A bicycle odometer (which measur(rsb*zv *y-wv vx60. vk 9tzdhes distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. Wzv6 0ds(vxv y r.-z**9b vhkwthat happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates dt) w-nlser/+ at constant angular 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 magnituded le )+/w-rstn of either component of linear acceleration change?

Could a nonrigid body be describezi)/kr8s 3ysn f7 shv,d by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a r+ s1 gdn85q x,gasio*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 wb 8yqfa c8/g jji*r(1wjk yz10ith your hands behind your head than when your arms are stretched out inj8/* ayb10(j zwgryiq1kj f8c front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at thqx0, z+ 9zsrdje pedal cranks. In which gear is it zz9x+srj d 0q,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 a large front sprocket? Why?

Mammals that depend on being able to run fast haz k +oi0l1n6vkve slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advao lkn1zv k0+i6ntageous.

Why do tightrope walkers (Fig. 8–35) carry a long, nar-4,a m *nv+92 aztjyj0tifhdtrow beam?

If the net force on a systo 6cc 7bj1t7ur/ tg8veem is zero, is the net torque also zero? If the net torcevct 7o /rbg86j17utque on a system is zero, is the net force zero?

Two inclines have the same height but make different angles 4j3n auws6kx6b6ef1-g a6rz sj 0 8odcwith the horizontal. The same steel ball is rolled down ea usgaad3 c4eb66 kjsx jwf-6 06z8n1orch incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneou r;k ;x 4ic(shrnoz77asly start rolling (from rest) down an incline. One sphere has twice the radi;4oz n77ski(c;r ra hxus 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 energy at the bottom?

A sphere and a cylinder have the same 5 .xhk9ytco6)rkp2k5snjg 9b radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total.r5s hn9bckkyj g2p5) 9tok6x kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are tue4*h,qewrh. ird4/conserved. Yet most moving or rotating objects er wr*uh t 4heqd/i4.e,ventually slow down and stop. Explain.

If there were a great miguca0pn ofke78awq(f d330f og2 bu1b .ration of people toward the Earth’s equator, how would this affpe cg2f du 3qoab(0f1w8 70fa.nuo k3bect the length of the day?

Can the diver of Fig. 8–29 do a z2 p ywtu 83rw43 j/tnperm69bh3xb-i somersault without having any initial rotation whe hb rutpw-3xm8 3b 9yj2346z/rnwt iepn she leaves the board?

The moment of inertia s.n*v3 t r/9s6 cqfsuz2dv(5t wjiom) of a rotating solid disk about an axis through its center of mass istod* tw5s 2s(v f.u i)nvm63c9/qsj zr $\frac{1}{2}WR^2$ (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 hx xrdlu1slw :3ev6:4:o lfv7outstretched hand. If you suddenly drop xo xh43 :vd uvwlrfl7 6e::l1sthe masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identi 3rp2a3b mkl3ucal and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is which ukp23mb ar33l.

The angular velocity91ofv/m ,e3 gv z0,vpkgb: o prcybdhz0u,70k of a wheel rotating 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/7 :30o yvu1bpve0pbgd,0 z,rgkc vkm f,hzo 9 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 turntable. Whatolq3 5f /)7vpaiq 6doc happens qidq63 afoo )7cvlp/5 if you walk toward the center?

A shortstop may leap inr9 nn *- zsngl+y3bx4fto the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opngyn*x rz-sn4 l9 fb+3posite direction (Fig. 8–36). Explain.

On the basis of the law of conservatdbc zhc1s1,6- )k,f3if8nxud2s tbb nion of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the secoxdbzintfk,n ,b-6u)b s1c1 fc8s3hd2nd propeller can operate to keep the helicopter stable.

  Express the following angles ix *(yqe 6c:ubfir s0mj41pz/rn radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, using the/.xosr+d 3wz7zeabux 4qr11 s information inside zu arbwdr1ss4 oe+3 zx7x/q1.the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed atz *tf8s tlze 3 x/+6zjs70zqir1wzl, w the Moon, 380,000 km from Earth. The beam diverges at an*e367z z,z0ttljlr8 ss1w iqz +/w zxf 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. ok-(7*8nlfu 7y wuhaiWhen the motor is turned off during operation, the blades slow to rest in 3.0 s. What iu(yl7fn 7oua-wki8*h s 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 the i.t 3s06qv c hugh0uh1ball makes 15.0 revolutions,h qv10t6 0uih suhc.3g what is its diameter?    m

  A bicycle with tires 68 cm in didaud: e6q qqy2 //v hfh),gwd;ameter travels 8.0 km. How many revolutions do the wheels may;hv: 6u)q/wq g,qd/edahd f2ke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm.iog1tw ig -jw-(enn lzn(2//w Calculate its angular velocig- je1//(nt i-gowzn2(ln wiwty 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 (Fig. 8lu7r6w(cjkmmkct ;t 8n/x: d;–38). (a) What is the linear speed of a child seated 1.2 m from the c7 dl;8rx6 kcw/ mu ;m:tjtn(kcenter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) igbugf3cvgr5 *mqmv a8,+kp 5im6-,a in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed*tfiq 5 rjjp n.9ne0w8 of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the azm3vfx:nar 1bof7)dj )wt,t, xis of rotation is to experience an acceleration o,dartwj 7: )tvf,o3 bzfmxn) 1f 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unif xk22 z::rmflwormly about its center from 130 rpm to 280 rpm in 4.lzfxr:: km22w0 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 zza.1 ,ctou8j6 ddk e8$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo sp m-txezo7:o)ap7vm njm badm9et: *e0(4i8a yacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylindere0yzioap:)b4mnv md8m eam:7xta7*ot (j9 e- 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. Through rr b 4.7delkxvon0z7s0 l( wk,(nu.ekhow many revolutions did it turn in this tir( . 7,.7n(4bskxzk0n dov0rwllu ek eme?    $rev$

  An automobile engine slows down from a.cno ng;dk,:sff 87f 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of ;1-)ckl wgbdk flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions kgwbc)1kld; -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 diamett:oc6n agg7k awdk6wl-c:r72bfu) k 7er accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have tra:k 67gdbkcnlwa-t2 a:w k r 76fc)u7ogveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/minz sdsa2-0lxlbv(zrs d*kp( 3 ; It turns 1500 revolutions before it comeslpd0(srbv da kz2-* sslx;z(3 to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the cgbyrp9 2-d8xwar reduces its speed uniformly from 95km/h to 45km/h The tires have a diametegbrw9-x y d2p8r 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 thbm7e8bbr -h/ qy/w)jf e car reduces its speed uniformly from 95km/h to 45km/h The )qw-b/re/ hjbf y7b8mtires 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 jk*+ yg36dtpl po6fz; weight on each pedal when climbing a hill. The pedals rotate in6z 3ltodg;6 jyp kf*p+ 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 vj. u sqj7wngkg n/9j*ri51/ eN on the end of a door 74 cm wide. What is the magnitude of the torque if the force ig eq75 nkj.*ns/v9jwj gr u/1is 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 shown in Fig. 93psz *i60w)isjwcie 8–39. Assume that a fri0 6psi9ezj i)swcwi3 *ction 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 massless rod which gxnmq4a, vu,n cc-rz08u wmgve6w(.+ pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and directimca+6z8vnxrc m q v (w,4w. neug,0ug-on of the net torque on this system.

  The bolts on the cylinder head ol ;r vw3ckgg ls/+4uk:f an engine require tightening to a torque of 38 3 c lvrg4:lsgk;uwk/+$m \cdot N$ 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 aw fdu. hm76f.ffd cr)9 10.8-kg sphere of radius 0.648 m when the axi9)7r6u.dhffc.w mffd s of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of ino4j.z:dj brfy,r :2o s8+czkcertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have s8:zc.b2ck +o 4:,fjydjzrora 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 thin, light rod is rotated in a hode(jd*). f+worm af ,j)d.do prizontal ajmjo+ *fpdfd)(.r)o wd.,de circle 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 5(axr *e v5xxf3n+:b cjiyew.a bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 13 f xwjxbiay5rn.cvex*5:( e+.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 6wo5dow5 2 tcqFig. 8–43 abo2w5q56ot wocd 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 whb j c2t5n/qr5i y fbr/7lu9c6loa,:c mose 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, uniformd9be hi4 t3ufr m-yus 6g1(9hc cylindrical 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 mhtg1 ye i94 3uuc96b(d-rfsh 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm ans yjo)7y pj7l(ri3(;8viiehjn j bzb5h-u3 y8d a mass of 0.580 kg. Calculz biu3jv b-ri;pi 5jhlhje)y7 y 7y8(j(38 sonate
(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 - j8:0ntvi 7dkntun r0big4o2a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and o.4 a .wcps;jbis 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 .o4.jcs;pa wbof 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 rotarmok e:h 33f8ating at 10,300 rpm is shut off and is eventually brought uniformly to r :mo8ak3 3ehrfest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kgb o0uka6u 0g gd0m4ve8 ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forr(:l k mdlh96ey,nisg1wnfwtw s64fk6vz. 6 9earm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelen zni(6l4swlk 91,y:rk s6m6 gt fef.6dwh 9vwrated 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 shown in Fig.lxvg l763 5e-sen z(yr 8–46. yrl 3esv7-znex5 l6g (
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of t /kk kfdxaep) j;,r5ol,sn *;uwo masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turns () 9 inxxqjqp,wg.we7 -revolu9 wqnie7q.x)j p wgx,-tions) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of:dw y( vaaj2e 9x4)p,cwiqj3g inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 2aj au*4s0 )ofd80 $m \cdot N$ 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 wi7yu:9uwlrh* tf(f j :gj6 5pxithout slipping down a lane agw: :li 6j5(y*9fhurtpuf7xj t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respec7jw/ uitpp0;owbw4jnu- y- ; dt to the Sun as the sum of two terms;jp ;o-nb/0tu-p w4uyw iwd j7,
(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 g; cnr+z7rt n(et;tvdde/wake)f 5 (* m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume iz ;twe( k/rv;tc* d7r5nt+edne f()gat is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from .e26rly,u uyd.q /hgvrest and rolls without slipping dowedruvh6 lyq uyg ..2,/n a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It4 x eamq2q+fs)o9dtf 5 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? [Hint: Ud 2+s o9f)tae qq54mfxse conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rovf*m dfwyf:;4,t:/,wcze sgc tating on the end of a thin string in a circle of radius 1.10 m at a cwcsv, ,ffgm;de*wyz:4tf: /n angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg un ;rtikg 1q;:otiform cylindrical grinding wheel of radius 18ktio;1q :r t;g 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 his7bj:hz btui58 side, on a platform that is rotating at a rate of 1.3rev/s If he raises his abj 57:tuh biz8rms 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 Flykysia b25 /a gqd5 (2r.3dcvig. 8–29) can 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, what is her any(a a25kd.v5/bigdryl3 c2q sgular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin r)0ot; 9)eqz a :hr)i1vypzmys otation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of)ps) a);qehz0viyy m9: t zr1o 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 axises n g:/ v9nv+krjb1mh+opjr/ 7-nxk3 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 the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it? e3 1v9r vjh gn7on+j/rb k:n+/px ms-k   $rev/s$

  (a) What is the angular momentum of a)e )9r:44mwhn psdmy n+g(yry 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.ex:jshi2 d+a 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 ofrad-k8g)mxhly k*4 l)ig+.uov4 xk 6 l inertia I is dropped onto an identical disk rotatingi yh4m)k gxa- rx.8ko6l 4 dglv)u*lk+ at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.40p6dwmz 0 kigl3-ohx / $rev/s$ 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 standsa x 2w)o)/zh(7w gqzps at the center of a rotating merry-go-round platform of radius 3)q)gpws/2zz7h( w axo.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 ansa;bwik y 77woy- fqjln,,ha(+n i,9 qgular velocity of 0.8 y l;jnqwqhf w,,a79a+i,7ib oyns (- k$rad/s$ 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 in0xneip ln q7. 6bm.ytbnoj(2 wq;1yj1to 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 carri6(t 1om1xl7jbw;nq2i p y b yj.0qne.nes 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 omn3fc)*:iw hiun c,im . hpd0*76yvv 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 iw ,h(x mh8hy1$ 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 ah ( voagy4bls5mroe,0.yk (;n platform, initially at rest, that can rotate freely without friction. The moment of inertia of the person plunsa;0lb(og5v4, mhreo (.yyk s 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 6jhfm g589eov-ml2lz edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and -5j m lfhgl e69om2zv8has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope dm6d zo)bu f3;lying on the top edge of the spool. A person grabs the end of the rope and walk z3 ;uddo)6mfbs 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 samxkj o4g w p9kjwpm/f* i-y:,7ie side always faces the Earth. Determine the ratio of the Moon’s spin angular myow j-mk* 79k,:ippx 4/iwgf jomentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/9uq c1mwx 4wbf ny-.)t$s^2$ 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 cylinzd(k5qt cjn*68 lvp ypa3h -7zder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s inj7 * (pz6k yzhct5a8-ln3 qpvdterval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg andy( 12fve bf+nx +mnb1h diameter 0.07n11f ++yvhn(e bxm2bf5 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the ;v;yl zri t7 ,iqxt5wlx7+w4k 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 j7d hd.u;0w e,6mpmb;pbk*s s.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 process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries oh;d;60.k *7u smbmespbdw ,jp ff no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobileg-3l/s f8pgse 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, and should be able to travelg-3 spe8f/slg 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 as8l9 1rkn 7 +;wqqqs fvz37e1chhszr 3o ;,idan $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at s exl0wxlg,mvt,c:+ .vpeed 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 p,gy9 kkvq*4kcan 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 rk*,vk py4g q9kelease, 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 56 jkvgq5dmjmg d/fr r0*p.c.radius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climbdm.vg6 .0 5 j qrk/fp5jmdg*cr a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/syli :r4 as9;gg on a flat road is making a turn with a radius The forces actingri9:g yas;4gl 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 i7/afe7ctzlx99w ybbh8+ :er lnto a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, x wezre/8+lf7b9a t7l9bhyc: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 hew(i gs554bph2,uhky b,w nw) dr arms as thin rods, making reasonable estimates for the dimensions. Then calculateh4 b sd) 5(,ny2u,p5wwbkwhgi the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylin 0( th*hi.y .lnvzo5 iblh:e.sdrical plates, of massne0ihihlh*y v ..z:lo. t5(bs $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radiudaw7islwq qq9qqg l* /;1w3u;s 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 point of the loop without leaving the track? Assume ;1 uq gwqq7*a s9q/qlldwwi;3$r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do n8ar /h5x0nhwfot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unif lm+mh*x(lb t1ormly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular axhl(m+1*mlt b cceleration 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