A bicycle odometer (which measures distance traveled)yhfere su/0qb)u -)v3 is attached near the wheel hub and is designuy-euh3 s)f 0r qbv/e)ed 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 a point on the rim hax8fltj/ j 84*y6 luim/9zia amo).ez wve radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point h/ m.aja yej9mlflx)u8*4z8/o6 zi tiwave 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 am c77zqgy1xh:)0 m a+8hvbfwc single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torquy6boru jx) 13ke 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 sis-vy m2ztnbs4f rhlx 8y:8di;e 08kr5 t-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help zn8el s fdysm05t8;vrx4b i2kh r-:8 yyou to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedalmua4 ds c 2lu:c 5ykm7d-wc(3ee scw, +hrdd7. cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gea y5cesducacku7re3dh,w 2 l- wm+ dsc :.7(md4r is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast hav* .goy6oz+ grve slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamzogyvr+. g*6oics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lo-q6fnub0u7ja h p 1on8ng, narrow beam?

If the net force on a system is zero, is the net torque also uk35hc2 tzlr)p ro 3)qzero? If the net torc3h)q 2p oztur53)krl que on a system is zero, is the net force zero?

Two inclines have the same height but make dif1k(gd w8bdg b 9f kqd1i(1kn7lferent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? kd9g(dfn7 q 1 dgi( bwk1l8kb1Explain.

Two solid spheres simultanenv7g 72/ z lyetg6qh**rkolcca) t0-bously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches c*-h) q6lk/tgv2on0 ztc7y 7rle*gbathe 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 cylind :qnsbb-- xs.ema-i t.er have the same radius and the same mass. They start 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? Wq -s-mbi.: .atsbenx- hich has the greater rotational KE?

We claim that momentum 1(fc1 2bsxa( tlsdt:nand angular momentum are conserved. Yet most moving or rotating objects 1nb(xs:cf da lt1s(2teventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equatz r1,cupk.jp8s jnd9fgv;/tf2 m j :/for, how would this affect the length of t ;pjjj. cdp ufkvn1fmrf8g:/t/29zs ,he day?

Can the diver of Fig. 8–29 do a somersault withm z*z9tc-znw* out having any initial rotation when she leavetm *9zz-nzc *ws the board?

The moment of inertia of a rotating so*vo: i hoqft2+h97ec olid disk about an axis through its center of masq:eo*htc9hf o 7+2ov is 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 eaq eb wiy:mf.1b0 vcr46ch outstretched hand. If you suddenly drop the masses, will your a4ebc.6fiv m1qwb 0ry :ngular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is(5gzt gaf//pin ogkhk,7b )6of, x;dy hollow and the other is solid. Describfd7(5i,x/ kzhkgn /o)6atpybo fg, g;e an experiment to determine which is which.

The angular velocity of a whe7po+7p jd5abssp6-j k el rotating on a horizontal axle points west. In what direction is the linear velocity of a point on6 pj7ko+j-db5 pssap 7 the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the ed;x y.p ,unpio4(jfyj *ge of a large freely rotating turntable. What happf y,4yup*jn;pox( j.i ens if you walk toward the center?

A shortstop may leaph-y:q 9lnrcqa (t- w7m into the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickn9(cy:tl 7r qm-w aq-hly 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 anguav s*tl* ger,4lar momentum, discuss why a helicopter must have more than*l,gtsae rv *4 one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in ranap:k+j k8 fzh, ww.m.ox j),5mue hy4dians: (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 ama n. vb2kaa++oozing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in ra b+va2no+ako .dians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 kkg37c/q; 0 1kccal 4fkyypgx+m from Earth. The beam diverges at ac3g/+c k kp;ckgx1qaf 7y0yl4n angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 8g2iwpz 0zg5 psl-j9fx4 s mg5rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What isz8 z g2 ix9g4wlms5ps0j-pgf 5 the angular acceleration 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 ball mak:o)iguc+brty di5g:* 7b:q hles 15.0 revolutiiby tuibc +lgr: )hd7::o* gq5ons, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How (7d gn)d liu)wm9:0k 1:kpyuhcact(jmany revolutions do the wheels m9n 0djt u(cl (:ukcd1ha :wyg7) mip)kake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotate b7c4;6el jenvs at 2500 rpm. Calculate its angular ve7 b4;vel6n jeclocity 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 revgaek8m05 t;z zjm:5 izolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seateg jzki:0a;m85zem 5z td 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 itdxrpi0/g o,o *s orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spc(q rfii4pm7;bjc +e(di ;/l keed 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 pa sra62z vcfmd-p)j: c-rticle 7.0 cm from the axis ofdp)z vcsj- 6a2 rmfc-: rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 13tbp7 tj vuv g(v*9ra6:0 rpm to 280 rpm in 4.0 s. v(:rjvgt6a 97pt u v*bDetermine
(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* 7o6 d5tm(hs jkjw2z2+m r jnezrxd59 $R_1$ 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 spacecraft pe0i l-j 3pg7 h5kebkr9ut 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-jig k7ekl 9r30bp5he 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 2+*d4chxb 90t shqkwhto 15,000 rpm in 220 s. Through how many revolutions did it turn in this time? sc0xwhb4 h *d2t9kh+q   $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. C 6da8uep(w4+s( * fu /g.kmuvwd;j2weqn2h su alculate
(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 stresa9l*fwgaa +7 wses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.9aag7fwla+* w0 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 unif 7rdxt;ep e*g+b;v oyi si*v/5ormly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in thisb5r;eo;+ gvvti/y p7*s*xide time?    m

  A cooling fan is turned off when it is runni;y nf5w;: 2 f7 0q4q )cm4yvffeippetwng at 850rev/min It turns 1500 revolutions before it comes to cf :i7nmfyeq;; )2fvfywe5pwt4 0q4 p 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 revolx z 1a-,xd-llautions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a ,lx-aa -zxl1ddiameter 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 6ctl).k+ e es-9d dxli5;g(hy x5 revolutions as the car reduces its speed uniformly from 95km/h to 45km+dl si get)c. -k;edy59xxhl(/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

  A 55-kg person riding a1iujo1b:n(y kh)ie*m bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle okmb)(y1o1h j iu ien*:f 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 thew8hhrjdk83o+svo u7 . magnitude of the torque if the h7dk v 8u3wrsh8.j+ooforce 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 about the axle of the wheel shown in/l)2yohmb8ohw 29 o+buq:nxz Fig. 8–39. Assume that a frictionmly)2x ozqbo9hou wh:nb28 +/ 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 massless1;ndewi13d ;c q.fxnq rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculatxf ;.dqwq; nc1i1dn3e e the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engi. r-hjwis) .e5 wl*ytone require tightening to a torque of 38 w- st j.elr*wyho.) i5$m \cdot N$ 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+w7qu9xvemh0v 2 x*8obcte+u the axis of rotation9w0hvbo*2mx +cv+e 7 xqu8t eu is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamee m3u.6wp2zkxe(l um+ter. The rim and timkmx2 w+ u(6lue.e3zpre have a combined 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 rotated inx tyv77uh8 lv.wb w.+y a horizontal circle of radius 1.2 mht 7bw+vx78 vw .lyuy.. 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 constaj:seu0.hd k q0yge,/ont angular speed (Fig. 8–42). The friction force between her hands and the cl0duj/hq 0s ,:ygoeke.ay 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 y.bms* o ,3g8yyfh3hpthe array of point objects shown in Fig. 8–43 ap,hy gfbysmohy8.3 3* bout
(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 atomsv ao41xq2er1 7b ge6yo 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 the correct rat3: i,9yejllew e, engineers fire four tangential rockets as shown in Fig. 8–44. If the satell 9,e3ilw:j eylite 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 cyzs 79l -fthgxk 24bd):y8sodblinder with a radius of 8.50 cm and a mass of 0.sl2-h ) 98d dof:4yk7gzt bsbx580 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, n1i gx ua40t1laccelerating 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 smaag9,/xf8wu*s5a r w o u6qnt8ill hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite eag 89a*wn/6i f 5atu,swxoqr 8uch 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 +75nq hvrzvbrfcy8 04e 7jgrzer-:6aoff and is eventually brought uniformly to rest by a frictional5-nq7ee +jyvzrr bvz874: f r6agh0cr 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 aci+h;nz*k7 ucve6ep 9 zcelerates 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 s gpd vj; nqz7 w8q+:krxi850vball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. T8dg8v+ki;r:0w7s5 j vp xnqzqhe 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 cano;an cdmm 64*n be considered a long thin rod, as shown in Fig. 8–46.nno6am m;4cd*
(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 twf0;hc x pn xbpty 4hbd648-k: ,dc1wmyo 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 fourg f 2,:qzn:) shlb.baj full turns (revolutions) and releases :aln.z s, b:bgj2h) qfit 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 inertiam d rvaovn3xs,8+px9n)3d:u j 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 de8ojhsvt u682+wx g3x vvelops 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 7ajojveu+f -2 .zbwc6without slipping down a la2 .j7uwao+vf-c j6be zne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to t;:ztbb-tze ;n v 7n*yshe Sun as the sum of twne 7: b t-n;sy;zt*vbzo 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 7.50 m. Ho)t,xu7cpv f1- )dso(f)zde- *ibhqg bw much net work is required to accelerate it from rest to a rotation ri)td-v zg bh qf1d,ce-o*()fp )bs x7uate 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 (pa xznb 0qt:8a,9khikg starts from rest and rolls without slipping downz(qb8inpkhx aa: 09,t 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 tiv*v+h-lz gddaq l..,lp. It starts to fall and its lower end does not slip. What will be the speed of.l,qvlzldhv ad +- .g* 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 akuirvx x / 5mk3w 0i)fv)bavp04wc(f8 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of i)wfv04a c wxvb( pm)vuif/0 k5xk83r 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentoxainomfpy )w.*q q089 :ftf2h;w k /oum of a 2.8-kg uniform cylindrical grinding wheel of radi90xh o )/nf io28w.*ffqmpo; ytq:wk aus 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 rotati)/kou igz egb))un-/af9 ve 2rng at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8 iku)) 2- ua)bnz/eegvfog /9r–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. a3jm3lem50bvl5m h0h44 sf js 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1. as 5 5m4j elvmhjh34sb0mf0l35 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+xejwo lx96c2yt0qx-j34d8 pvfsk/u her spin rotation rate from an initial rate of 1.0 rev4+uxf j-yk8v 20 dotsqp/j3e 9lx6cxw every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis througf wuia+o/8- ykh 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 th/-aiofkw +8u ye clay sticks to it?    $rev/s$

  (a) What is the angular momentum ofb 8i)i 3 xxv6ex5 q3y,h8kyocq 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 Eara1l 9 g+optyz8th
(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 droui.j;pjt5agp* 4vbz7 pped onto an identical disk rotating aa57*4j zjpptvgb.i ;u t angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.z7 * rs wg.:zt oa8d7/ufdbb4qt:6yyp4 $rev/s$ 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 thebw*/f/8/iu)whfw 4z 7g5f0hb ubf* ui;j yslh center of a rotating merry-go-round platform of radius 3.0 m and moment of fz8/4/y ufw0;w*u5l) bibhf/g b7 uj*hfiw hs 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 -ye 622g-bxav. cgimg cju72zfreely with an angular velocity of 0.2 xg zg27v.y - -uc6ijaeg2mcb8 $rad/s$ 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 a *b,pl*9gvj38f rswrtbout half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the *t9bs*8r 3,fvwl p gjrnearest 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 exceswm; ij rue967ts 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 oq46npu 5mz/r:;lj ot$ 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 6vz h g- kdy/:elyv,5cfreely without friction. The moment of inertia of the person plus thzvyeh6 cg dl-/k5: vy,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 edge of a 6.5-m diameter me,clc7ta0g8l 5 q3g mikrry-go-round turntabllmkc37a gq ,0c85ti gle that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the grouni lci.5gq1w,w4k;l9b rgua:td with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipprt;15 abgiu li,kw w9l.cg 4q:ing. What length of 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 f*8 69ysb v.iijdo 1okiaces 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, trea6kiiy8os d.j *boi9 v1t the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate ofzt5hd2b3/ j wh 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 mmo v9hj i:,y.;s 1jvo*f zqws9 acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torquoq.w9svyh*1 sf v j9o ij:m;z,e delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two d uz)f*q ig6hx:lvpg*tm6aga26h -8z solid cylindrical disks, 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 tha -f)6vlg8zzhtx6h g2:mup 6 a*dg* iqe 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 angular speed of the rear wheel 1dtc )fga t1 l:-k,ysd($\omega_R$) 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 wi2dnc5:zo.pss4jhd dcq+ 7, rtth 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 unifohc 5.d2p+tcsr o dnds7:z 4qj,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 automobilen8o2bfai7ozf3*phr z9dk;p/1kr hb 1 is for it to use energy stored in a heavy rotating flywheedf9hfk7ro*oh 1ki bp n82a zbr;z/ 1p3l. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates-)j,fuarp y0u 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 hor a8:-ji4gvs c vyuyz t(o0o8-wizontal 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 massxf d3s2k0xjgx -3ny/s M and length 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 sfnxyxgk3dsx/ 0 -32jthe 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 vertically85sn6mx f ,y5xcj 9ycq on the floor, and we want to exert a horizontac5n fx,5jyc ym698xqs l force F at its axle 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=4.2m/s on a flat road is mazkholx5ru 4j;, 8/rl wking a turn with a radius The forces acting on the cyclist and cycle are the normal fork ulhwz/x48,rlro5 j ;ce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 o9qn*jtze/9w(d) j hu1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5e9djj*)outh w 9 /qzn(.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 bodysa* (;(qrb b3z2 cxaoh as a solid cylinder and her arms as thin rods, makin hb2abaxs cz;*o(q3r (g reasonable 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 clk91)k g5u tcstxj/5yq utch assembly which consists of two cylindrical plates, of mass 1yqt gst9k5) ku5c xj/$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 r v//b hhes4vxxq7n7) dolls along the looped rough track of Fig. 8–58. What is the minenh b/7x)hv 7vxsd4 /qimum 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, but dl* bkhzc8ffy1 ;l t70co not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85*hzn.ah 3qr.d( fpo)m revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have qahr 3*onh (pz)f.dm .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