions did 5lk imj -h10guit turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm*h l-plr rz3z4e vb*+y i34gyt 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 thew*af tq:8wt .hppaf/z6f1w* d stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 mawf dt/a8*. 1hpqfpzw wt6*f:in 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 ah/oe 3p)y1k;tf zlyh9 yx;vn6ccelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this time? )6ky vphhlyz/9fxet1;nyo3;    m

  A cooling fan is turned off when it is running at 850rev/min It zx8ntmrr33va36jj 9*o 1eprm ygu4+qturns 1500 revolutions before it comes to a oerqy1m9vj83 +4apn3grxr3u mzj 6*tstop.
(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 car reduces its speed uniformly ;/thqib0r 6vc from 95km/h to 45km/h The tires have a diameter of6qrh i/;bcvt 0 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 itsd 3wjzgv* ea f-n81e7mqu1t/v speed uniformly from 95km/h to 45km/h The tigq-juft1e* nm7az wevd831v/ res 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 putstg8m sulx++txj1o2/ x pqa/s; all her weight on each pedal when climbing a hill. The pe st 8jt+qusgx21/p xlo/+x; madals 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 2f s0pza kssusp*z q.+ a+/f.-9p anlwdoor 74 cm wide. What is the magnitude of the torque if thzsnfus q-af k/9a p+zswl0p *..as2+pe 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 ofmjgsg4l iqfx7d)fo ad685+ z8 the wheel shown in Fig. 8–39. Assume that a fricgf4o8)q 6mxdgzsi7d8f lj5a +tion 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 massles)1f 7 qrx1yqaas rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude a1fq ax qy)a71rnd direction of the net torque on this system.

  The bolts on the cylinder head of an en7p-cjo dpur ;4l 2;rilgine require tightening to a torque of 38 op u;jp- ;i2r4cd7 rll$m \cdot N$ 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 spher b )x*h;hpjsnf4t*i e(e of radius 0.648 m when the axx jh)h *stpfn( i*be4;is of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameqxy+g(v +j vp9rzq,b (ter. The rim and tire ,q(pqg+rxj9+b( vvzy have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a 1,6kz zh,0c,udf6eiiw 4du yjthin, light rod is rotated in a horizontal circle of r1y,0hiwdz zeuki, cu4 f,6j6 dadius 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 a7+innhj:6f7 r d r2s,ntbru6n potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N to6:ri,n ju rn nntf6dbs77+2 hrtal.
(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 objectr6. ucvtil 24 j +tzl(jhk8xf)s shown in Fig. 8–43 a 2v fk tr.(i64+jc)x8tzulljh 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 atoms :pbsdxjx7 *,z-cz e.l whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct gh)mqzg8 brr0-f 1zh6 rate, engineers fire four tangential rockets as sgz1hh g6m) z8rf r0q-bhown 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 a radius of 8.50 cm and a mass o2m:+yme2. qhdb tzgi*xg6 mrj 17u4(wj oqx 7zf 0.582jhm:yoje 6 +m4gb d7izuq q2(rx m *.z1g7txw0 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 bateaj1nw/p myx)kk t j*h6sr :24, 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-de4gn1 rn:wk, g g87ekhriven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk o,ee hnng4k: kw1gr87g f 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 10,300 rpm is shut off and is eventuald)1z1c )(sdlj, o+hlg llfk+ply brought uniformly to rest by a fr) k(glhj11f+o ,pl)dc dzls l+ictional 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 7cy7w3at9 uq0dx .p xspi91 9sf $m/s^2$ 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 jkwq-/c 4x:mvah3 9gw ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated unifow gv:amwxch qk-9j/43rmly 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 xojm v-1 hsb;r2+z0lncan be considered a long thin rod, as shown in Fig. 8–46.1x b;+v2h-rln zo0mj s
(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 consivldu smrp80itl1 g2g e:(h49wsts 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 wjqjq+ 2p 0zl3.dr cur7ithin four full turns (revolutions) and relu3 +rqcjdj0 2qz.pr 7leases 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-1or1yj:8m8cm tqn7lz hj:v w 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 devfkyr,plgpdd65y(w q 0e2) ym m3;mmk1elops 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 withoup2g91.oraodyf)zt5r ohj- j*dh s i/ ,t slipping down a lane at 3.3 9rrtzos1hoy/hg5di j,af -2p.ojd)* $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the;a*:epop 14c*gm* chw+:kf)hq ei r fl Earth with respect to the Sun as the sum of two terflck*copa*hg)h1 * 4wqir: +p;f me: ems,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net 6v7h.rm9 s ux p)2pfzlwork is required to accelerate it fu79v s)zxmrpfp. 6 2hlrom rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts frno6ei(j+ q5x ywz x; x5,kuf9som rest and rolls without slipping5n u;6 o 5,yfxw9+xiz(sx jekq 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 balan+- get3t6igz )7idox jced 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 ti3i tdg+) -tjz gox67ehe ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg babvyma(; *pj s7 e2x i*zrn y+yz1ji2l1ll rotating on the end of a thin string in a cirj;l2y biz(* n1 ivrzypm*e1y2x+ja7 scle 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 2.) 4r1 ek fpb(-1we/2 hy7ss hhol0rbjf8-kg uniform cylindrical grinding wheel of radius7-bs b py2 rf( 1kjhroe0w1 hl4f)h/es 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at:6t8tbb riftk(q kzo n57rm f8+i.jl6 a rate of 1.3rev/s If he raises his arms to a hor l+ ni8to7k.rri6m6btfbkzj( :tq 85fizontal 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) 6 w-lp* yvam4av)8z xcn/vo7zcan reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, a6 y7cav*/8pzwx-zo vl4nm) v what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase eg:k,4p);ld(yew2yl:v2c,j al dttd25 sd rrher spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a fslk ;el,t)d,ywae l42djtcv5 :r p (rd:2y2 dginal rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at asmj n+bd1 ihr)kv6w6;t-ijg 1 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 ;1jnv d )hr+61g iwskbi-tj6m 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 momentumhj8wa3(g 9w bi of 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 of the Eala o5)xlh2)p p8 gtf ,xgvgzz(-q+u6urth
(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 disz5* lwq0 qmti -+i+td gnntu8(xa a6(ek of moment of inertia I is dropped onto an identical disk rotating at angular (x+lm( ti+*ztwg adn-5tina8 0e uq6qspeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at lwl -y z6;y c ebuvo/m9551zmy8w+sdy2.4 $rev/s$ 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-go-round plk 9 sqp2ifww e.:3z -o)seq1hkatform of radius 3.0 9.q :wezq3fwe1-hsi os) k k2pm 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 velochtck /;kgv y4bgn 9y29ity of 0. vhn2cby99k /ygk g4;t8 $rad/s$ 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, a 8h l87snni68 rkytiufq6up.)krj*: losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass catqnps6j niukh :8a.8k fir8u* y7)6rlrries 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,,h 5o-jf*jygvy 8m4ftv 3khi 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 zj05hs2, iwhy$ 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 y w9-1fdf9s5),s xqz1qyxlqp can rotate freely without friction. The moment of inertia of the person plus the platform ilqdwys yp-q9f,9xfx1q5 sz1) s $I_P$ 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 x qrd1-xjmz;ht5 d8w1 of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inerti1 dw-dh rxx1;qjm85zta 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 e /2*av4sml( rm b)qmgvbit*,gyk 2 i.ond of the rope lying on the top edge of the spool. A person grabs the end of thet)/(kgr o by b2i am. sligm2vq*4m,v* 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 such that the same side always28zp pql)ag--a3 y0ovijrqk 8 faces the Earth. Determine the ratio of the Moon’s spin angular momentum alo-)2kp 3arqziy8q0gjv p-8 (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 rest8 oji5ct:zzm: 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 of a ifm-8/o qhn+6oy ae6b uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant anbyoe6m 8h /-oiaq6n +fgular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two soli s0r*dj/ /p1mmuc tg6md 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 calculate the linear speed of the yo-yo wheu1cd0rmmm / pt*gs j6/n 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 spedksa 2r13eli 1:,q gijed 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 greawz rw)x a4*z7e 37ia9vny3zogt, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutraw7z aozzw*ygn9 r3)3eiv74x on 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 u//jiggy d )v,use energy stored in a heavy rotating flywheel. Suppov i/ud) jg,yg/se 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 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 illustrates )8on-bka pvy7an $H_2O$ 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 og + ul3 yb4dyul(a jt6bc6dghe 3:b9-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 3/cic6sif wp)b*z-7 q tg * +g6roldfbp*ud(m can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gl6u*dq*f(t7 g *mirpwbf/6dg+pz cib -3)ocs ravity 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 thebt) kz84 r :wrslok5v2 floor, and we want to exert a horsk5 4ro8lk w vz)tbr:2izontal 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 speed v=4.2m/s on a flat road is making a turn with(si a1t-lg:c78/ ozpcko,/xxth/w qw a r 1sc7/ (wtolg-kx,h8iaz/q wpc:xt o/adius The forces acting 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 1.5 m at2(; fvx)o0bbsn-k e jnd begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm a2j )kfsvo(n-0 xb;etbfter 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 th rvm9vr1d)9aer d-3tk in rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular stk-d 9m rre var1dv)93peeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of wz/ elzqy;lgc 3)(4;auzfs)sr:mn ,qtwo cylindrical plates, of massgem)y,q)rlz;sul3( ;q 4 f/c answzz: $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alo3h. n u* j.6gjxxt)ucgng 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 point of the loop without leaving the tg .cgj)6h*.3 tx uujxnrack? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do n7 bv/a-gfxm-jiin5k 9ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car 0 wa/ f4t 6v*ihsng/wtreduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the ang6fa/i/wwn4 h0sg*t vtular 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