A bicycle odometer (whu/x t4wjps4gh6h1 7 vvich measures distance traveled) is attached near the wheel hub4 v1w/76htxhsgvu p j4 and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rimizjv).eik ; pnz:*:n aymg f)+ 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 oi:ga)+v mjk*in )p:.e;zyf nz f either component of linear acceleration change?

Could a nonrigid body be described by a single value of the ) zyk6sw g+h5kangular velocity $\omega$ Explain.

Can a small force everjb(qwli:tila:c 9,. a u/ e2ki 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 head thanvn.5d6/qx3a pi9 fsj y when your arms are stretched out d3s6q vif.y5/a9n jxp in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear whev 1 u*ny65x(c-flv xzdel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large r ndu-6(1v 5fl xxvyz*cear 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 have slendeyci/3v )x9q.,iqumv uj 2 kmay5cve*1r lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why thiq */vvxe.m1ju2yu m93q,5icai) ycv ks distribution of mass is advantageous.

Why do tightrope walkers cym/ wn:+4w te(Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If th,lym 0 v-vfzy9lj01fye net torque on a system is zerof1, j0z9mv l0vlfy- yy, is the net force zero?

Two inclines have the same height but make diffz5 h2q.)je 9 lr(v mcavcj1z7ekav9+ merent 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? Evr q 9e.95m7k +czva j1lemvahj2 cz()xplain.

Two solid spheres simultaneously start rolling )e3 sldgj g-/m(from rest) down an incline. One sphere has twice the radius and twice the mas-dm g/3g )eljss 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 cylinperfc 2n:u,)ider have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greaterp, cr)i2:eu nf speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved.;bq8e-5dxc hx Yet most moving or rotating objects eventually slow down and hc;e5-qxx8b d stop. Explain.

If there were a great migration of people toward the Earth’s equ;.e mxua31i jhva e6p*pf9km.ator, how would this affv;mpmj.k * xafe u9p1eha36i.ect the length of the day?

Can the diver of Fig. 8–29 dod7asd yg 9cr(m g w-w9k;qrx*1 a somersault without having any initial rotation when shedrrmagdk -c;w*9g1sqw( xy79 leaves the board?

The moment of inertia of a rotating solid disk about an axis through w+*h x kwrcnxtx7+*s d3 4j2fvits center 7wrxxk d3v 4w*nc+h 2j x*tsf+of mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting jm7w 1l1xl g7won a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity incre7g1w7xl jm1 lwase, decrease, or stay the same? Explain.

Two spheres look identical and have the same m 4l .w3bzhxj7uass. However, one is hollow and the other is solid. Describe an experiment to deh7zlu. w3j 4xbtermine which is which.

The angular velocity of a wheel rotating c g.ig/n4i(s non 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 easg/s(nn i 4.gcit, 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 fre/ h1bulxy-p(mq;wk n7ely rotating turntable. What happens i/ybmp(qlh ;-ux1 nk w7f you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quicklyzw*w6 km+ ug68ttjm0/n wwiz* . As he throws the ball, the upper part of his body rotates. If yo*mwigk*wm+ t6uznz 6 tww8j0/u look quickly 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 momentum, discuss why a h.(tt r) sdttvdte,f )l nz63h8elicopter must have more than one rotor (or propeller). 6)tntt e (htzrfds 3vd t.)l,8Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in rae++ 6 elphuo6sdians: (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. Cl0gqrc:dp )/c,.qz 4 iaylp3xalculate, using the information inside the F0c g4lp.pdaqcz3, i/xry l):qront 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 at thezfsb 4j5 uri)5c-a4xi Moon, 380,000 km from Earth. The beam diverges at an ajc4zsi f 4xrbi-)u5a5 ngle $\theta$ (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 2sv :vj /nw:*tjcu5 ttxhqf86off during operati/ q:*v8sv6xctjtj 5h u :2nwtfon, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on *2b0yj xxqy j-a level floor 3.5 m to another child. If the ball makes 15.0 revoyjx0 2 *xq-byjlutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travelsq4z p :gfbo7)h7nfr0kb+a:g5e qls 0e 8.0 km. How many revolutions do the ohza4ef+0b:g5) qqber 77klgs 0: nfpwheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate it7ggg-.kkp i nc,,ib r7s angular velocity in bg7cpkgink, .-,7ig r$rad/s$ $\omega$ =    $rad/sec$
(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 revolutq rvevdms3/cvu; yq1w,6y . 3 py0p8rdion in 4.0 s (Fig. 8–38). (a) Whasdmyqc/p y.6pe8d,vv3wvq1;ru r3y 0t is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit asn*qk5tf//q4x)k dj pround the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spewe d12,yv/kejzva; ;m ed 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 axim9udnzv9bj:n b6+ i6c s of rotation is to experience an acceleratio9+un:66 cmd bbvjzi n9n of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center frvtl7y 7eq;( zpom 130 rpm to 28q(77evtyp lz;0 rpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius2p m,)tj)h ftk 2 e.vm ;fyjr;auy,4nv $R_1$ 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 0y/hpe;i 8nd 3 n9cgeuaboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelden9e gni3 0 ucyhp;/8erated 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 r,moxlefym)u* w 5-bu4est to 15,000 rpm in 220 s. Through how many revoluti*)um 5y wb emfxluo4-,ons did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 u7g3*epz4 r8fr.6oyvonz a rrx)e3 dn3;o. forpm 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 sw+q ns- ung )6 as.*lnrgus*6vtresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 compw)rs-nlqg*g .usa 6 n6nu*s v+lete 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 accele(pt yb ln9zy51 v i+bw:s2,fkgrates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on tl+py9 t(f:bkv bgn, 1iszyw52he edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turn.1gff-1ii : 5 o,dxzrrxsh/lv s 1500 revolutions before it comes t1.ho, 5xi:z1s ri dvg-xrlff/o 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 car reduces its sp 9 59ic )hur6svfqidsu, k 5uee;+j/zreed uniformly from 95km/h to 45km/h The tires ha5ssduuch rqiue) eri6+,zkj; 95v/ 9fve 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 revoluti5 ,a /v1irin7gdemd. zons as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diamet/d71g id nvir am5ez.,er of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbing ty m(k0u5rbb 6) .jek:jev)bz a hill.vu6r0zk)j kb.5t )e yb(je :mb The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. W*wbro5jh d (jl is6v+(hat is the magnitude of the torque if the force is eh boj6( +wl5(v*rsi djxerted
(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. 8–39. Aspbseeoml7u*-. 02obw sume that a frictio *0mwusoee-b. b2pl7on 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 t 3h*ct2x2b zb8+ioyvshe ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then *tih8bboc2vy3 z2x+s released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder 51 ,qowc)j2aou(hhru head of an engine require tightening to a torque of 38 h, o h1u2uwqj c5o(ra)$m \cdot N$ 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 momentb7u+t sy.2es2ocm(l0jxbi tj3 +9xec of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its ceno jsle+.7y3 mc2sxub ib+tj2e t9(x0cter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim 3a z2l.rpr z,8lpp4cp and tire have a combined mass of 1.25 kg. The mass of the hub can bl l,4rp3a2czz pr.8ppe ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a hori2vsf x j;)w9tnzontal2x 9n;tfwsv)j 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 constant ang om2/vgyzx a-;x/*ighular speed (Fig. 8–42). The friction force between her handsvxox2*gm ;- zg/h/iya and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array pwrn q*f7*k9nof point objects shown in Fig. 8–43 about*fnwk 7 n9qp*r
(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 x 89p 49vhz eiuzpb-7owhose 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, uniformcb3bvu mdoc9oy1v 52 4 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 mc35vv 2ob4d 1ob9uycrpm 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 of 0.elipant1c57ltlg u 7 45 84uey5805lyetpt 7cn e il4lu8ag4 571u kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, acceleratinmpspb.swfd z(7t0o - 9ua 8dcd( :pbf)g 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 meuzel:l5rv0nf c 0;2hh rry-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 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 torh;:l20nz0 hufev 5lcrque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brn :u4ujg (hdwo0 rt+hj2avdzw9 -*cr7 ought uniformly to rest by n j ou:(wvj*da gdc7t+0 9zr2u4hhw-ra 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 accenf4 0tgqjs dhg/6zu.*+a47sh 5eupa dlerates a 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 bal+5pulybdt)e,/prxm:6 a+*bbgr l 4dgl 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 uniformly from regymlbr d,exd/p5*4l+ubt6b:rg pa) +st 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. 8–kbeg*4 r0ztpv; 751y z.g+d)a v vmxdj46r*xdv+g17bz ) kdvj p 54gat; m.ve0yz.
(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 tpy41ewbb 8j(kwo 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 thh(w quf6fmqe :h8s 4t*e hammer from rest within four full turns (revolutions) and releases it at4hh:t 8mqwf ufse( 6q* 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 inertwj2u) d z)zmxg-; 1zcqia 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 dkg))m- cz ftyx b1j8u4.y fb-aevelops 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;o*) r53p dhdfbaehk a do1g8v6br)9d rolls without slipping down a laneh f;d9kr)8*dbdd vehb1r 365a)a gopo at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as thnb0hw 5o/: 7r6jz mwz1 mg icrw5cs)t;e sum of two termwjc/s7wi 50 m5zz;:h6o )bwt1rncrgms,
(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 hasl3 vb, resy(*t a mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution se(3l vb*r,t yper 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls wita ryq8 m(06zvk5o,vq phout slipping down a 38 0aopk( z6r,yvq5 qvm0.0 $^{\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 vertically9r7bt 6 /blfd(,zcxd q on its tip. It starts to fall and its lower end does not slip. What will be the sfclddb96qt ,(/ bz7xrpeed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball 18 r0+u,)xdc6zb5py1nz qqnqll :kbx rotating on the end of a thin string in a circle of radius 1.10 m at an a)qnqcnk6b8x ylp+5:d,0bzq1l 1u z rxngular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical gr9sshz2 t 1xb6einding wheel of radius 18 cm when rotating at 150 hs6eztbs291x0 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 rot wqiq,t+79gxb8f3ze m ating at a rate of 1.3rev/s If he raises hisb+7q 89zig mf3qx,wt e 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 2l)fsy0f gkh9.q lamx5u /q6m moment of inertia by a factor of about 3.52 u9f. /q6)syl ah km05xqglfm when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can ilnsi7(y1 ;)s bijh6a wncrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a finalal1 s)by 7 ;(iwihs6jn 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 a 3 0jbucu 6um;la9cc6s+h;fav frequency of 1.5rev/s The wheel can be considered a uniform buc6smcfu;acj +0 9l ;hu3a6 vdisk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure k4yvib56um 83uwldz n)8m5 yt 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 yt(e0i7x3.s exjf f;dls 5l l+ p*vw6hwj.nh3 Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is droppe*1u4 d-e ncvekd onto an identical disk rotativ k4ne1e-duc *ng at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 :0n ysg4pn.x;.xkj do$rev/s$ 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 centev8e x 2:mfq1hop,)aupr of a rotating merry-go-round platform of radius 3.0 m and moment omuqp:18xe v,of ha2)p f 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-bfrn(v4*p u0,zz (yrwgo-round is rotating freely with an angular velocity of 0. 0rwybpzur(,*n4 (vfz 8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collat*) buwjm 8qa5pses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what wobjwaq* m8tu) 5uld the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess ofz-om7ywlf. t / 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 x, x3t ;nmfc q:8p5drz$ 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,l3+wa ,f osq1na:8mpl initially at rest, that can rotate freely without friction. The moment of inertia of the persa1m:8+s fqal wpn o3,lon 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. i- ckcsg s5ts5)psy015-m diameter merry-go-round turntable that is mounted on frictionless bearings and ic k5g-spss5y0c 1)sthas 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 hkuzf)qy6)7s hbvhurt:1 d3* rolls 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. The spool rolls behind the persyu36 sb :1kh7drvhtz)*q)h uf on 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 tfh.q rz5/y4oeh 5jr-s6bs9gqhat the same side always faces the Earth. Determine the ratio of the Moon’s spin y6 hzrog5 /ssrq-f .ejqb95h4angular momentum (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 fro 1taxe;vcvcyug,9;d 5m rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shapeaqx sfy,+b7w (kf4f 6n of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular a7 syfq(6,bfxwfa + 4nkcceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylab7td5 kyr.)x indrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass )br 7dk5xty .a0.0050 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 angular speed of the rear wln 5/se/j zb4ykdy)0wheel ($\omega_R$) 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 gre,h yt-q9jgq e* 0r7hzyat, 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 y7g-e * ,9h0ryzjq qththree-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 use 7jk96yg dq(k pioeh(.energy stored inked ki6(qj o.y p9(g7h 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 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.vq h2h.qh9gk,ckag. an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horiq/r gxy);- n t bkv;:;wiforuf1/et0azontal 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 can pivot freely ((tgxx fn7 )1j8vvs4uui.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 accelnt 7jvx)8v su(4g u1fxeration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the fl,, 1jsq1hkix woor, and we want to exert a horizontal force F at i ik1qxh,,s j1wts 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 f8na22dh y;jeldn(5rv.jhx r 0lat road is making a turn with a radius The forces acting on the cyclist and cycle are th( d2jrrajhd 8len2.; nx0 yv5he 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 x )a )(vadnta*of length 1.5 m and begins whirling the rock in a nearlyxdna a a)(tv*) horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’c6y 73shvq0q gs body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skatehsg y3 7vq6q0cr with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindricaape jn /s84v,qzx3* ki1i2)h:1splt njg8 agil plates, of massg:/1*vn41 8 qp,inss) apeg 8 jhalxi 2ti3kzj $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the lools px8vs .gnr+7/yb0 d;pod o(rm0 c3jped 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 rm(rb8dx c7s o +r0ys .l3ogpnp d0/j;veach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, be oj1o3oo . syw j(c5r+2a-)vomr+:1ehnvshg ut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions adeh zaias*+; ;s the car reduces its speed uniformly from 90km/h to 60km/h The tirdh*+ ; ;aaszeies have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s