A bicycle odometer (which measures distance traveled) is attached near the fd :a+;snx 6tbwheel hub and is designed for 27-inch wheels. What happens if you u tsba d;n+f:6xse it on a bicycle with 24-inch wheels?

Suppose a disk rotate q) (pmq/eg3pra q7(jqs at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the poiajgqmq( (e)7 qpp/r 3qnt 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 by a single value / a7vm+ yb:ivjwm/l9uof the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque thaz,igm00mv :zrw f f4*q) cuby4n 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 b o,* j8j tys(xolu)2uyehind your head than when yjj( ylo8*,) u tso2uyxour arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the 5oa:w )ldnfvcwb*z- 7 rear wheel and three at the pedal cranks. In which gvzw5 w) *ano:7-fcdbl ear 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? Why?

Mammals that depend on being able to run fast have slender lower legs withu v h(euicb;lf3/ciq7fc/yq 50w,)m k flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain q3c lucfh;c/uq(m/,0 wiyib 7kf)5v ewhy this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a8au1.gwn fl u6 long, narrow beam?

If the net force on a syst-ig)k* 05 ne7h glqso3 kyxo 1*mvbj*iem is zero, is the net torque also zero? If the net torquekqx1o5s i0ib7 gey*v3k)**hgm o-lnj on a system is zero, is the net force zero?

Two inclines have the same height but make different angles with the hordn-w4vv-mggm ,(jr : 0tit6naizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball ati4:- ,6gv0(gd namm- rtnvtj w the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incliprh::zdl 0pe*i r7l4 z6m*hs xne. One sphere has twice the radius ah4 z7p*ep0l:6 :rldrmx z*si hnd twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the sam:u7fa8 7/mu:l ys. w1fdygtype mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater sy.f1tsu7:fya7 g8l/ uyp w:mdpeed 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 co;q1 hvbvx e6,hcfk o8:.gange6l9 sj (nserved. Yet most moving or rotating objects evene1ec6nsvf :8gl,6v;k gh x9ob(.j qahtually slow down and stop. Explain.

If there were a great migrti q)p5oypg 18-9m jcration of people toward the Earth’s equator, how would j5m1ri)q-poy gc89pt this affect the length of the day?

Can the diver of Figvdt i3n*k8vv 3. 8–29 do a somersault without having any initial rotation whe t3nikv3 8dvv*n she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its cep0z9,ft/hhd l nter of mass iz,dp0/l9 fhths $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each ouxrgjbjh41+y 40 79ef10ninlier o1 di tstretched hand. If you suddenly drop the masses, will jig+y ii4frn0le01o 9xd1e7jr nhb14your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and vz4,2hecfsy9 o 83 ufathe other is solid. Describe an experiment tofu3o2esvz9h f ca8,4y determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points wt*;m kv 8bac(s npo*5fest. 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 ak*c a*mnt8 f;b5o( pvst the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. 2y p *aod6ok0rWhat happens if you walk toward the cend o oyr*p0k62ater?

A shortstop may leap into the air to cat( w4hnt jc* k/xt,giynz4:wb: ch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite directiw z,*y 44c(i jhg:tbxkt/:nnw on (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discu;l.pv;;u: ec.nc, s wakdw8hnss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways wl;vu8,cpdan:;.;w sc nh k.e the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (wm2op(g) l/d( o z7chma) 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 67 f:eqihi0if coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of ehf qf 6i7i:0ithe Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Mw ; ),flcty/s+ klbnh9oon, 380,000 km from Earth. The beam diverges at an ahy)+k l,/ns;9lwtfbcngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender royyth*c *wl98u tate 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 bladu*9lw*tc8h yyes slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anaqbg2+v+vmd id r52d4other child. If the ball makes 15.0 revolutiga mvqddd24+2v+rb 5 ions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutoj18u/x 7jg-z( mmge pions do the wheelsgpe/mjug7 zj- m (x8o1 make?    $rev$

  (a) A grinding wheel 0.35 m i8h5 ew2h9t( *)o,bhhe6cw7 bl l4k kmb v4qgnbn diameter rotates at 2500 rpm. Calculate its angulaw 9bohckwkmh4v 4 26h8bb t),*n5h( 7egbq lelr velocity 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 . 1ij:e7k m5mxthn1 bv/udo,uin 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the m7nmeo11j, 5.:/ ivhkub tud xcenter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of thet a r7xsg .s8/p*ovaj: Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speedns+g( zmt6ahe/0 dk7f 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.:yb my5xbw ,o*0 cm from the axis of rotation is to experience an acx :mbb,ywy 5*oceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uni h cjzkvk1 55;*j7zrbwformly about its center from 130 rpm to 280 rpm in 45k jvj*kw7 zz1c 5;hbr.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 radiusv,hz nq,ytx1+v or;cpl)m/0j m c6.cs8m ;ght $R_1$ 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 sppxm 7oz8 g ht*xg:94lracecraft put themselves into a slow rotation to distp*xg 7 ormlxgz8h 9t:4ribute 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 15,000 rpm i0g-)pk vjxk m)viws7; n 220 s. Through how many revolutions did it tur v)kj7;-0 gx)v mpsikwn in this time?    $rev$

  An automobile engine slows down from 4500 rpm (l:hb7krgy649dod pa 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)y47c5fm yrk ob6tk/f highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 c6 fr/7b5 tckf)yko my4omplete 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 diameekf ;u6 ).cc:wxcni p(ter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have travelicw xc.e)n;fu kcp :6(ed in this time?    m

  A cooling fan is turned off when it is running at 850rev/mini*7c k4 qyo 0x gh6t,)vmd9cac It turns 1500 revolutmchq4ox,kic)g69tdc yv *07aions before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as thr6bui 1jg xndu(yuqje*12+5m n .xlq7e car reduces its speed uniformly from 95k2gyn16d*+qj 5rx.ex u bj q(mu7ul1nim/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 k zx,x6wp 2ot*the car reduces its speed uniformly from 95km/h to 45km/h The tires have a wz2txp 6k*, oxdiameter 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 p)qo+hr(6i q(-uc shccuts all her weight on each pedal when climbing a hill. The pedals rotate in a ci-irq6cshhco(u )c + (qrcle 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 alfg6jzf:(h o- hw* zj/ door 74 cm wide. What is the magnitude of the torque if the force is ex wfl:hzzf-j(6 hgoj /*erted
(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 the4mf ;zg u mdxx+rhrto2:8x6,n fr:5pb axle of the wheel shown in Fig. 8–39. Assume that a friction torque ptr+x df m86b5:gz r4xnhu mo;2 ,f:xrof 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a 9f4deznik z7a6e x9o oh9 mmn(v,38xtmassless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of di,vt7k xf z8oa6zm o9949(mn3eh xnethe net torque on this system.

  The bolts on the cylind4 rrmf 1wq-t,.0zmvhner head of an engine require tightening to a torque of 38nm ,hz f1tq w4rmv.0r- $m \cdot N$ 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 +w.6t(ng wc5yo3x y 6 dcrs5nesphere of radius 0.648 m when the axis of rotation is through its cengoy rx6 y6d(5snc+t .enw w5c3ter.    $kg \cdot m^2$

  Calculate the moment o , l /m:bxvkk.5i rbixw6ma 2nsge4+f+f inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of tml2bi+iw rke af ,v5.kgx/smb x46:n+he hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotatt ybap;3us;gy58r/,f 4gx ulr ed in a horizontal circle of radius 1.2 mysp,b;gxu 834ulg5yf / arrt; . 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 whee bq6xq- w,eb,d* bq+zgl rotating at constant angular speed (Fig. 8–42). The frictiob,ed+q, -zbwxq g6*qbn force between her hands 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 oh2y64 ks,fve wc13ntzf the array of point objects shown in Fig. 8–43 ab6ek4,h yvncws12 fz 3tout
(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 whose total m9u(avm0kr sgm3 m)k3yass 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 cyli +jile+/zuo2k b; qgt7ndrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mas+gztok ;7i+uq/ jel2bs 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 r qdj00qnl3z t.eh .wq)pfncjc2 8* io0adius of 8.50 cm and a mass of 0.580 kg. Calculate 2 d o.)cn*0f8qqj0wtjp0l.qnz3e chi
(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 from40-w1nx((zq lbkcg pg2lxx yj*5fw z 76m nh.k 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 smywsdw(5l 0 vt.all hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opy0lw .vstwd( 5posite 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 aw0 (6t8(ncpwp xbw*d.7x mhda gf-1d gt 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torqueghf.pbx0 npdmddg7w*x8 (t 1w(6-aw c of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6- 5ur nqma6q 2n6ls :ww*mzzl+0kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which z hxwx2;t42it) mnxieg8bc+ : rotates about the elbow joint under the actio84 hwi;:t mz+gxbxet xin22c)n 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 in Fig. yf,6m q0)jqch, lxm r;8–4mlhf qrm j,;)q06cy,x6.
(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 consis4868lh7k pqlt j l*lu1d nz,apts 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 fr0 xg-w9 dsfhgmse,e,zxr*.m8 om rest within four full turns (revolutions) a e*,g.-msx,xmedzfh9rg s w0 8nd releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment or083qsyiq q2 yem4y3r f inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torqug9k8yst o;c 99fa u)q7swjl,q,ca1md e 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 wizycme8 s3/*mkeop a 47eri .w(thout slipping down a lancm *ep r /.3 oeimzekay4s87w(e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic e;bn )q)z eknhpen- ;2gnergy of the Earth with respect to the Sun as the sum of tkhpbzeenqn g 2;); n)-wo 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 of 1640 kg and a radius of 7.50 m. How qal;9wb/d-q imuch net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assuml-/w 9i qqdb;ae it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest arniw:yk ;vg*d2rd70 hnli( )qcy0 s 8bnd rolls without slipping0s k:biqr2d(y*0ygl ; ncihd w 7n)vr8 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 vertically on its tip. Itht5(-7w3 xa dbry utx* starts to fall and its lower end does not slip. What will be the speed of the* 3au7 drthxb(x-wyt5 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 r cnbv6*;unj+ d q;(sdltj-ca;otating on the end of a thin string in a circle of radius 1.10 m at an angu;d*;c-b dvjqnlct s(j6 +;nualar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindricrwd9;w b m( i3ody:it4al grinding wheel of radius 18 cm wtddbwo r34m( i9i;y :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 hhb0l/ ed y*es +nh(bj5is side, on a platform that is rotating at a rate of 1.3rev/s If he rai0bnehd el(*b yhs j+5/ses 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 ones ;yx9zi/bf h6 shown in Fig. 8–29) can reduce her moment of inertia by a fz h6xb;f9si y/actor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation ri1aykh*gc-/z ;c(y no ate from an initial rate of 1.0 rev every 2.0 s to a finalc ay;-/*ych g1znoki( 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 fr 1 on-iky . ;gm)mzjh/dz/ 48snbxgz:hequency 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,o/mi8xm 4-;g:)k./s jghzzn nz bdy1h 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 skao(+vh-3 sk:eh:cl+j gxql8suter 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 momentumq.xtba ewitf yhed*- z3u009o,m2z :v8.rmuh 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 inertia I is dropped onto an identiidp 0pp s+0sv.cal disk ro.dp00spp sv i+tating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atzm7jd-/y* rxq 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 stands at the center of a rotating mer1bq)g w 54l6bmu pts1g9vvc)t ry-go-round platformm)g16gvb wv9s pl4 t)q51butc of radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity ofqh6 0;u,sndo tbb r7z-c oo)g; 0.8quo bh-o 6z)t d on,c;b0g7r;s $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about;swe/ f tyk/8t 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 mf;tt 8ske//w yomentum, 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 exces9i+s -uykrs d0s 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 s/ eb1aj t6ll/7pvxs*$ 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 can rotate freely*tvh t/nmksdd047/e kj6wdw* without friction. The moment ofdvt*k76jm/0 sndek/h dw *4wt 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 thyer8t01b,ab+e lt ne )4dd y-ve edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia ofa10n ly- r ,ytbebt) e4vdde+8 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 rollsr +m-2xf phfu kfht 1o(, .z3wo8xh.mw on the ground with the end of the rope lying on the top edge of the spo.o(w ro,w-.+fff81 h2m 3xhkm phzxtuol. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the E /9y q o;o +;rmn5 q(m.iohc,cc/gyybmarth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (I(r5 oyomcgcch oyqi;q.m/,9 +by /m;nn the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist acceleratesc, h:/8 ick tugs-g0ge from 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 shape of a uniform cylinder ofv1 lo))en;vck radius 0.20 m acquires a rotational rate of from rest e)lc k1 )vo;nvover a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid c-r-k x+m0 nwp( a4jvwjnzei2 1ylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use consn + 14pra(2n0jwi emv-wx-zj kervation 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 whef rz r0k4io5g)h-p 8kael ($\omega_R$) 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 o1c+ sau. 5lnqlur Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momenaq5u1lsl + .cntum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stqd /8khscva1 7ored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, u/k 18sv 7hqdacses 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 al*t9oq l9rnln14vm/ 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 horimr/g /6k2ew ,cq 7yztezontal 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,olrka nhf u .0+wbo,hy:)b6c2p3u ah freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the r,:fuo 6)ouhh30np,b.2y wbl ca+rah kod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on t hm 74izq9*,ubja8qdy he floor, and we want to exert a horizontal force F at its axaq i9jub7h8ydzq*4 , mle 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 speu2 ed:rnz(uy4q5 t((okvb)m w ed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and4o5zvwq()mubntd2( (ekr u:y cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of len vrq rln* wgclg2)f+;8gth 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque 8w;lf+ v2rq *nlggrc) required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s bod/bqb8; bf efz*y 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 skater with outstretched arms, and with arms held tightly against her body*/bz;bf8b fe q.   

  You are designing a clutch assembly which consists of 8yq1b f;l cwga),vf7stwo cylindrical plates, of c 17qf8;swval) b,fgymass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rateih 8v5y6lash* -b;y dius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaviihv5-el8sbtyy ;*a h6ng the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do n1+-9mbg mfgspx5f (nz-er(a eot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions a 4),-obz1 o ppcsuiygmk8c*7 ss the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angul po gpy7b8oc-umsz, 1cs k)*4iar 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