A bicycle odometer (whi:ji; zo rsadjf0 nmqky tj6*x;(i5-o6ch measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if5kn-t(qi*a f;sjx oo mz;yij r6dj0:6 you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant afuixyzs:h9f4.o4syd1c g .j :ngular velocity. Does a point on the rim haves:y :u.icf 1h 49sjf4d x.zgoy radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by4bt j/ z5)nviit9mi( g a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torqcgd rl2ikij**+ 5f vt*ue 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 difficult3nrnlv65 9i*gtx q0xe to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A n 3gtvx*6x re q9in50ldiagram may help you to answer this.

A 21-speed bicycle has seven sppfd h -ybxas6itl8 n(pppd0(dza/h 9b* 8;nk) rockets 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 sprocket or a large front sprocket? Wp i8-9p /h dpfd (a*(8nhl t6nkdabx;s0pz)ybhy?

Mammals that depend on being able to run fast 6za0p*unu 0if have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34)*up60iu z fn0a. On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a longgcjzntc1 yuzw4 ud, s1 os,pj8b 900agb;4wl(, narrow beam?

If the net force on a systmu)g9xi bs:*sp:4qc 8 kcq/i;sszg 4gem is zero, is the net torque also zero? If the net torque on a ums4*qg;) /qc8 gk iibs4sxg9:pc:z ssystem is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizonc xqn6c+ 0:jgzwz4lq75r g wx6tal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? E:cj5 67+ x6xgqwl4cn qzrz g0wxplain.

Two solid spheres simultaneously start rolling (from resvnzi0; aq v5z+c,0tzpb u;n 7ht) down an incline. One sphere has twice the radius and twice the mass ofz;57+0tv ncza ,uv0n; pihq zb 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 radius and the same mass. They8 i*+np+w929q)ycglub smvb d start from rest at the top of an incline. Wqciup98+l n9wsdbv*m2by )+ghich 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 conservedk.dvw:jkna (dw57 u y8. Yet most moving or rotating objects eventually slow down and stop. E:k(vaujn kw .dw87d5yxplain.

If there were a great migration of people toward the Earth’s equator, hox3q.rih6 jh51z(ay o gw would this affect th16gy35a(jhr z h.xqoie length of the day?

Can the diver of Fig. 8–29 do a somersault withxri: . ogakite .+j)-t4;sk wsout having any initial rotation when she leaves the board?w4eitsk.- .ak)sxt+ ;oj :i rg

The moment of inertia of a rota 76td 3h +7fjqz7u)w-xntv)o gccl 8kpting solid disk about an axis through its center of mas-x7 u7fn zqg+l khcd8v)cto)t6w7jp3s 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 holdin:6l ,.g 4fu:cjbpxqyg g a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angula6yg:lj b qfcxpu:4,g.r velocity increase, decrease, or stay the same? Explain.

Two spheres look ide7y6/ru9b8ab4qn.lim*us t oh ui, o4entical and have the same mass. However, one is hollow and the other is solid. l69 /rbo7a o,*u thm4 ueuiy sin8.bq4Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizonh)yl04b uuu 0cln*3gc tal 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, describu ll)guh*034 yu0bncc e 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 turntanlh 6n*cyzm27 3ar7csble. What happens if you walk toward the centl2 rn7 37 synh*a6zcmcer?

A shortstop may leap into t*k3x3b znc sr8he air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look qu8r 3szkxb 3n*cickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular mom5*g pk,sjf8 qoentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or*s kqgpj, o5f8 more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a a(;iv9 gsfs)n) 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 c,(az3ndom:j b pu e;5xr1poy0oincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as se xmbz;5p uaer y(1jndo:03o p,en on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km frocrl0das m7(.sr6 ihc o;,p j)om Earth. The beam diverges at an angle ls) m,c.0i6rr oj(oas7chp;d$\theta$ (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 motogb */ 8appoa0lqp8)t p4y2yqfr is turned off during operation, the blades slowyy8tqpbp*q 8 20ola4f/a g)p p 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 m4g0f(r 0ayd.yj2fb q o to another child. If the ball makes 15.0 revolutions, what is its diametfbdf0g04 y qroay(j.2er?    m

  A bicycle with tires 68 cm in diameter travels 8.04hi+q ay kcj0. km. How many revolutions do the wheels make?q. 4yck 0h+jai    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 250qs m1-s y1pytbydk2 d4)g28vx0 rpm. Calculate its angular velocitx-m d2d 8 yss1kqvy 2y41pt)bgy 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 make)ak w;(1vszifc:q q v374fwres one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of az c;(1rwf7w)sq v:qa feik43 v 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 Earp2+rd -prz rpind03h2j y7 i/wth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of(/wd ukgx he;/ 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 rq np:h u26avse 5z)6 iji;g6vuotate if a particle 7.0 cm from the axis of rotation is to expe6ue2i z)vvj p gh:6q5asu; 6inrience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acciof-/s8grn kh swufs-l) -8w- elerates uniformly about its center from 130 rpm to 280 rpm in 4.0 sr hkw gis--n)wf8lo8 suf-/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 radius5xcr w 3ceaie 3h*w22y $R_1$ 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 spacecraft pu.ko rv5* cc2rut themselves into a slow rotation to distribute. v*2urorc5c k the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000znesk0 /j7y87- m nrd 10d9nh1 iwobl;g5ddrm rpm in 220 s. Through how many revolutions did it turn id d m9d51ri /r0 ybdkn-e7m;no1g 0zwsl 7jhn8n this time?    $rev$

  An automobile engine slows down from 4500 rpmh3k81yau e16lh sehs6 to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets 3 o5 jc/yg5pnhn5 wdu.7mw:juin a whirling “human centrifuge,” which n .pjugw57m3oy/hn5ud5:c wjtakes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformlvaz4vbw,) a4n y from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in zb4 na, vw4)avthis time?    m

  A cooling fan is turned off when it is running at 850rev/min It turt0l 6manslt.b- n+js8ns 1500 revolutions before it comes to a stol s.m+bjastt0-68l n np.
(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 redu n t5j6bn rzbpt71n.d*q:ac9nces its speed uniformly from 95km/h .an r n65: 1*nzbb7ttnq9cjdp 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

  The tires of a car make 65 revolutions as the car reduces its speed uniformly fg:a:o2jdw 8rzrom 95km/h to 45km/h The tires have a diameter of 0.80 :gz8d:oaj w2rm.
(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 ridin azh.3zxzj7q0b9w .bug a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle ofbxh3zq .. 079az zjbuw radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What ,+f 38dg8hy1owrzaztis the magnitude of the torfzrt 8zh+d wo,83yga1que 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 thylr bu *e7/ask3s1,d ilh: 2tj:d8l tce axle of the wheel shown in Fig. 8–39. Assume that a frc ussrek:tajl7/iblyl *3:2d81 dh t,iction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of m fy, -dieh,.fq1/ldg p8uum r*ass m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the ho*q.1umulf, fye8dd i,r/ghp- rizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a 6h sj/ 5 vv eqxqznt4o:v4w*/ytorque of 38 4tqwv/s * :hnvv/ey xq4j oz56$m \cdot N$ 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 z3ffz2 8pda3zv(n l,cof inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is thr2dzz3c,f (lp8zavn3 f ough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. glynb*0jy u.xjy * uoq4b (*5jThe rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignorejn u***jl(uyy 4j5o q.0bgyb xd (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is red/j1,m6 i;,(mc ypc(lnd:bicy)gf aotated in a horizontal d :n)/g;pby1 af(deiym ,c,j cml c6i(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 a bowl on a potter’s wheel rotating at wpjpb8h; 6kl7v v0n)econstant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N phwp6 kbe 0v7v8 l;)njtotal.
(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. 8–4.du9ix a2-l rso 7qrg.3 about -a.drlsg2 .7 oi9q rux
(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 ae)l49syf np8c toms 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 spi7gpyw mo75xk 9y22wa0 4wts sjnning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 ss902aojpwy 4g7 w yxwk7mt25kg 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-gr6(n wlzzj7 cylinder with a radius of 8.50 cm and a mass of 0.580 kg6rgl7(wz -zjn . 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 from re4 rm m (mco;5) n7bgvqf.uza*lo(/pmust 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 meloe*zu9jp wrzs )) (9arry-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 of radius 2.5 m and has a mass of 760 kg, and two chiz(u)aelw9z rs ) ojp9*ldren (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 eventua-n/po,ux: fd*j(i rv -gljjlif9s)w +lly brought uniformly ( fsxud/-g-*l+j :i, vf ilw)pjj9 rnoto rest 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 o.5m8cxajj,gw hy sb;rcc70m-g n9y3a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by tf/6kf(z25zl 6 *:cn6w8fs zkuvip o kohe action of the forearm, which rotates about the elbow joi lzkw n2:fcv 8ok5 *6zp uzsfk6foi/6(nt 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 shown j1jpao 3m6 )jqin Fig. 8–46.a 1)jjmj3q op6
(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 consist-vzer)3.h rxu 6fqfc.s of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turcua;46qk7ifhpz) t7 t ns (revolutions) and releases itahpk )tf;c t6iu47z7q 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 ineoaysc7k 3ge n*i6: ll(rtia 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 ab5ektmsdu-; 0- ll;qtorque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping doyjsx 5n6/c.gewn a lane at 3.3 ynxcgj/ s.65e$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as th/ rno;sxc fhce/5p:1oe sum of twoc 1xn:5crhfepoo/ /;s terms,
(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 6 e 0t.iwp3e dd)j4wyaof 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 iyd dt j.)w3ae4w06ep8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 ka(ngcsj/n :ej rd x4/3g starts from rest and rolls without slippinxcdjnng//: (raj4e3s g 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 balanced verticr3zf 99 kmbzg3ally on its tip. It starts to fall and its lower end does not slip. What will m9fk 33 rgz9zbbe the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of6u:- mxhcak r fviz *g(p;)t*k a 0.210-kg ball rotating on the end of a thin stri(auvhp-i)6mx ftrkc*:g kz; *ng in a circle 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 2.8-kg uniform cylinm9 mi nnubftdk k6).o(cy2g5f 6::luedrical grinding wheel of radius 18 cm whn )96ctb2 :u(kmein ld:.6ufm okf5gyen 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 1h hai- i,gly8hvx*+ob ; wg,rat his side, on a platform that is rotating at a rate of 1.3rev/s If he h1 gi*h8y, vib+ l,hw;- oarxgraises 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 o e:a6;ahdqlt-)tzc;meu 3 )bs f 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 positioz;d ms;ac-tt q:bue)3el )h6an, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate frt-o ,k mx/f9nk7rdi 7wom an initial rate of 1.0 rev every 2.0 -rdk mxk/to7i,f 7 w9ns 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 around a vertical axis through its cen9xrrlan,lepm 2:+e + ater 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 c,e++9rlra2xem :pan l m, 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 momentumih*wlyyub 0 85c onj6xn/,uq-l9el*p 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 ;etg5ndk)94rjlk( uas3uc 9 zmomentum 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(bwp) 9 bq05ivjth w6q inertia I is dropped onto an identical disk rotating at angularh b i(wqpvqt )jw069b5 speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns r bw2x;cmcb 9t6. re)fat 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 sgx7h l7f q(yjt/2ef;nqhr((jtands at the center of a rotating merry-go-round platform of radius 3.0 m angjfqnly7 (2(h(h7rf t/q jex;d moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angu: hsxkw: 0np0:8denu z5w4 cfslar velocity of 0.8: pw xf0ez4 s:h :s8kwdnc5nu0 $rad/s$ 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 dwar ek6;bv4jbvs vzg -5i*7i xcf(f, losing about half its mass inbb- (*cef7v6 5;vvx g4ijikzs 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 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 wi3x b(g xfis f1q rykx,;(fldp:4. kt1lnds 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 e*vwa1e1d2 oa$ 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 4wg;w cz47t2034t(nm m*efsqz juoa ican rotate freely without fri4es zo( 23mn ci4;0mguwqwf az*t j7t4ction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m d fib1y6hws1qz:65prs yky;q3iameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia o5hz6 y 3wqbqk16 y:yp1ssi f;rf 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 rol 6to5wzv94xojls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. Th9xzo t4j w6o5ve 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 alw5ls*cr r;.jmm t:buy8 )maqv4ays faces the Earth. Determine the ratio of the Moon’s spin angular momentm*.4ju :st8byrmv m qcar5)l;um (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 re4.cyjlb6 f44uc 5xpdk j4omm* (l/smj/arz) wst 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 uniform cylinder of radius v1yg 2r*c +anprbn4-h0.20 m acquires a rotational rate of from rest over a 6.0-s interval ata42+* cp vy-ghrn n1br constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eae9i tylk9vf29szw 5di5i t- 1qch 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 osvi 5tt5w -qf91d99ii kezy2 lf 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 an18 fm pq3ep71 ulpten-gular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, 4h rh /p,(z-gqz9lts)j/oin; th1anfwere 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 9qhih ;f- p/l1tt,zza)o /nsg(h rn4jrotation 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 ocv 2u n r7y,g.u/a n/ngt*mv4low-pollution automobile is for it to use energy stored in a 4vr2 nmyu./vocg7u g,n *an/theavy 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 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 a-hq4z4p o q9;zv5jcqy n $H_2O$ 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 rk 43u,j+xk1b szrtk-) is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and leng *ig u gywk o7j2uh-a))6r*crcth L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the oaw) 2u-j*g6r*yc c7 g rk)huirod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically oilt +kre6ae; 9n the floor, and we want to exert a horizontal force F+lke;ati6r9e 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 sp(*eaio9v stbfs p3),l 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 thftb(si9e 3v,s o pa)*le 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 and bevja )37w 2,yndy)su bkgins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s7ay,)b3knw2)yv jusd. 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 cy v)igpfvyx:4l25 qq n+linder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held t2vnpygf qvxi 5q+)l4:ightly against her body.   

  You are designing a ;.kq 8z2jojrfao1,cifik:d1clutch assembly which consists of two cylindrical plates, of mas1o1 8i.:z,faokkq dfj2; rijc s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 8ywh mmz ,kh,1r rolls along the looped rough track of Fig. 8–58. What is the,kyhm8mw1z, h 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 $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dos yjt6mk(mbs94s 2 t/m not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unif jmy1gwb i1k sd,p-37jormly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each t s31d1 m7yjb,j- ikwgpire? $\approx$    $rad/s^2$(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