A bicycle odometer (which measures distance traveled)+c2,dq ecd o+u is attached near the wheel hub and is designed for 27-inch wheels. What happens if you uue,qc +d+cd2ose it on a bicycle with 24-inch wheels?

Suppose a disk rotates at con4xyna m* l/ rxm/d5jfk-cai/; stant angular velocity. 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 /j54 l-rxncyamm /dxaf;/*ik which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be4n) +gx9 seoue described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert .sy1 krob, h/ca greater torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind yourxy;r jn*kh od4* 9-4ips9lkdm head than when your arms are stretched out in front of you? A diagram may help you to answer thiss9;djdn h*kpxor9i ml4*4y-k .

A 21-speed bicycle has seven sprockets at the rear wheel and three at the px1naf 1) 8yargqja9;r ;e syh,edal cranks. In which gear is it hard1hfges;)xy,qy8a ;j ranr 9a1er 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 with fl k6oau8uob d:wdq- b9;esh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of ro- kbowb6a89d qod;:uutational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) cal:; pxlzditj4a+55yo rry a long, narrow beam?

If the net force on a system is.ok;4 b icb823ssbuza zero, is the net torque also zero? If the net torque on a system is zero, is acob8 isszkub ;b432.the net force zero?

Two inclines have the same height but make different angles with the wz 8y(yk mpdc2lpyps /)6 jop/7,ps)zhorizontal. The same steel ball is rolllpz8 s 2kw7s,c)/pm)p/zy 6yyd( jp oped 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) fkb-- olh- n6acp3 ap3down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of th- apop3 cl--h63akf bne 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 msq qs w5((..yrvth xs7xnskz ,bwt5 44ass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater sn74q, (s( bstwqvzw4.55y sr.htsxk xpeed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angul p;k ei,xi or:* v(axk(qo1m.car momentum are conserved. Yet most moving or rotating objects eventually sloo q*;1 k(xka(rmi.vp :ie c,oxw down and stop. Explain.

If there were a great migration of people toward the cc9c.yzueh) z2a.s ; aoou9t *Earth’s equator, how would thiy9.to*h2azc; u 9szcc ou ae).s affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation r +)yg+ij6sr pwhen she leaves the giry6psr ) +j+board?

The moment of inertia of a rotating solid disk about an axis through its cena -gdlp*g9cpk4nv6i uml572oter of gllga -u*i6 d57k9nmvp2o4p cmass 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 stk csh xbx 86md6gba0.(ool holding a 2-kg mass in each outstretched hk a8gxc(b.6b mx sh60dand. 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. However, one is l35fvyw/zn(/ s pvwo8hollow and the other vwnzwpf//(s8 3o 5lyvis solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on afye/40 w kkob; 3i2z9d7nl6v0wja wit 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 decreasinw4 ;tizk9w 0 f02i lk63jvdw/oye7a bng?

Suppose you are standing on the edge of a lar,)nno1.gzg8 p 7lwurfvx5dbua4 i+3 xge freely rotating turntable. What happens if you walk tou g grdwn3xub5 vlp+41 if,7) an.zx8oward the center?

A shortstop may leap into the air to catch a ball and throw it qui/3xx lc)h (k qi4b;gbyckly. As he throws the ball, the upper part of his body rotates. If you loc/;hqygkl ix3b4b )(xok quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss w;r1/w-b/jq7 3tnv ih) agqol nhy a helicopter must have more than one rotor (or propeller). Discuhqjnborl/173 ing )qv tw/;a-ss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 309*x2f5/y5kc y:h )qn.g xjzh or: cxco $^{\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 Earthublvuo2yg7yv / sg,e j(06-a9onm1 gm because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular 2o0l j eu7g6(-n bvy1g/m9ygmau o,vs diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moo dzx:6l (egph+n, 380,000 km from Earth. The beam diverges at anh6 gpelz+:x d( 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 themk ( 1 esa 757k8w8ytd(lppfbw motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angulp8tw8aedms ( yf1kw (b7kp57lar acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a leg/9jhy.lxhgkw,t 4-h vel floor 3.5 m to another child. If the ball makes 15.0 revolutionklwgh.h ty9 g/,jx- 4hs, what is its diameter?    m

  A bicycle with tires 68 cm in diameteri3h8sosnzp5ua3 g6mo/ 6ba (j travels 8.0 km. How many revolutions do the whe(h jna6azis3p o5m 8us3o/6g bels make?    $rev$

  (a) A grinding wheel 0.35 m in qa5ha w wijjv-a5q7,upb 6f(1 diameter rotates at 2500 rpm. Calculate its angular velocity ijuw5 w7qa-5ij ap avfbh,q1(6 n $rad/s$ $\omega$ =    $rad/sec$
(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 4.0pzia zx /3-vg 3ei35mi8 4ikpz s (Fig. 8–38). (a) What is the linear speed of a child3m8 ii/zi5 zappvi-x 3zgk e43 seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Sunqcv; l:i6 a0g os;(cfn    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a 1p,tm uayh:s - .vbph 1vj8q-ypoint
(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 7.0b eco8bjr d*h xm(3l23 cm from the axis of rotation is th dm3x3c*o(e8br2j blo experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheen , xn13pxdaqn-4i6j rl accelerates uniformly about its center from 130 rpm to 280 rpm i n6xr q ,ni-4ja1nx3dpn 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiu( s4 hak v8tsrmt 6nolzi+64q2s $R_1$ 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 Apo 6;mtmbt0zz r-9)x pdzllo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of theim)z tr60px zt-zdmb ;9r 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 in 22rdq,,4d ga9 ig0 s. Through how many revolutions did it turn ing4rad,i q ,dg9 this time?    $rev$

  An automobile engine slows down from 4500 rpm to 120)e h dhm;xz3 c8vcutw:e0 40s05dsjas0 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whx.g:5 j4h / q x-a4dr8dxjsg-zlo3vjhirling “human centrifuge,” which takes 1.0 min to turn through 20 co45 :j/8 g3vrao4z-d j jqg-ld x.hxhsxmplete 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 diametem lx::n;zd vm1v/h/rjr accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wher/j m nxm/hdlv:;:z1v el have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/ngvx-b63a4ac6r al c1min It turns 1500 revolutions aana43c16br- clgvx6before 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 the c;;khp(oq)vl*5ki cr y ei9s8b cu8n4mar reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 me89sh puvk;( c4)q*n rok5lym8b;ci i.
(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 )yunptxn- *6x 0dt:cquniformly from 95km/h to 45km/h The tires have a *t )0x q :6pycdtn-nuxdiameter 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 allva 2)crvbm8: g her weight on each pedal when climbing a hill. The pedals rotate in a circle of r cbv2mvga)8: radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. oown9hr;*7zrd *jfn9xf/k(3 3o q mqaWhat is the magnitude of the torque if the force is exe7woq3dj rknoqf; fmor**xz (nh9/9 a 3rted
(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 axleo)ymvz 2fww+ebl5fkp;q n0) s )kc8n)/g0sck of the wheel shown in Fig. 8–39. Assume that a f;/wkyl0nzek) vfs)pc)wk5g08cq+ 2sm )bn o friction 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 of a ma8 c;di ezw1v9xg vrxa :2q).h3w hk+xsssless 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 the net torque on this ;shwxzewx1v cxk +a .)h8:r dq9g2i3vsystem.

  The bolts on the cylinder head of an engine require tightening to a 2do j9p *9c:pty8roeu qi5si6y 2ol+y torque of 38oru9 i+j* y o5plo96 t:icyd 8qp22yse $m \cdot N$ 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 o*50 :2 2 6 st2yq,ipj5ktezheos lbum3aki/trf a 10.8-kg sphere of radius 0.648 m when the axis of rotation is throu e :kyt2i*6b5a u5 tsm2o0 kp2h/q rlzji3,setgh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7r)y5zg1m1;luw 1 q .o;j ov:/k wrlji(u. yauq cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (w/ q zjl1;. ) u;orj( 1qgykw .v1u5ma:iorulwyhy?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a gn x o6t35lte(2gj9epthin, light rod is rotated in a horizontal circle of radius 1.2 m.g5g63 jon lt 9p2x(tee Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant a.oppd t-16l/llo8 z vmya0;ypq*ko 9pngular speed (Fig. 8–42). The friction force betweloz.paomt-ol ;y 901qp /*yd8vppl k6en 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 o9u/o)2xyk;o / 1;ang m goefqlf inertia of the array of point objects shown in Fig. 8–43 mkn;/u xg2lagof;/o q)ey1o9 about
(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 mass isu7uo sun6x5. q6f+fxr3yf6 va $5.3 \times10^{ -26}$ 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, unifo8 cv4van7, r.r1kkuor rm cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satev784nu1.rv ca,kk o rrllite 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 cylinde yentp q7k, eeni9yw7e2*a/j8r with a radius of 8.50 cm and a mass of 0.5i ewy7 q9e,e tna/*pk7nejy 8280 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings60. 4snhjmizv a bat, accelerating 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 andlmc ,2x+ 2. 9pgm+xqtjtpen7h is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniformnq+2 2m.pjpchmxel+ gt9x,7 t 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 rpu:j +xvtg- xm:h2/b (jkc 2oulm is shut off and is eventually brought uniformly to rest by a fricti2gxumj kutolh/jx :( +v-c b:2onal 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 accel(e wscj1q2a d2wl3t(berates 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 bapta-vam/nf)6io1 )(e b2 7evz y+r fqill is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly frbqv +6iyevf ( rp1n2o/ma )ai-f)e7tzom 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 thi50nter zta-e +n rod, as shown in Fig. 8–46.t- e er5za+n0t
(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 t.mw:4yxmi s(qq 2qs5xwo 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 ajk.p.be4zk ; from rest within four full turns (revolutions) and releases it at a sp4.j.b ; zkeakpeed 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 inertxw 09ger7c - comdz6+wia 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 .sgn91g 0nqqrof 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down a la qk:: 0*cix6 n)uvp mwnhlo/l78 umo9mne at 3.wqhck 9n7 m)umo vli 6ump::0*xln 8o/3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of-)zv 0 en7k8wd,m4 n5d0n vvttethk+n the Earth with respect to the Sun as the sum of two terd,v+nenztn ) k 0-8vn75em4htwk0 d tvms,
(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 wz w:0bf*)h.o i0n/tlbvy gt0mass of 1640 kg and a radius of 7.50 m. How much net work is required to acceler tbfhi*v0/yl o: b0g0) wzwt.nate 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 kg5 edyb 7k8(r3/zmwmrs starts from rest and rolls without slipping /ds m k7m3 rrw(5bz8yedown 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. It starts to fa ag.hgb0 )cs/bv8v,wh ll and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conses)/h,bgha 8wg.b c0vvrvation of energy.]    $m/s$

  What is the angular momentumtq;c z (w pd2m68q(rym of a 0.210-kg ball rotating on the end of a thin string in a circle of radius mw(cp qr2(6mtyzq8d;1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of kq57)i 7;kvpdwi- li la 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rkv;ik ql7 l7-)wi5dip otating 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 a ratebs 9+5 zr1f.e; rdy)ug 7yzvjc of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.+ y rvd7jus.1cyzzg5;9) bref8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. m( axq,o3f,yb 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her ango(xbq,f3 ,am yular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an ay+jaqa- 8a8b initial rate of 1.0 rev every 2.0 s aa qja 8+-ya8bto a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertic u*n4vgaxie6iez2 yoo gx;vx8-7 01r lal 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 flate v7g 6v*unxa48yiog2-zroxi le 1x ;0 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 8r6yi/m8z: g*mjgvwwskater 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 the0a13jtmm7 vse 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 ofepfe-bx u/qanp8gl s./8-hhko v x3)/ inertia I is dropped onto an identical disk rotating at angular hgus-o8b8x)lfn .hav/- x /ep 3pq/ekspeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns 33n.zwreps2 b at 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 merryg*2srsy j ;/wlh6nx e;-go-round platform og elrsw;s/6x jy*; nh2f 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 ro f(fu v o30uajr/o,4uji4.eqb- jswk,tating freely with an angular velocity of 0.8(k-je34.s0b ,ajuo rf j/w4ufi v ,qou $rad/s$ 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 collapseko +8vfed kuo.yj9 6.jjka,.ys into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(roundk,kf9ov.ao68edk.j yuj y .j + 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 wi t 5lle(2jq,;iu)/aj5o ipxdynds 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 1 u4 nt )llpb5e.8qfts$ 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 free,l6 ms, z,bs:u t;horlly without friction. l:;l6 uh s,b o,,tsrzmThe moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round x6coe st;juygwytztm1a50/v ) 7 v7c;turntable that is mounted on frictionless bearings and has a moment ;gc z1u;6)tat m0xy5 w7c y7oetjsvv/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 te*11nqmm n )quhe ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the ) enm1qm*uq 1nspool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such th mbm+9kt//t4d lxc9mc t:sypsc :0 ;egat the same side always faces the Earth. Determine the ratit9 tc/m :d;mpm seby4/+0tclgk:9scx o 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 the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratol +ywurg ;5ebx 3cel/d b239ye of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 u)e 74 vbud q/s;0vccpm acquires a rotational rate of from rest over a 6.0-s interval at constant angular accec)seub /upd04 ;v7 qvcleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mj-nug1ojw sw (sl;c3mk*0nr (5 y f4fdass 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 fl4d (30*5ywjsc 1fnu; gsr(ojwnk-mit 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 speejk8:70p f 5 hrx2/ey1a utxgqcd 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 timeszpzr8psf 1a2w2 ,fa.z 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 o82s a2p,z z.pffrwz1 af 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 500 fv na,hfcaglgw .9a low-pollution automobile is for it to use energy stored in a heavy rotating f,galgvf0h w0 na5 .9cflywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates / y+gnaz5 kd3jdj6g)n 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 spea.v yu) lcuizl7 9h:+led 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 fr2 mv*br,ix fsxj5pb28eely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54,52*rbxiv8 jmx2bfs p. 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 rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radk8v7f.nueln 5n h- )rcius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against wr87kfn. n -h)5unvlc ehich it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling wic0r:18: mdbx qe yasd2th speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cb8s :2 1:rdaxe0 qcdymycle 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 ro4wubhha 7t47 u8fn t;cck into a sling of length 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 aft7b84huftcn;7 thwa4u er 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 th44s-0-f kex xzkgb5z b.j.lnl in rods, making reasonable estz.-x4 lkb.j5x n-lgs kb0z4 feimates 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.   

  You are designing a clutch assembly which consists of two cylinbak8/1v dg9px4 k)/yggo,fd adrical plates, of mas9yv481,/dkad)b/p k fgagogx s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped y 4iy/nytx:n .+bnhp7: vu3 ovrough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if itp37:binvun .+n tyx4: /ovhyy 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, bsrtjspib+:; e m i:;x7e3ngd-ut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car red:(su 2xo4:a bx xcqo.vuces its speed uniformly from 90km/h to 60km/h The tires have a diameter.( o o:xb:2vx xcu4saq of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s