A bicycle odometer (which measures distance traveled) is attached near thee(y:tr, d4hc6 fvboe 2x.uzh7ta: sr6 wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-ih 6uz dfyb2teev :(sx4,a7 ro6hr.:ctnch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the z3wu zslo.6y q. dj.qk,02m ejrim have radial and/or tangential acceleral0s equzm3dk.yj q6j,.o w2z.tion? 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 component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocityidxm1z5 hlz;- $\omega$ Explain.

Can a small force ever exert a greater torque than a ew )a : f1tdtfin6k16 3ubvg/alarger 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 58wcff/*rdsf5d fd v) your hands behind your head than when your arms are stretched out in front of you? A fw5 cvrdd )f/f*ds5 f8diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at xyo 3/1xvz6ue 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 a large front sprocket? W3o yuze/x 1x6vhy?

Mammals that depend on being able to run fast have slender lower legs with fleo5 nk),f;l7n.aqbi* atqnl *i sh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why thi fbln nlk7*5qaa)io i;*t, n.qs distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrluy v x:b-0m ot7(+klsow beam?

If the net force on a system is ze0 8s yiafzr92wro, is the net torque also zero? If the net torque90w 8 arizs2fy on a system is zero, is the net force zero?

Two inclines have the same height but make diylq;oec .5 cyzhq;6d 1 saj8j:fferent angles with the horizontal. The same stee acyjs. e 6;1h8zqqclo yjd:5;l 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 (fro4kpy7) av*r7 zksbi *cm rest) down an incline. One baz c7s7ripvk* 4 )ky*sphere 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 starfa.0imh:ydnx-. re;3p 4ova s t from 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? Which has the greater rxs 3 da.n-oiv0yr;fp 4.m e:haotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving.e k h-1r2pm;l9r7ca ehre l/f or rotating objects eventually slow down and stop. Exp/l p1m29ehrk; e. errl-hf ca7lain.

If there were a great migr-wbbd-x1/*fd 4:g ksc ri lqxfqq 0p*:ation of people toward the Earth’s equator, how would tqdk14:0i-f:lbsxqq*x d* wrpgc/b f-his affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial )y 8stw9pb*9 heeny 2*0add ugrotation when she leaves the board?ue dhy)b0 *9sa9wye2d ptgn8 *

The moment of inertia of a rotating solid disk about an axis thsa f;z 3)peyf2 m-8swjrough its center of massj)maysw ef;s28 p-f3z 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 outspwh6c+ nrl,+g*rjwm iko7 25xtretched hand. If you suddenly drop the massesljh rn++xr57c 2i6 m*,gkpoww , will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identiccdt(h6 sw+fa+ al and have the same mass. However, one is hollow and the other is solid. Describe an experiment +ht( wsafc6d+to determine which is which.

The angular velocity of a wheel rotating on a horiz)5 e xlx(ycv h6u).i gev1vq9pontal 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 tangentia1hue5x)vpv6lc xv iq9e) .(gy l 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 laro22 t8d1vkmqsge freely rotating turntable. What happens if you walk toward the center?12d2s8mq kvto

A shortstop may leap into the air to catch a ball and throw it quickly.qx,2o -+ g7bzq(r,k7k+y9ac 2gi fploglum 2d As he throws the ball, the upper part of his body rotates. If you look quic-aqko+k, ,qpi lg7ly 2bgr9 xo2cg zdm2u(7+fkly 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 sh 92(4yy1*nfnxafh tmomentum, discuss why a helicopter must have h4a(* yyxh219n tfsfnmore than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a+l504rhp nopr) 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. Calcu8b rv5ukk g2(zp18pb u/jwd3ewm6gx/nv()fk late, using the information inside the Front Cb/3(gkre21wx8z(6 vun p b vupjkm 8 gf/)k5dwover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km frmqp)e88fb ppl1i6v w po( (8jjom Earth. The beam diverges at an a8lj8iqpw(fo bp)16pj(v 8e m pngle $\theta$ (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. zgq o8aydytw 1hrr/:bz*: :n7When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acy8 hz/::by g a7drr1*ozw:tqnceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the bbxs*kh) 5fz+6 b .xcuoall makes 15.0 revolutions, what is its diamsxc *f xu.+k56)bbzoh eter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. 0 xi de*6qjnz1/pvd;yHow many revolutions do the whep;x *jne6ddqi1/v z y0els make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotv5iz1vg gxe bsp0+ /y8ates at 2500 rpm. Calculate its angular velocezi8gx1y v5/0gs+ vbpity 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-roun9wj7ajbd+ ah9rk ly*p3 x2r*rd makes one complete revolution in 4.0 s (Fig. 8–38). (a) Wha9j39p bx+rd lr* wry2jh7a k*at is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the 4 rv4ce-2pi hb.xj1f )9iiprrEarth (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 poin-jbnz-iou d z +k3)fop0ltj*3t
(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 ro;, a(qfu ,g :i91wzjlhbf dr9ztate if a particle 7.0 cm from the axis of rotation is to experience an ah;dbjw9zf il:u gra f(19,,zqcceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abobzm9vw 8z iel*ufizzr 8 u1)p5,60ijdut its center from 130 rpm to 280 rpm inie d5v0zuf imi6j )89wp1r zlzz8ub*, 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 radiu1zl1oqxnml,z 61m +w (g ; tzqud6jo2rs $R_1$ 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 jjet*8,,pp1 l3chr g-it/ uioaboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy eveipluhrge*itj/o3t81c ,jp ,- nly. 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 unifb d:e xqz0 2rg0+sv8kv o9nv9pormly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time? +br o svvg0vdnk9p 0:eqz9 2x8   $rev$

  An automobile engine slows x+ 2zcii1xf2 /c3jwad 7kg/m fzq)4v xdown from 4500 rpm 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 th hb/(el9ywau 6e stresses of flying highspeed jets in a whirling “human centrifuge,” which t( ub9ewh /6yalakes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter acm(t ,etjv(z6 csp3u izk1uw20celerates uniformly from 240 rpm to 360 rpm in 6.5 s. How fai2c6u(,skevt t pjw 0zu3 mz(1r will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off w6 8k.4.dglguq jzuorn54uk n4hen it is running at 850rev/min It turns 1500 revolutions before it comes to a stodjk8u4z5n4kn6lu4u.q r og . gp.
(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 reducm9sbv fm yr12gz +*qup6)z7zkes its speed uniformly from 95km/y+k*)z7b 612 zm z 9svgrfqpmuh 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 rev)azj;orxc(/n klm46q) ;vzfc olutions as the car reduces its speed uniformly from 95km/h to 45km/h Thqr n c/al m;vjxcfk)6z(;z o)4e tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bis1puryr)7;u .ef x0qzke puts all her weight on each pedal when climbing a hill. The pedals r1rz pfsq7;ue0xr u)y.otate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm x a;2u(- d bfzj 43i.elvbf3j7 nx4cyzwide. What is the magnitude of the torque if the force ilx -3 dajjyzv2cxnf e;bu74i.3(fb4zs exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Figb7up)5, asly c e)*mriih91bj. 8–39. Assume that a friction torqu7,ebyp5 rub hcai)mj9 )1l*si e of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass t8+ us+hp0x ytm, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Iys+0x t+u t8hpnitially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tig1 l5.kiqtf,in htening to a torque of 38.if5ikq1lt n , $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when cflg*q v h,o*je +r(1cthe axis of rotation is throug,g*qrf v1l(e h+c* joch its center.    $kg \cdot m^2$

  Calculate the moment/l/2;r*lp rje:w fb lm.k ,tzu of inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. *t; plfzk :mrer /. 2ll/ubw,jThe 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 a;sjg o yc5l yk/k28a(p horizontal ciryc82;l gk(/5o pk yjsacle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular m pjwbw: y6-b7 kt:k*wspeed (Fig. 8–42). The friction force between her hands and m*7:ww:pk6tw byk- bj the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point h+-ozstwav3ak 3tj jsz054l1 objects shown in Fig. 8–43 t a +zjl0ovtwa 1sz4335khs-jabout
(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 ;kk 9fzuh9ze ,tr-he-oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the9slq( yn dk(k5 correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass ond (ysk5 lqk(9f 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass ocypkgk 1rc7*i (ue-0g f 0.57 yi*1gckg pecku( r0-80 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, acceler0-)emk e5c 2mmwjs*hdwvk;s ;ating 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-ron ,,xc -i3bv8 ormnzx5und and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictn-i vob ,8x z3cnrm5x,ional 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 eventugyvm4zq nd ozil:9 ox*t, 9y-*ally brought uniformly to rest by a frictional torque of gvymni,o*-4:z9tod9 * xz y lq1.2 $m \cdot N$ 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-kwt9rs-p ;+0nu)fwp hqg 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, whic/kp uyh5;zo 1yh rotates about the elbow joint under t 5 y/ok;uzpyh1he action 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 e1;mgcs 3)9bjy ckkg .Fig. 8–46.mg1gcc.;3 ky s)9 bejk
(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 co6/ep-m ww6 ivsnsists 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 from rest within four ful(m+avs ikjz 4zdyz -.sz,5vz*l turns (revolutions) and releases it*v-z4.5vm+(yza,zk szdjz si 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 lnda,v-m)n7.qk o0- a j1jojx.4 uogxinertia 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 2nd6/4ogtgck z4sc7u6 ypx3 t develops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass w1p-tem tf*8z 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.3 1-e*8pwttm zf$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to t6kn;p2fhm xt g)c;p gt2wd( w(he Sun as the sum of two t6c;)m p xk gp dtf2gnt2(wwh;(erms,
(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 muc5ec-erg. 5 wx)3o etzpl8k9jv8 5 kwxph net work is required to accelerate it from rest to a rotation rate of 1.00 reozxkjp9-v3e 5rcw p.8x8ge)5w5l tke volution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 w; + vye,gfj-c5 o l,pm:hu;escm and mass 1.80 kg starts from rest and rolls without slippingegcl5, ;o+;pfe,s :hyuj-vw m 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 balaqv( qczyox(; t)yzt n1tn.0)anced vertically on its tip. It starts to fall and its lower end does not slip. What will be th y)tqt (0co)nz(aztv ;q1.y nxe speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of anpg7f/jxc ;0+iludgq7z( z -hk :sf8n thin string in a circle of radius 1.10 m gz:+7sk i7h( xj8-nngc0zf;lupfdq/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 gri n87)krhg ma49 3flhavow j0x3nding wheel of radiusl)a 34hxra fkwjm0 v79 3on8gh 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 rot; tz9jzscn02b2ej ,gc ating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rog0t z92z j2jesb, ;ccntation 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 hgw7+m gk p:m9f;h 5 eybks6rv6er moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck posis g;69+gmykbh :vfrm7 5pk 6wetion. 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(s4e fqz(h76 iil2fzziq ebqu(hg +,; initial rate of 1.0 rev every 2.0 s t(qfb i zgq 7qz,h(zeu4(isi6 le+ ;h2fo 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 arouy, zf+3 xgf.kx3 y+lm **tby.yiodfj/nd a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay z*+d.jlob3 xf* m +y xky,t iyfg3.y/fsticks to it?    $rev/s$

  (a) What is the angular m 0*le sv;5qm4- rmkhyeomentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the+.oq2;w, y+ca l nbuip 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 imck-, nfsah1h6du ay-- qnv2 .s dropped onto an identical disk rotating at cskyn-u1.ha q,-nh v-2am fd6angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at uv9) v6yczqu hf ;vw(22.4 $rev/s$ 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 rotatinx16 znjc* wom.g merry-go-round platform of rjxo wnz.c 1m6*adius 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-r3sh dl*tkwp)k4.tb x8ound is rotating freely with an angular velocity of 0. l)p4stkk t. bwhxd*838 $rad/s$ 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, los;6, josyuso /bing 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 as/6osjy,ou ;bngular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involvvw.nf/h bg pwm *-w 7myk7;r3re winds 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 6 h*dau(:ai/rx /ah gl$ 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 platszep 0/l+a dxxu6- s5oform, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus thxxpedas+6zlos -/50u e 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 edge8dxysl;z4vrf*i1i 8-xy k 9za of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and h rysiz;d f -x89 yxl1avkiz*84as a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the2m x7d5 hoh;rm 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 th ;dmhm 7xo2r5he 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 sucvqh5zu gm,m92zw wg /(h that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat t u vwh/5zg,q mw(mzg29he Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratej)kj)bf:4u r+px..c /gema)wdoq2 nz 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 o 329ifnq w,lnxa0 uvi-f 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. C,f2l3- wni9nu0ia vxqalculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diskswrzslxl05h ieq0 x1 ;9, 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 the end of its 1.0-m- lrw0 i95z0l;xqxe1h slong string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, hoafljp8h tnd)b(i7r7 4 w is 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 witfj(t -bqwn -.h0 m/ r-*backrjh 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 sta(n0 bk mj-q.bch/t-wf r ra*-jr 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 ahimq x: ,yo8v:ath h9* nbn z:x:1y1dxutomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “:xh,dy 9bzqi xvyhxhn:on1:m 1: at*8spinup.”
(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 illustralfa2c xb0 u4l5tes 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 su7blqzlei;zvtfa 6tq4 csf-/v.4*s 7z rface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we h n n yz5zek, wzw)b/(,d8hok3ignore 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 3wkoz8/nybh,n,zd hew z k)( 5acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on th4h brff9+ p*i8gcc 0ene floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55 n*i80 c rfchgp+4e9bf). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat roa qcdfyvq;vc a56a6di t1un6.,d is making a turn with a radius The forces56. q;id6aq1 ,fuvnca6y t cvd acting on the 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 all2u 8 nky+nf0 sling of length 1.5 m and begins whirling thl8lnnfk+ yu20e 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 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 thin rods, mak+)uhoptqv8 lv +h iim (78yelbawv+1 0ing reasonablep0mqv7vl (tuv+++8lob1a i 8heiy)hw 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.   

  You are designing a clutch assembly which consists of two cylindricalpmp e,ut5 .2c*ouapbquw- j 83 plates, of massp.t c23 ,up* um-8bo qjweau5p $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 lja:5+ kou5)wn o(,t;gl*i tryv zr0rk ooped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the)au+ot 5ortrvl gk*: 5kynj,w0 ri(z; marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but .mv;f f 9y j)q8c(178v xx7l gzpivw6btdnu l:do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its scrmjm ey,,;j 89 sg*rapeed uniformly from 90km/h to 60km/h The tir mg,mjr *r9 8sayj;ce,es have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s