A bicycle odometer (which measures distance traveled) is attached n-7,5ai :hpadh- u u c8mni gm)r*gqn.year the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle q7u8m a *pn, h -ha5cng-u:.m)gy ridiwith 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doezk,jt08-u- u m xlrn5es a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases unifj x tkn-rmz0,-u85eu lormly, 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 bodyp vh)r,/jt:( sjrr fg+h ,sfo .uz*3qh be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Expwkc 9hgkk:oc32tk / )plain.

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 g:oa8.e j:c tythan when your arms ar8ao :e.:t gyjce stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at th* 40ybe+5nhu dpkdwz1:t9dm oe rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear spr o1 h* u 9kd0dep:zm5w+tynbd4ocket? 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 rqpu9 f7st22ito run fast have slender lower legs with flesh and muscle concentrated high, close to thef2 qsir79t2 pu body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carr9u;tmcb m +hh9y a long, narrow beam?

If the net force on a system is js)edsfon nn,8ib25 (l1osi2ib3o t7zero, is the net torque also zero? If the net torque on a system is zeidn s)b25i2sbei(l , njsf7 n3t8 1oooro, is the net force zero?

Two inclines have the same height but make different anglw uv4*ji8,sg3fzq(p aog ;xw4es 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 greux8,(g4zq i f43g;wav pw*jsoater? Explain.

Two solid spheres simultaneously st1xv ig4dn8hs7u6s m)wart rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline fxvhgm68u dnis7)1s 4 wirst? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the sa-wxl i e,(nwz5me radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom znlw w x-i5e(,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 consf o. y0oa9govpzz6:-)qe-zlterved. Yet most moving or rotating objects eventually slow down and st:zo -v.q f-tzao0)z6lgyoe 9pop. Explain.

If there were a great migration of people toward the Earth’s equator, hvv2v:1uqv fa4 ow would this affect tvu: v1fav 4qv2he length of the day?

Can the diver of Fig. 8–29 do a somersault without ha;thqkv *x1 -hcocx5nbe6eah -y2k 27hving any initial rotation when she leaves t x hheo7*e-n x-c61tb ac25hq2kv hyk;he board?

The moment of inertia of a rotating solid disk about an axis through itac/1vc jle mx8cf *+p8yhs2 *ds center ofce*fjasm 8ly+* v2dcx/1ph8 c mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool ho7(z)vif *qyj 3xec0m nlding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decc 7 yjif*n3x)zq (0emvrease, or stay the same? Explain.

Two spheres look identical( 2.0 titdrawb and have the same mass. However, one is hollow and the other is solid. Descr(tda 0itwr.b2 ibe an experiment to determine which is which.

The angular velocity of a wheel rotati-5kyton*-rg7kuzi b 6 ng on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreu--ktgr5iznok7*y6 basing?

Suppose you are standin whsrgoj*7ti6e* 8 r1ojx lb*;g on the edge of a large freely rotating turntable. What happens if youeo*rw*i6 bs*7 8g rh1l jtox;j walk toward the center?

A shortstop may leap into the air to catch a ball a(9e*i zkoutp); jsn) und 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 roep)tz s j9kun;i o(*u)tate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, disyqw50ue pyv jrj8j 58-cuss why a helicopter must have more than one rotor (or propellrj5 5pvyu0-q8eyj8 j wer). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following3 bamc0u ap4 s05::q h.yupi,qyeq(kr angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, usingcek 6p*z,nt gk5 rgn98 the information inside the Front Cover, the angular diameters (in radians) of the n6ge 98k,gcpt *knr5zSun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Ey y5k.tf.auq/pn .- (yhgqib*arth. The beam diverges at an angle h g.tp5 y.akyuq*( -/iy.qbfn $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blenderbc)czg3m i3/s rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow downz3/bgi cs3m c)?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m *1 sjtv5 usbo rhygza6gs)kn v2eon,0 ;;;jy(to another child. If the ball makes 15.0 revolutionsbz5o;hog6k (svy1ynat ;2;j0 r, eussv gn )*j, what is its diameter?    m

  A bicycle with tires 68 cm in d7 jxhl.gmy d48iameter travels 8.0 km. How many revolutions do the wheels mak.g7hdm4 yxlj8 e?    $rev$

  (a) A grinding wheel 0.35 m inxd6 zj /bj0,f0hg,( miyiprx 8 diameter rotates at 2500 rpm. Calculate its angular velocih6z8yxd(b 0mr, 0jf ,igj i/xpty 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. 8–38). ( a p5 87/oahivei)q rbsh8*b0n9pqby8 a) What is the linear speed of a child y erio a 9n qhip5bh8)0sbqab v*/8p87seated 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 iwbtv 8xx5k5(dts orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poiow*60(gny8o0h n fr1y,swuiknt
(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 (zv ck5s gf+3ra centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acc5c fzgs rkv(+3eleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 2g9kp-qfm1; kck/i) g3 o(dz,wxlm3xy80 rpm in 4.0 s. Determi lkz ;g,9wq/ pdmkf x1omy)g 3(i-c3kxne
(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 jhcwl *y)-:y2+hl wcp$R_1$ 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 put themselve;8w /0 (iz c/jeldginys 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 cylinder with a diameter of 8.5 m. yncld/egi(0i z/ ;jw8 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 acceleratesio loi+ w6:z2 s*jh(ra uniformly from rest to 15,000 rpm in 220 s. Through how mi *a(wrzl2i hosoj:+6 any revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculateihq jhdho,tk587 83g *xszq 7r
(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 ofsq)(c jw j0q-2q 6nswd flying highspeed jets in a whirling “human centrifuge,” whic60q d)jn s2jq-sw(wq ch takes 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 uniformly from 240 rpm to 360 rpm in 6.5z7t;j*tn 3fl.r 2xnw m s. How far will a point on the edge of the wheel have traveled in this time? jn rzxl7*n.fwt2 ; 3mt   m

  A cooling fan is turned off when it is runnio 8rs-a.hl3oa ng at 850rev/min It turns 1500 revolutions before it comes to a stohral3. a8s-oop.
(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 s jqldz4ke,vsl;,ay yn-j+ w0( peed uniformly fros0k;4yw ydj vql+l,-(j,n eza m 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed x s*0.:fc j0luf tpxo8uniformly from 95km/hx. u*:o c xlpstfj008f to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her wq-72mpvc9p zkck3lf 7eight on each pedal when climbing a hill. The p2-k 7 9plk7czqfcvm3 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 (cq8ey9/ y(vrz2 laamthe end of a door 74 cm wide. What is the magnitude of the torque if a qy2/yeamz8l9v rc(( the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel gjp/ir7;wfbwhd( z /*,n i.m rshown in Fig. 8–39. Assume that a fr ,ihjw//g.rrnb( pfdz;mwi*7 iction 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 of544wsxj du) 0c, vhcbs a massless rod which pivots as shown in Fig. 8–40. Initiallbd 54c,0scw)sv4ju x hy the rod is held in the horizontal 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 re6 l- skxmxho(z n6x9ih kf72/fquire tightening to a torque of 38fso- 9lx( 6 k2x hifx6mk/z7hn $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the /y+(/m9 siddm9uw)w +jp;wu x tj(5m+s jxifq axis of rotation is w;wd/(mxs+mwjpuq )5tx umis( jd/+9jyfi+9 through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7stp+( xl 1mb vgfyl608 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass8gfyt+ sx v60(pbl1lm of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the rmf4 p9p23v*2 xbh2bbwfa.ogend of a thin, light rod is rotated in a horizontal circle ofg2r fbb 92pvb2am4*o .3wxh pf 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 bou 3xp-g(vja 6cwl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N tjxvu a(c 3-p6gotal.
(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 iner1*+v* .q1d c:lhyjxp,) bcg a5llljwptia of the array of point objects shown in Fig. 8–43 ab 1*.ga5lh)blxj*:vcll, dyp +pc1 qjwout
(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 atzm8dc0hlyvsy8 0 :lg0x */n nqoms 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 cylindr 7tx i)tp8ov3y.no cil,//nnvical satellite spinning at the correct rate, engineers fire four tangen t/n/c p8voii. t3n)ly ,7ovxntial rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cnfy xnib 3nwj+2g*kw( izd 9/+m and a mass of 0.580 kg. Calculajwnd/+ fzb ixn(+yin 2*wg93kte
(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, accwsde--deg y3 4elerating 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 anil7m.9 zh+l pad is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the mer7i l h+m.plaz9ry-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 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 eventuallr2lbz6.c :.+xv)asr;kg it,gjip wm 1y brought uniformly to rest by 1s cwjxbv tl:mi.g g6pzrr2, a;i.k+ )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 accl;xhggl+/qe : elerates 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 ball is thrown solely by the aclvuf6r 3i -cr0tion of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is acul60rr v 3-ficcelerated 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 thactpu2 f4oci;l7+y*n) v:fo p in rod, as shown in Fig. 8–46.*uao2;c+47ylpv )fip nf c:to
(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 twy ),+gcpa vb:gi gd7 a6wn-my+o 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 iutc yx* xi(4 .isavfo; 23g+ethe hammer from rest within four full turns (revolutions) and releases it at a spe2ti *si 4fivxu(.a3 yeo+c;xged 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 inertia o/z3sgzrbsqe*f5x 85r k0 mehu 3ox+9qf $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops3z6kt+dhr-,c u 3 vswb 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 rollfnb+ji9iwqej6nv0c * p +ar5)l -zuq2s without slipping down a lane ap*n+w-qv z9curl56ib e+) in0qa jjf2t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energh evzw1-g.w v5y of the Earth with respect to the Sun as the sum of two termwz1.g5v v w-hes,
(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 164(w9gdg;.)su vr: 8 mjkzuol(c0 kg and a radius of 7.50 m. How much net workgk w;)mlodvuc(g.(:98uz jrs is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rok(a/ bylt rz q.*8+ukest and rolls without slipping downklrk( y8oq/au *.+zbt 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:o ufezk*c9a,aon6i 6 vertically on its tip. It starts to fall and its lower*ckeoa,f :9a6oiu 6 zn end does not slip. What will be 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 of a 0.210-kg ball rotating ood) 4uerja/ e*n the end of a thin string in a circle of radius 1.10 m at an angular speed of 1da)jreu/ *eo40.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifor)k0teo(wvq m lihd-64m cylindrical grinding wheel of radiusdet kl-hiqw)o6mv(4 0 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at p+ zc .,7 j1+lnawrwmka rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0n++z,a1rww pl m7.k jc.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 r qu 8 c6k(yd7iq6d-ylof inertia by a factor of about 3.5 when changin iq-dl6ucrq6 ydk 8(7yg 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 increase her spin rotationz3so/,1 b+z*sy,ol(9u d yk(yoc6qqdopkf:h rate from an initial rate of 1.0 rev every 2.0 s to a final rate uo+q3*c( /,b ysh9ky(zl 1ydsdo,p6:o fzkqo 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 r i*38.g e1fa rqe4u2j(fswtcr otating around a vertical axis 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 th.fs31 waqegf4u c*rejt82(rie frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular m,p;mzka2y9 db4, w ;k(wri*sivgmzgc y/*5hs omentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of thc69e rgq qp8em)yu 7:te 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 cylindz-* 4k 5epyxl3-rsbzerical disk of moment of inertia I is dropped onto an identical disk rotatinge 3-x *zplyrsek4z5 b- 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 kja-t0nj,uybvf;5 k 7bc7p7 ojw,:vo $rev/s$ 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 csfoa r ;kd m*7ia()4vvglg13 at the center of a rotating merry-go-round platform of radius 3.0 m ama13olkg 7c(*ra ;)sigvfvd4 nd moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular veloley-m2r1z,cb73yp ;,ka hg iq city of 0.8r,m ,y;ebqh12igcy 7a z-l3pk $rad/s$ 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 intoew+-pvu6cf r7i /bb: r a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% cf:+7/r6b-eu b ivrpw 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 winds in excess of 12dhx4(. /ps q1f bgsp:m0 $km/h$ 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 wv ah5 m hjqck:vx8+h9:6jjp k3jn.x 0xv5ct6$ 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, in7, +zjq/ky jgqitially at rest, that can rotate freely without fric7zyg+ j/,k qjqtion. 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 staqvt/y5 n4aoh ne.(k1d nds at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and a t4eoh v/dqn1nk(5y. has 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 tl*t+hhkp wc +b2 unefjl o2g 0sl5qe: hh5+(x-he end of the rope lying on the top edge of the spool. A person grabs the end of the rope -5kjf+heulet x(s22c onp+hb h:lh w05*lq+ gand walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that theem4tycj 7w 7/ ly*typ :b;s:gi same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting theyl: tcge*yjtw / y74pi7:msb; Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/96 o9quqe4 hya)copjyxcm (0 ,$s^2$ 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 0.20 m acquire8r vcp 2qc fxhj-ar)bfxvw d 9)52)7jos a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque deliv8 )oxab9-rwjx qj)frp 57c 2 vvhdf2)cered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of 2n7mcds cj 6ux2+q9gm mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub ofscc j27mq mxnug 9+d62 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 angular speed of the rear wheel ( znps6)8k ,ch t*hf1mz$\omega_R$) 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 x s22wp2bi)4 vjmdr3 eSun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution everym3bp24rdvj x2) ies2w 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 off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stor9 w 0p41wr3q/jwou xdhcdsh;0ed in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniforxs3r0 c o4hj;1ph9wqd uw0/wdm 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 illustra9 wfkk p505p6nv,i rjctes 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 rollqi(gyddk s,1 :ing 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 length L can pivot freely (i.e., we ignore frictioraql;o o5.v26d9s f kxn) 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 vfo 5k;lsx 6r d2.a9qothe 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 floor, ana s)buw 0qmssj8jz /;6qqn;0md we want to exert a horizontal force F at its axle so that it will climb qwj /sau0 0m8nmj qzs;qb6; s)a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is maki20*co6 bzcg nw)6ymmfng a turn with a radius The forces acting on the cyclist andccw2n gym6o)bzm* 6 0f 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 qkrvv.ah zf7++k* 262bjas iuinto a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circb.6jsv k 2akh*iu +q+ fr27vzale above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder anuf3jp.v,zlqzvn * ,o -d her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a sp3-z, v.n lz,*qfvjo upinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assemblypip. ikt6hs5vmt;o :kg 3/ wz: which consists of two cylindrical plates, ofk; :vhz/ig.tkpm5p 3 : 6stiwo mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. p7mc imm3+ c.lt,73ooke1o / nkud /8hejz(xs 8–58. What is the minimmj.3k1 dosmz7oecle8/u o/k7,+ n cim(htx3 pum 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 do x.s7cvrx:f(3c q0 07xwdyqmbnot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its spep( vm ,t41 n;yp30/rco ejnunbed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each n,4turj(1p0 3pen/ ob;nmv cytire? $\approx$    $rad/s^2$(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