A bicycle odometer (which measures distance traveled) ; o1(i-5up vscovp5/kqu1 jhpis attached near the wheel hub and is dp ;qivc o5 /jhvkop5s(1- 1puuesigned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a9juno5 sn h13v point on the rim have radial and/or tangential acceleration? If thvu h5n3s1n9joe 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 des:t4c vy pd5c*bcribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a g3zs;.x88ic )hav(k3r ymoa i mreater 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 your head than whdq:mlq r6;lro)ej8f ; +rqrn6:uj ub 2en your arms are stretched brr:rqd qqu)lj 2;6uo:rl6n8e;+ fj mout in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear md** owr(j er19) 7eg/2y+vnw oq rigbwheel 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 smalle1+ov9wj ry 7i* d r*mgor2ebng/)wq( front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast hav3zkc0/x g 6wvee slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is acg/0v ew63x kzdvantageous.

Why do tightrope walkers ,gap65rjc (x*c*-mgqtku8o d (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net .pk-ye+o jml( 3+ssyvtorqe pol-s++ky. vysm (j3ue on a system is zero, is the net force zero?

Two inclines have the same height but make different angles with t-r7/bdu hn ki.he horizontal. The same steel ball is rolled down each incline. On which hk -d.nbr/iu7incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling ( - 8 8g-kd.kzefot7/eia3fhclfrom rest) down an incline. One spld/h3 ak.iz-8o efte78kfg-chere 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 rado07 .on yh+ifx6nqb0j p )4cigius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? W4n6hfb.0 i ijxpnqy)cog+o0 7hich has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum acci:x q my:(x5nd angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Explam (yqcxc:x:5 iin.

If there were a great migration of people toward y ua. p:f6q-emwne5 d)the Earth’s equator, how would this affect the length of the day?)d q:ew 5ua-me. n6fyp

Can the diver of Fig. 8–29 do a somersv-ym/ 8kx2 n4 :vakbmt nb4b7bault without having any initial rotation when she v -8bm24bm7xytnv:kn 4a bb /kleaves the board?

The moment of inertia of uqs 2+7f9mfe yuv17i /4lanrsa rotating solid disk about an axis through its center oe19u f7yafir 4/q vssnu+7ml2f 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 hollwz 3brg30ir4ding a 2-kg mass in each outstretched hand. If you suddenly drop the massir03 rlwzbg 34es, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and ham (wfd)k; .5p:7lefmi 4hw vcjve the same mass. However, one is hollow and the other is solid. Describe an exp5p; (:mj7v l4hcw.f e )fwkmideriment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axl+c+*lfox gwt -j0uoow 56blp5b ,f) iue 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 l*t+ lpio uw xfw5,-0boogbj 5u)f6c+ the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on thew9wsqbu,1g( c edge of a large freely rotating turntable. What happens if wu1(ws c9,gb qyou walk toward the center?

A shortstop may leap into the air to catch a 3 j9ci,3ubrheball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposibj3 ichure93, te direction (Fig. 8–36). Explain.

On the basis of the law v*sr8 ,1dzayriv0y3 y of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the0 *vry ,8rdia3vzy1y s helicopter stable.

  Express the following angles in rad0.csi yk01qun ians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, using t0 :b) kahve;o6nm4m rnhe information inside the Frov6ahm kboen)mr 4n0: ;nt Cover, 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 from Earth. The beam diverges ax m:do+m.u0r it an oxiu+ m:.mrd 0angle $\theta$ (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 e* qv8 y4/zpi/ fy4zw:+aumjs rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the an a w q*y vy/:zu4i+8pe/m4jfzsgular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a leve /9dkm fj c9o9p8r(moxl floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diameter8 kf9(c9/xjd 9ommp ro?    m

  A bicycle with tires 68 cvcg-.ul k + p;f k:(q mcv;)xfe0)twwem in diameter travels 8.0 km. How many revolutions do thew0 e;e p.-;fcl)qw mfvc +)kx:tv k(ug wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate ijj*g zyg ulhw ;ld8z5253r dv3ts angular veloc8 2 ; 3h*dgwu3d rj5zgljvz5ylity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes)zn ckjk186-lvf6krz one complete revolution in 4.0 s (Fig. 8–38). (a) What is the z rkkcf kv16z)6-jln8linear 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 thm gajf 4-jf/9le Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sp, nv uh4-9+ )m br-s*pivdmqdaeed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a cenlm kstep:v0(bv,r3 )ctrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an accelerat vk,m:3)cbpstv re0(lion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel ackn:f7 f,)v esscelerates uniformly about its center from 130 rpm to 280 rpm f:esknf )s7 v,in 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 radius;h *n bmsy4owklpm, *5 $R_1$ 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 y ;,)b3tp(ie.j mcmq iaboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spc;tm3biq.j,i )p(ey m acecraft 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 15,0qc;fpm:ri c:un,1ma000 rpm in 220 s. Through how nm r u0c;c,mq: ap1:fimany revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500wr*y2susdo qrg( 9 y(- 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 the 8gzxtbxl8,lhrnw7 72g8yxd*3mh h5m stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 conzlx88hhymxdxw,mg tbl 5 8 73*hg72r mplete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 k7qrv9o /ro bk6k0; turpm in 6.5 s. How far will a point on the edge of the wheel have traveled in ot; ok q06k9/brr 7uvkthis time?    m

  A cooling fan is turned off when it is running at 850rev/min I-w1lr jmu-ct( t turns 1500 revolutions before it t-wj-r(1lm uc 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 car redu*1-d 6 -cbxdylwdzaj*h-lfh 7-j3h6c u gr9kn ces its speed uniformly from 95km/h to 45km/h The tires have d 6l3fkc- d1z-jnh-u xghdah rj*y7wc6- 9 bl*a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its spee+ghhtr(z2dd1y 2 nr bhsuvx f9 :d65g(d uniformly from 95km/h to 45km/h The tires havy udhfh2tvgbzr+ r d 9h: 5(1x(g2ns6de a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbing ae 6 gpm6/ibv5v hill. The pedals rotate ingm5vpv/e66b i 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 55dp3zjqzn /,aq((m s+z N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exespan3jmz,dqq ( / zz(+rted
(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 s -arp62dx9 wighown in Fig. 8–39. Assume that a friction torque of 0.2dp-rw9i 6xga4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to thk2 r 10awdourn3;cq)c83 l yff4.vfene ends of a massless 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 sys4nc v3o8 w1ffny frl)2c;qu 0er3 dk.atem.

  The bolts on the cylinder head of an engine require tightening to d 7os1pnx+ v8/lmz-w:vxmbt *a torque of 38o lvsmdbp71/wnz*tx8xv -m +: $m \cdot N$ 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 k dj+h52 jis,i-r xpre4svgos r.63:n the axis of rotation is through its cente s:+ 2si4vis5n6rho-g px, .3rkjdrjer.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diql47g.y wgy4g ameter. The rim and tire have a com44w7 glgy .gqybined mass of 1.25 kg. The 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 rot;wk 4 dgev.zr)ated in a horizontal circle of radius 1.2.4v)k dw g;erz 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 coepmrp e27re3 5nstant angular speed (Fig. 8e 52p7r 3pmere–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia ,35izs y,yjx*nwi c 2yof the array of point objects shown in Fig. 8–43 abwz, yi*3 yj2yn,5ics xout
(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 whose0vj7ov*t u:+cevdn/ g 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 the correcty l969 ia cr)(7as0m0er up-qb)zvrqt rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellitc9b m ar l6ryvtzr0p- 09qa7)su)( iqee has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50nrdfga*z;5: x cm and a mass of 0.5;g* :rfzax5d n80 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, accelerating it from rest to h )j0 w 89xtbtyyx* 3upr8jj*w3 $rev/s$ 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 cpv(b4c4l, d8qvt ( um5zvq,vsmall 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 t8c,(qv45uv pcv m 4q(,bzlvd of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventua2qyz8t7rc gp. mb:8oplly brought uniformly to rest by a frictional torc yp8 7r.gptz m:q28obque 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–v ab2rp nl:ngo69 rq:845 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown s cywaig71/ nc9olely 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 un7/ayci9w c1gn iformly 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 shown8 lkrp:+a*l.cpo w 0ap in Fig. 8–4rp+lp*k:.0 oacw p 8la6.
(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 two mass 15ar8 v4+ ce, hzzvsg24vz4(ivgld qles, $m_1$ 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 -rq+k idwn) i;hammer from rest within four full turns (revolutions) and releadqk+i wi);-rnses 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 inertia of j46vzzipl zd7qsmv p n 49c0o.70g c,d$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops jal., *xsr4fd a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping5c d (lqd*r5nr down a lane adcln* (55drr qt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of49nf3plap/ wv e44sg* cgr8sr two /ngwr 4 *s8rsp34v94pega lfc terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of us,zvm*mw r r0lnq4. nq0w4q 27.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assunr,z4 m*wm4vrwq.l0nqq 0us2 me it is a solid cylinder.    J

  A sphere of radius 20.0 cymkbg +/jpd6- lk6jb ,m and mass 1.80 kg starts from rest and rolls without slip,jb6kmp 6dklg+ /j-ybping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and its*n:7iiixo/y gpn l2 *i lower end does not slip. What will be the speed of ix2go: * /n7iypl* niithe 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 a thidm egs9w g yu.innt;38o-5fzj;pk-i* n string in a circle of radius 1.10 m at an angular8e *;ts-jnz3d.iwp of-9g5 ;i nykugm speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mom-/lh1+x7zxmgpxd zqhi-. s(x entum of a 2.8-kg uniform cylindrical grinding wheel of radius 18g+hl x/(dz six-m p7-h .1zxqx 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 axs),x:u v2ic jt his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed )ci : ,xs2xjuvof 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) c,5qydn*z09h it +ptful/2a r ,suxme/an 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 posi /,/e5+ rfpx nqshz0ymu* 2alt9t,duition, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can inh69 mj4 xube9q1 re8rucrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate ore jh 98r4xe1u9bm6uqf 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 rotating2/+ -pi aq;v a7mssn;w tvy /6nbjby2n around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The pott2i/ya yjmsb 2b- v7 tnv;n;nwaq6s/+per 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 sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure s*zf*;.2 apjbj5equymh vo7 m3kater 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 reu *7be +lykkb:w7h)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 cvk i2xcc2,td66,adg inertia I is dropped onto an identical disk rotating at an,x2 i,dadc66cvkc 2gt gular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns 9jacj ( 2ofrx4efi0(0b rthrr4hf:/kn( wgo, 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 rotatime*7-dv 7wk b)dfu 2vlyrylnat.ox*j*y)p, ,ng merry-go-round platform of radius 3.0 m and moment of d.fl)bovypl7yty2r,7d-*mvn , *e uxkwa)j *inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity o qz( /psx1ue7mf 0.8p71 u/s(qzm xe $rad/s$ 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 half ivdp8kyqz.+x 3x.n qc 2kr qma. 1v:;o 0dadp5ots 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 kvq +dpdz.xd;q2 c 5:nmaov3oxyp arkq 8. 10.the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excessj k7fwgxy;3smgl i c46z3s)1 i 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 o5:x tejsb. 1y$ 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 rotatx(zf afokhq)-;.e 4doe freely without friction. The momeneo) fdf ka.x;z4-qh(o t 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 dagmtwhc,d8 c6g u(:3 - 4gm3 :xzyikhxiameter merry-go-round turntable that is mounted on frictionless bearings and has a moment mgk :xg:a(utwix 3 6h8yc,4-h3gdzcmof inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of them7 oj99u:)htd 3/inpnxwji frff - s8; 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 pers-8op hdti/sjw9r) nxi7:f; mffnju 9 3on without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such th+uxv/gi+x xy 7at the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axi7vg+/ iy+xuxx s) 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 rate of5 obg4 pc.x;vo 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 rad6j e9dn-4 apvyius 0.20 m acquires a rotational rate of from rest over a 6.d pvy-46nae9 j0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mas:qw2fzel +q*f-d) 5nc5 h ofjls 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cyli 5hfd f:2fl-qzlwjo5 n*cq)+endrical 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-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 angulcjcmg5m -85mp ar 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 our7gy +tlse gwi,f z 68(d.6ivyx Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotatio(8itx+w zl6y 6s ,d7g .fegviyn 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 automobzxqwq f.3+ryc 0w 1ba5ile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car hacq+br w 5.3qy0w zf1axs 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 illustratvucl 3gg;44d ues 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 *fz*k2uxk ri1ms e37q(lqy4nat 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 lengi *:k*afu.5+ gr 68zwhmg ctjyth L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of *.hg zar8j5w+it fuy*:kc 6gmthe 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 verticalhhe -wh6e:fx0 )zb381a sne-pl.kmcb ly on the floor, and we want to exert a horizontal force F at its axepa hbe3hsf)hz:- bw 601cmkl8nx.e- le so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4rty311j0b- ba8c.cmifp 3avr .2m/s on a flat road is making a turn with a radius The forces acting on the cr.3 f j1ib-paaym0 1vtbc 8r3cyclist 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 sp0*6 ls03z;nzrcel5 wra b (;o1t b jtl51hbrfling of length 1.5 m and begins whirling the rock itb;h s* 6z0prl0 ; 3e1ol ca5n1tlr5brz(fbwjn 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 c;.xygx nfiu9jv+r; 8k ylinder and her arms as thin rods, making reasonable estimates for the dimensg.uf;xrykij+; nv98 x ions. 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 cylindri wvy1o99 7vgjical plates, of1v9y j7wg9v oi mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looi411yln c -ghaped rough track of Fig. 8–58. What is the minimua 1nc-hig1 ly4m value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, b0p((+eeoxue3i kjc 3s 4ka s-rut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the 0p s6 .frc rlqf2a hpac98g25lcar reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration2gf0rq86 rl fpachsa l5c29 p. 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