A bicycle odometer (fl4+l,* or ng:l 3svkc dhu;3lhlr)6zwhich measures distance traveled) is attached near the wheel hub and is designed for dvhu:;nl+ osr l3rl3 zk)4chf6 ,*lgl27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant anguli: qf k2ilto(8xm0zo:ar velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, doeoifki:0 8o2(txm q zl:s 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 velocxoibhg-i w80) rr3qy: ity $\omega$ Explain.

Can a small force ev7zwzh0 :u;rdami05v /ebkd9 ler exert a 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 igevds,vbm*smd6sr cp -gk+.;)c 7 a6 hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer thiscmgg6 svd;k)sv+bs76d ,r .a*pc eim-.

A 21-speed bicycle has seven sprocbcr(fyn 5i li+d-h 37ckets at the rear wheel and three at the pedal crancifir nh(l b73y+c5 -dks. 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? Why?

Mammals that depend on being able to run fast have slender lower legs with 6v v5oc+wr- rvflesh and muscle concentrated high, close to t rv65vvco+w -rhe body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a 1:(xt - s3ccej8 lybdolong, narrow beam?

If the net force on a system is zero, is the net torque also zero? Ifb0-yf/df; w pa the net torque on a systa0 ;b-f yp/wfdem is zero, is the net force zero?

Two inclines have the same height but 6m3qkskg-cm 0 k r;v3rmake different angles with the horizontal. The same steel ball is rolled down each incline. On which im3mv g rk kq0-;k6sr3cncline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simus dze+w0f ur8c1bu(xxq*a++iltaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greatex8+su*crd (xzqufw1+ei a0+ br 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 ,/vq1 l aztsu(bb9z b6start 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 16bav(tq b9/l,u zsb zenergy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are consers e ikam,*4+0dfk9fjgg 7g8y uved. Yet most moving or rotating objects eventuall, s98g* f edjuyfamgk40 g7k+iy slow down and stop. Explain.

If there were a great migration om1l 2ro c6t.xuvb -tx1ftj5 y-f people toward the Earth’s equator, how would this affect the leng1cy6t bxu521tmf- or-xv lj.t th of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation8r64eh8 v1jrbujo u /t when she leavesrjout8 b86j/ u4v hr1e the board?

The moment of inertia of a rotating solid disk about an axis through its cuk,oh)a cqb/ s i74*hdenter diho/scqk4 )a,7*hu b of mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a, sn vu;p8(5 /txstjsc 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity, 8xt us /tsp5(s;cnjv increase, decrease, or stay the same? Explain.

Two spheres look identical and haw+ex b qti2 vv;6wx4w 1/de9j 6ekjt/qve the same mass. However, one is hollow and the other is solid. t6d 2 qv4ww 9e6qkb; x1 xei+jvtjwe//Describe an experiment to determine which is which.

The angular velocity of a wheel rotati+h ujh7z)s dp)ng on a 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 t +zd7h p))jushop of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a larco hz/qwy i1:cfo 91/bqps)d: ge freely rotating turntable. What ha woyqs/1cqhi opfcd)1 9b /::zppens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it h. -a,qfkom+h/ lemk54do o( ,rnyqj1quickly. 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 /,14eom5qq+jr(o-kf anh,kyd.hl m o in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angn5 2z4i4 j sj,cglp*tqular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss onei 42 lnq4ztcjsj gp*5, or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in8r/e3603y.vu1 eyguud cpddm radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an a u/o;f,j2 ekf i+5bputmazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the/f,bu pouti 5fkej;+2 Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at thgd0g + pt-n.g7ewxw m9e Moon, 380,000 km from Earth. The beam diverges at angg+getxw - 09wp .nm7d 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 6500dfu5:yg 8 u*un rpm. When the motor is turned off during operation, the blades :gnuf5* yduu 8slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anll3-o9paca7dao b6i2 other child. If the ball makes 15d l72oa3-baco9 lpai6.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions.d h zw / p;axy7/y)rds-lb:lv do the wheels make? s ;d/). rll/y z7pwda:xb-vhy   $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500qzlhc6*-n v-9h nc*twfx k1x 6+deui; rpm. Calculate its angulx*+ix*qz9k vudlfh 6t-1c;we6 n- cnhar velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38) w495 pk b -+j5g/eynwxj4nlth. (a) What is the linear spebn ejlt/pw 5j-hky45+ n94wxg ed 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 Earth (a) in its orbit around the Sun *dw.wim/cx oe r:p r8*   $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speer eo:dp(r.3f8gvkirx 07ri6.hx+vts d of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the fot 3 xd1b/ *5wx/lafzaxis of rotation is to lf 1/5x3tzxb/d w*aofexperience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm t , /wfcl, j7 ,8ixsn obl;y9jygv/n;wudn5vg 1o 28 nfd5g,gy1icujvb;x ;wyjn/7l /9lv 8 os,nw,0 rpm 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 radius2(cbj d g+4ok.ebt,xosn0;rx $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spaceiiw /dvh x;s+-craft 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 spacecraft can be t x;/dvihsi -w+hought 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 univ5w+f o tmb6(kformly from rest to 15,000 rpm in 220 s. Through how(o k5b 6fv+tmw many revolutions did it turn in this time?    $rev$

  An automobile engine slows downc(. -0vja7; ; cdhi*bwdwpchn 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 the stresses of flying highspeed jeh7ykqfpkg5)f- ui 1r0ts in a whirling “human centrifuge,” whicr )y qgh51 7ikff0-pkuh takes 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 accelerates uniformly from 240 rpm to 360 1 -,yxacje jp m*0kf,mrpm in 6.5 s. How far will a point on the edge of the wheel have trkfm x- m0acj,,pe* jy1aveled in this time?    m

  A cooling fan is turned off when meuc.q. x 0i7+x oaye:it is running at 850rev/min It turns 1500 revolutions before it comes to a stop. xcu:a.x0ye e m+o7.qi
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly :u ekjcp,: q6wfrom:w:ej cup,kq 6 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed ujpc.1nr.wjlon58 z9d niformly from 95km/h to 45km/h The tires han n9cw1 o j8l5r.djp.zve 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 pedaf*(8j s .uu3vt5xmkhzru-8i e l when climbing a hill. Thezek-8* 5f(jsixt83um urh.vu pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is; i2xde0-oth u the magnitude of 2x ;0hi eud-tothe torque if the force is 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 abou(6jlwv)emk)bw0d gz4* qe mc*t the axle of the wheel shown in Fig. 8–39. Assume m*m( c)4g*0w 6bwv eql)kez jdthat a 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 massless x ne6/p. zb;z6z.orrnrod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then reeznb;po. xz . /rrz6n6leased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require)kyu wq8i ue3, tightening to a torque of 38e3, yu8kwiu)q $m \cdot N$ 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 sp+ov 4 r63abd0.648 m when the axis of rotation is through ao 6+34vpsbrdits center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diametey)v(45 o09 dtuhrxy5 bg(tauw r. The rim and tire have a combined mass of 1.25 kg. The mass of the hub((trtu5x4gu5vyhyda b)9ow 0 can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thizw8g.29x vk9fth od,ss jio /q0i+o2kn, light rod is rotated in a horizontal circ0v s+ 8d/9gsxo,jioi k zqt2 fh9ko.2wle 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’x)sy/ fte0s5 ;yd8w1sjosxb66icbw 7s wheel rotating at constant angular speed (Fig. 8–4xyb 6 6jseic/ 8 d;1 sfw)7wsxst5ob0y2). 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 of the aruybz3y.o.4 2wtmsg1cf v4vxg;a;ze -i ot 9z4ray of point objects shown in Fig. 8–43 ab cbo tt;z .sx912g aw3.y v4uzzi4vy4fg-;emoout
(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 whos9orf 1t qef- )w6fi 8b,fzl:qxe 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 c, i7o9pv50yul r*ynl yym zt3(qh m+*norrect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the sa9n ( yh nl0*i vpty5qml+uyo7myz,r3 *tellite 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,+u;dl8 *qw5ka9mg/hntmo onoa 3eg( a radius of 8.50 cm and a mass of 0.580 kg. Calculate ,g9g 3o ;* /oqlmw nemh(tonad8u+a5k
(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 bsn68hf nhr86yat, 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-roun7dgozdidp y; dt:mmftgb1 t* 6i,+f8,d 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 e6t;df,gg yp8+*b tz7tf d1m: d iomdi,dge. 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 10tn/ s.7p )mtqe,300 rpm is shut off and is eventually brought uniformly to rest by a frictioqmt7 .p/n s)tenal 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 accelerates a 3.6-kg bal7n;/n gjubd(( 9no cjgl at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which e02qub/bhv 7)ubql ) nrotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniuq7hqb )/2unb0v) bleformly 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 r be3/y6u7xpnc* njj3 -6fm ioyod, as shown in Fig. 8–4c3-n7j yyj3pnuf/mxe 6 i6b* o6.
(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 masseslfj -zr q+n++nhjig)9, $m_1$ 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 thhyshte y(- v 10qi(1-g s*ebjhe hammer from rest within four full turns (revolutions) and relee(*iveshyt1y bh 1-0-( hgs jqases 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 inert pjv9 wey59n+ria 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 oflr69/zqchi 2:bbcx l + 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 ;myedc0 5 twr6 t5lml0rolls without slipping down a lane ate;6055m lt0w mtcydr l 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth bx)fr h:9mmf5r..la o with respect to the Sun as the sum of two terms lab)m xorf9hr5m:.f.,
(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 o8scri)usu95 zd8, fh zf 1640 kg and a radius of 7.50 m. How much net work isz8fss)9zd58 u uh,icr required to accelerate 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 kg starts from rest abxid7:c o 11j3wq xc/gv1zw*xnd rolls without slipx:wdc b3x1v7/g1q wzjo1i *cxping 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 l(ntl 55j6-oecf5yottip. It starts to fall 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 conservatiooej5l56 ( ftn-lc5 yton of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating p889 /y ekfaci;/wygu0yv (gzon the end of a thin string in a circle of radius 1.10 m atc/vfyz y8u98y; /ka0ggpwe i( an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum oz4yx7ljd /pn n2uzi. :f a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm whdnyn xz4u 2:l7.j pz/ien 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 rotatin :uqavu ih:.z;g at a rate of 1.3rev/s If he raises his arms to a horizontal posizqi:a;u.: hvu tion, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce 0yzfd:n .u +tfher moment of inertia by a u0. ff:zd y+ntfactor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1. euop a2s6l v.3c0zd:y0 rev every 232ypdseu6oalz c. 0:v.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a ve.f 1k)xlrer (ryj0hpq*m: 8b zrtical 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 thy 0zxm *:r(hr fejqpl)r8b.k1 e clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinn.+wfw+jgm8 bac+ld b9 ing 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 Ed9 r9yu-v 5b14bmzk wyhf6,ksarth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is d)( ufzru 5xvqlx4o9 4c+ to8riropped onto an identical disk49xuz85r4o(ut o xlfr )iq+c v rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 kihv1x9k:ef dk-m:l9k1s 6v m$rev/s$ 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 t, hrrjidx)8 9qhe center of a rotating merry-go-round platform of radius 3.0 m and moment of qr r9jxd)8hi, 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-ro; s8 2 6ufb5ovcyl( xfeot:*xzund is rotating freely with an angular velocity of 0.8t;6lscz(u2*o8x ff:v x5b eyo $rad/s$ 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, losin(0 s hinrgkt)3(y /gpb:haonl) 5csd 5g 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 rngh:i n3 gys( tr/5shop0)ckb5()dlaotation 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 excess o wevh(ura90iw/f7/ti f 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 3ta m05o69 xuqffqo kjq0j77b$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely withouteo2: .meo,w rn9u2sli friction. The moment of inertia of the person plus t,nw 9ore 2s:o.im2elu he 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 zkrdw3r4hu -cs bql1rog,(0- a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a1 h-4kwqgo- l0b3cs(rrzdur , 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 the ground with the *icqwk(y ,4o*.p -kxvhb dav0end 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 o op*-ha4ykk q,d0(w*v vxb.cif rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Dp sou7 8(s6dy:ou mqi1etermine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbio8i7 y:ou sum6s p1(qdtal 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 of 2ga. r2rv m -6va5umgaqn:if41 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 t hmnkm npkh5: fvu*/),he shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by k:khnhu)*m m5/ fv,pn the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and dfs d-),5srlu0ehu c9 g 8b0eaaii31pfiameter 0.075 m, joined by a (concentric0dgiif5u)e-a89sr, f10c p laus3ebh ) 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-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 angularx+ nh.b kjtiw3 xkqi :27ddv:4 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, buku)g3kec :fcgar5 g0,p i;et6 t with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a unifo3 0)krg cea5:p tkc i6efugg;,rm sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy pvu( +7-r k39e51ujbleuf2o,, hsoxtb)yytr 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 flywheel9uhoyrjx35utryvp(fub+ k ,l, )1b- s7toee2 “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 illustrat g c h**oj6 d0a1d4.svux-dqcres 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 rolli no9)xcy:e7vqp 8gko*ng on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivqqaabhw o8 dm ucw5d46+ ff..)ot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizon.cohfm.+8b wu6)a4 a5fddwq qtally 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 radius R. It is standing vertj we8/m29k qed*l:ma8r zuw*h0bp:vq ically on the 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). The step has height w0kpu 8jmee / mbq*8l ad*:zh 9wq2vr:h, where h

  A bicyclist traveling wit:x1* 3g pic ,qht+ttnvh speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the normal f1tcth3ig *x:,vpq t +norce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length y ;ys* x-vqhs*1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a ratey-vqh*s y *x;s 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 thiw.-l + bg:zou:qvn1bzn rods, making reasonable estimates for the dimensions. Then calculate the ratio o+ zq:bzob-1un:wlv .gf the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly whichv s2: c8igjo*o consists of two cylindrical plates, of mass 8jci: go*v2os $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rough track of Fpy,1w4bwf.pc.0r0uh l o(dgw ig. 8–58. What is the minimu.g ,hw1.clof (b wu0yrpdp 04wm 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, bt4+g+z el/k/sw6wq i1 nf +mjput do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car ra3jkjaal3bnfaoqjl6-n:4o 3 p4 8 w;g educes its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What 3ajf8w3a 6:bga ;olqpjn-43o4kaljnwas 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