A bicycle odometer (which measures distance traveled) is attacx u+awb 1bl 3:v7oeg:qot2k; ched near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-l 37uexq+:1oobc2 bvg t ;:akwinch wheels?

Suppose a disk rotates at constant angular vel+lo r8mmwmfbwzz , u /1 w+,cvt6zg5z /x0ywh5ocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either compone8,zb1zf m +hoxzmuv/ 5mw6zglw0+5yw w rt,/ cnt of linear acceleration change?

Could a nonrigid body be described by a si*2wpuz -ioi +bngle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger fo rh*dw mknk )jk(graa 8+.,3mwrce? 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 si ysvb0p-ji98m*yns5. 7say pot-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may90pypo ss5ab 8 *js my-ny7i.v help you to answer this.

A 21-speed bicycle has seven sprockets at the rv8(yro t6+kit kj-l9rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or t(ktov-i6yrlr8+ k 9ja large front sprocket? Why?

Mammals that depend on being able to run fv;fadlq; t4 3duc j)z0-do no8bu*:b bast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the b 0du;bdc*:u z4o dbv qblo ;n)ft38-ajasis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carr.zwv dz fjr(+pcf/ :m+y a long, narrow beam?

If the net force on a system is zero, is the net a5t1cue:n+ i mtorque also zero? If the net torque on a system 15ei+n:tm acu is zero, is the net force zero?

Two inclines have the same heighth6giyw 8w* ,pt but make different angles with the horizontal. The same swg*8p6h iyw,t teel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline..(pm rtsz p1v9yn;p )h One spy s9t nr.(v) 1hmpz;pphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start from5 it8sc u to3/wi-yi ::;ulct ht,ro8d rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? W 3ulcw:-85tti, dy:o8tcrt ; usi hio/hich has the greater rotational KE?

We claim that momentum and angular momenx* 0c c4x*a hlxaz8 t-ombc-f)uqg+;jtum are conserved. Yet most moving or rotating objects eventually slow down and stop. Expgh)4 xqlx atf+ c*cm;z*cx - au-o0b8jlain.

If there were a great migration of peoplei 7*rhoru n gpbv .51ga/n;- *eziun7p toward the Earth’s equator, how would this affect the lo i r*n7;pgnegz/-h iba7nr 51u*.pvuength of the day?

Can the diver of Fig. 8–29 do a somersault without having any* dwnl2e:lb 7b initial rotation when s2wlleb7db n*:he leaves the board?

The moment of inertia of a rotating solid disk about an axis through its cen.dn9ko7;me9oh3qp 3; es iqvfo7;lbiter q9; d l7ev3 smhfnobki;;.o i93p7eoqof 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 holding a 2-kg mass in each outstretw ahys5bd1d/l7y dc2tv 52oq0 dwb 5s2yqc0/dt dlyv 52o7h1ached hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howu:1/xh72lafp6 wl qowever, one is hollow and the l2w7pu/ o:qfw6 lah1xother is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal ouf( a7t4) oqub/aec2 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 ta27o tub/ uec4f(a)a qongential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely roi;m73r sj wsv/tating turntable. What happens if you walk tow /3mri7;svws jard the center?

A shortstop may leap into the air to catch a ball and throw it quimf7 4ke)d ou+nckly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs roo4 f)ken dm7+utate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular m+0n waah5r .i k4/lkbtob:x 6nomentum, discuss why a helicopter mnn.+wabkb6:x5/4l i art 0h koust have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles iig23 abtp6 rw(xlk )y, r,:xpbn 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. Calculatnh. qmuy7 1+bfgr xcqm 971vo*e, using the information inside the Front Cover, the angular diameters (in radians) of thmg7+u1fq9cxmvyn7 h 1 *ro.q be Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moh8rr.; vtxo 5eu npb+5on, 380,000 km from Earth. The beam diverges ato58 ;xb u+trn5 phvr.e an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turnedoy4qc s/ek,yeqn 17n . off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as th14.q eo , 7ycen/nqksye blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball0udv 2tvt,l0v;*xvr f makes 15.0 revolu0lv0d r,tv u*v 2;tvfxtions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.07ugkr f0i)k6 z km. How many revolutions do trfg6ki7k )u0z he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates aq7k1 dbapf z3 *8q 8rvpbcek,2t 2500 rpm. Calculate its angular velocity in b7vp,8af2*k 3d8cerq kp1bzq $rad/s$ $\omega$ =    $rad/sec$
(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 g5mem;(, ovmo/o wudv,kzx+oi0- j w*revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a c5( wmgk/vowjdm0xi mo+;u-o*v, ez o,hild 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 +qwax-on c :(tEarth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point .-;hqve15c 5vtx zau l
(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 cr3 in j69lnxp7entrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleratioxp6 9ri3nln7j n of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acce a sxid-*k7 yq *)l3zx8bwp(gelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Deter-)lx w eq*iag* kx(db783 zspymine
(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 radiuwkn.o sj5nn3u2qi)ck; 74mwg -n odo1s $R_1$ 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, astronautsc*nx pcj; d1f1 aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trindc* x;1 pjcf1p, 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 in 220 2w:p7q- pprjxd/,xu*e 3pcn qs. Through how many revolutions did ic qxj/d*wr7q, x :enp-3 pupp2t turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calc1i com05tqh0x h ml.cun*g2:f ulate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets iyyd7a6rmw p;,vq jb )oprd4 /,n a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete rejdro6; b pym w)qp4/,dy v7,ravolutions 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 6.5 sn2(whqxt 3; juaz sy1(j)cu p6. How far will a point on3( y;6j1hc p2nxuw( )aqzutsj the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turnsus,yyj.oav( 3sr;6w9c *g rky 1500 revolutions before it comes to a stopv*3k9yg6.,jc r yor wss(uya;.
(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 speed uniforwjot ;kl gm6l,k)q m2(mly from 95km/h to 45km/h The tires have a diameter6)g2k w;oqtmk (l,jlm 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 redtbdpic g. d:5q b:,ozg- uw/o,uces its speed uniformly from 95km/h to 45km/h The tires have a -oduqcbwog :gpib ,tzd5,/:. 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 bi+n6 cuwiflx jyv-*mgwx5 j /+(ke puts all her weight on each pedal when climbing a hill. fv6+w xnujgj (cim+lw*-y 5x/ 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 o-v1 o g9j,njzxn the end of a door 74 cm wide. What is the magnitude of the torque if the force is exerx,9jv1- nj zgoted
(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 re8wexg ,*)27wsg z njin Fig. 8–39. Assume that a friction torque o8*rxeg )zn,e wg2ws 7jf 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of masskw v8l:awe7z/ m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–407lwvwkze /a:8. Initially 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 en 6vl,+wsp qb5 3t0vlhvgine require tightening to a torque of 38 lvt5wsp h06l+vb,q3 v$m \cdot N$ 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 radiuvo(00m9-br:sqfp ce 9eve*a ms 0.648 m when the axis of rotatem ob cev9f0m0rv(pqe a9 :*s-ion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66viv of7toero ztdu79kz9* r +kr0,*;n.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The* 97 k 9tr en*zotioz7v+ ko;rfdr0,vu mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in o1.rhw o;at (ia horizontal circle of radius 1.2 m. Calcothrai1w( .; oulate
(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 ofe)+e-8pvhy j/ a oji8n 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 +yve h a/)ei8 jpf8jo-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 inertia2z lf l2/ 6.0rotcvoja of the array of point objects shown in Fig. 8–43 6ov zr0tj2 lc.2flo/ aabout
(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 who/3end:(pna / r m9jup5sroxw 5se total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite7ytxk( t;y49 mowb3f d spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellitedy3 tb4y(7;9f x mowkt 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 r2 3e.gv,: pdqduig b1aadius of 8.50 cm and a mass of 0.580 kg bdi.a:gv 2egduqp1,3. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it fjzi9 ;j7o knf:zu )rp,rom 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 tangenrxt 7ugg1zd6 ujg4wciy;/6;w tially on a small 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 opposite each other on the edge. Calculate the torque requi g1gx;gutyc6 6zw;ur d7j/ wi4red 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 eventually brought n16xpzj z -l.isu0 vo7m 7d(swuniformly to rest s10 zixmu7(7.-dnw o p6jvlszby 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 accelerateszp9o4em4qnvsz a7kprffu2 1 2+ 3 )ixv 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 throw(grv24 u/a0ky* u wovwn 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 accew g/*(y2rk4uwovu0a vlerated 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 condij -v 2md9paovs153gsidered a long thin rod, as shown in Fig. 8–46. - doaj p9v51vms2 3dgi
(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 of3u f sn jbywp.b85+ao) 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 accelernla7y,j gm9 +eates the hammer from rest within four full turns (revolutions) and 7,gal 9enym+jreleases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of is y xbj x7tm*io1142gjnertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 28028p dz*zbj-iy $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg -no 74b 4sucmsand radius 9.0 cm rolls without slipping down a lane at 3.3 bmoc4u-n4s s7 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic j ju59myj )u:dphuj,269b2a(bj5: omrguzgw energy of the Earth with respect to the Sun as the sum of two terms,g2 d:z(9jbm)b o9rjmguu, j 5j2pyaj6 h5wu:u
(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 16 ig1 sa+)v;h* 6zk8 dpz5mejtp40 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8)dghp*p zz6j 5+ vimka 1te;s8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg sta vt0+jg ; lddpo9.la6erts from rest and rolls without slipping down a 30.pdg tl6j9va o+;0el d.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 itfe- t0 5t(xzils tip. It starts to fall and its lower end does not slip. What will be the se ztxf-i0(l 5tpeed 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-kgrq1m(d- g3.wnptg ws.k +eb:n ball rotating on the end of a thin string in a circte3pgnm -:wwb s.(kdgq +.n 1rle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whed5* eclq*qlc9 xvyxzwzw/r 0-2g1 v3lel of rl31 gvw09cl/wvxrz5l qqz *yc x2-ed* adius 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 rotatak b z q9 ( y6ze8zfdx6zqs*u+*:*twuking at a rate of 1.3rev/s I qyz*as9ue86z:d*6 b*f x wktk(zzu+qf he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment0)1d2sd,g5kwhkp k 9 d7 mnnbh of inertia by a factorkk)07h2m pgbk ndd9dn,1wh 5s 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 rate from an ini*)c, jzwpdx99b)z-mnvs /wbx(*f+rk bemk 4 t tial rate of 1.0 rems, nm*rz vb) bx9p) * +xbw9z-tc/kw4fdjk(ev every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating ar)49kbt jg37stk0 rzv zound 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 jz4t39)s zbvg 0kk7rtas 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 skater spinnin 30 m3*d0 neoh61:banmlvrwlzp 75rnbg 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 t2 rv:n,q- 22b wp)-lhzoyk evmhe 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 droppef -xrd 1*idn k+hsjqz/4.ic5xf5nt (q m*cvp *d onto an identical disk sj n .i5r nc*v-m/x4xq*qdpfi c1t(d*5 hz kf+rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ax ptl:,h,a* h27nil rqt 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 sxg,it 29( r7tcdkk) gmtands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of i2igt)km k ( x97dgt,crnertia 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 d-hsu0lufa)6c y7iuq; cg,+z with an angular velocity of 0.8 fd,l hq0g ;-s+zccuuiu)6 7ya$rad/s$ 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 halfc7:f1s(6x 6uybd auao atx2 m9 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 wo(oxus2 ud c6bat16yama:f79xuld 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 wi1gf6 s hh3oz.25re xxbnds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass s +w05fl(vt; ;c0bozb6sbecf $ 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, thadf -ek4b5;x8m 4 b:lxkygpuh e8q4)hit can rotate freely without friction. The moment of inbbk:qhi 5x gel4uf ;e-ym8k8h4xd4 ) pertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m di gnj ,zun bea)(pi,)h6 *2bnptameter merry-go-round turntable that is mounted on friction p*jgut p )2(nb,,)b6n zaeihnless bearings and 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 or ,bl 1lv1vki/n the ground with the end of the rope lying on the top edge of the spool. A person grabs thl/l1b,viv1k r e 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 sar c) 2(g/)ud;swq qeluch that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to u2a/qw r d)sg)q( ;celits orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at j3 ozll3y su)4/cmq/la 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 tg)mm fej 95hoani( k13he shape 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 acceleration. Calculate the torque delivered by the momhf9 3mo(e g )ja1i5nktor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diske8.kp: 6 r6vd nxihta6s, 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 conservation of energy to calculate the linear speed of the yo-yo when it reaches td .t6xrepi:6vkh68 an he 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 yyeu:0q , lz9ris the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were r a pb2+q22fpkgotating at a speed of 1.0 revolution every 12 days. Ifpba2k qf+g2p2 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 aut ) h/c-+h.sdmbv8alvbomobile is for it to use energy stored in a heavy rotating -+hbc )l ashv.v bd/8mflywheel. 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 illustratet9frwi 1pt79g,*r mtr,ahh0ks 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 horizontal surface at speed v=96u juna /uwle1s7lic:yd+ 3 y5 stxpp.(m7ay3.3 $m/s$ 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 freelys7v9qx6 ,1uqfo(abl h (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 th,b vq1 ox9af sl6(quh7e 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. I*, gaaq/4bzcqmt3ja +t is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will clim q3a*q4 /m jcat+zabg,b 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 4*a jej2 gi 1z v-/odgr2nk3chroad is making a turn with a radius The forces acting on the z3 c*j a14vni/o2ejhkgd2g r-cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins whirqm0we32o wzroo75:3l souk( bling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rat5u ko3 7e( 0roq3wb:oowm2ls ze of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as t575 lm/fi8ry*a z/oa/hj )*huw ajkechin rods, making reasonable estimates for the dimensions. Then cal/r5) oc7j8 hia/jm5 f*lu ake*ayw h/zculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates,swc 318xnlw4c of mass 34sw1cwn l8xc $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 few tm.ss, w:);xjw3y 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 t,fj)t3 s:xww.ys; we mhe highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do nm-lq 7v ta49a5d5s zksot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniforzen)n5/rm ,is(yk 5mrmly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) Whr5 ) s(mznni/em, 5ykrat 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