A bicycle odometer (which mo.:6wydv2xyh j4fik d,)q 4 fxeasures distance traveled) is attached near the wheel hubf :w hy6x)xvifk4.oyd4 d2,qj and is designed 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 tejh-a6x 8/mpwhe rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have rad emha68pj w-/xial 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 thew7l+g 88qn x,)xyfdp y angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque thaunipc 1j q;ip,0l,*m/jq,dfdn 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 c*zfg*+y * /5y.0 cba8d.fq9+ rdxphkqdnu rkthan when your *f +r zrx8n0yydcu .qf+dk*bg/*9cq.k d5 pha arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the p nyb422 sq6ww ,enzt s+0o5ktvedal cranks. In which gear is it harder to pedal, a small rear sprocket or a larg0 sn zoqv ks4n2+w25tetw,b 6ye 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 5v +mxm1io q: yopu*w*flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotpi 1xmo:m5uq* o+ vy*wational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beau,*8e ib /riukm?

If the net force on a system is zero, is the net torque sq- +f) xkr6t6 +k8 kra-zvadhalso zero? If the net tor 8 kkfr+6)zxqs6thd+a -avk-rque on a system is zero, is the net force zero?

Two inclines have the same height *fu/gz fmhe9*but make different 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 greate9e**mfh g/uzf r? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. Oi :mydxz: ,4tyne sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater to m4tizy:x, y:dtal kinetic energy at the bottom?

A sphere and a cylinder have th(sb; : hadp.:yi+hwrr;i9dt he same radius and the same mass. They start from rest at the top of an incline. Whi9: hsi(rt:i +y d ;pb;wdrah.hch reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momenqywxz onl-o7 .;e 4er8qg7p )ptum are conserved. Yet most moving or rotating owl7o7 p -qye; )zgnx8e4.orp qbjects eventually slow down and stop. Explain.

If there were a great migration of pe0 ag mz508 :frix1z7ofnx n xc0h9dbz8ople toward the Earth’s equator, how would thi rnxaz:i7oh1 x fn5x8d09mgbzf 0zc 08s affect the length of the day?

Can the diver of Fig. 8–29 do a somersaubamyq6y6w8d9 rwvz -) lt without having any initial rotation when she leaves thewbzma -9r8qyywv)6 d 6 board?

The moment of inertia of a rotating solid disk about an w myffpo7m c(4v r22n/axis through its center ny2f(w2o c m/vfp47mr 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 stoo 4u-nby 5 o;avz.8tvw/c 2nqm1ic n3izl holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same.i1 8/in zumny2o;zcn-35 qcat4wb vv? Explain.

Two spheres look identi5( 8hnduxah-c cal and have the same mass. However, one is hollow and the other is solid. c8uh5an-d(h xDescribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. :)wutkwm r:o.:coq .eIn what direction qrt::e . wokcu) mo:w.is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large f4y:,a)cz e0l 9*oyj 7zszp rjwreely rotating turntable. What happens if you walk towarl 7y0*z c4ya)osz:ej9jpw ,rzd the center?

A shortstop may leap into the air to catch a ball and throw it quickkvu.mj )bff- q23uf6y ly. As he throws the ball, the upper part of his by.qu m v- bff63u)2fkjody rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, dis cjq4i54 ofon6lh2 ;v e2zs/pecuss why a helicopto zf 6 h4ci24l 2vn;q/e5jsepoer must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles ffp+qd2n/j0bin 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 amazing coincidence. y qhqpnlznrg1 k+tt bl,5 ,1-i+sdfq++-6apyCalculate, using the information inside the Front Cover, the angular diameters (in radians) of -n+f+al t,l d-ysnph ib q6rqgp,qt+1k +1zy5the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Ea(kc . g9xsynjusr4j8ulsie (idm((9 9rth. The beam diverges at an an uris. gsn ksj9eyi( j8u(9(d9m(cx l4gle $\theta$ (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 rotl,6: af1sa51vd -dh 1r+gau ;lfgcuate of 6500 rpm. When the motor is turned off during ope f;a:- 1g g1,a1u+s tl5radlhuo vfdc6ration, the blades slow 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 another child. If tha 7 +k.ruro0gde ball makes 15.0 revolutions, what is its r.o0 ua7+rgkd diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How m5gou/b1v+0q. s o rd/ngp)tcb any revolutions do the whecbpos1gu /n/vq )rd t+05 .bogels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter vb; k n3slqo4ofxd b4.ug-60 vrotates at 2500 rpm. Calculate its angulv-s vox4bn.3ub; kql64fdg 0oar 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.o y z8 thy35qh8rxkd5. 8–38). (a) What is the linear speed of a child seated 18 z5o3tyhxd5k 8yhq r..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 (aja iy88r:pr ;h) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spr4 hz+2coxlj +5(am 9l8 mlclceed 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 netc,zd,,:n ov/mx*m cm from the axis of rotation is to experience an av dnn:,et,cm m oz/,x*cceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accexww js:/jp:w:id mb+8lerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Detwb: dxj8mpw w:j:/+i sermine
(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 ymykqip p9pi0-9u -3y$R_1$ 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 put tac k/n vhgl0w*f j,:*dhemselves 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 thought of as a cylinder with a diameter of 8.5 m. Determi ak0nwf*:,vh*g/ cljdne
(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 acceleragk-s 66/toi uxtes uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in txoi/ 6- ug6tkshis time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5:/glr8k eupnd 39 +mplhyx8:a 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 jt4d7hcn6ki l)ets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachl76knci h4t)d ing 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-c io 06gt,ze gqj1ua9 to 360 rpm in 6.5 s. How far goi6 0,- jtgze9aq1 ucwill a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It tur ;:s1ky)d muyyns 1500 revolutions before it co :;)s duyyk1mymes 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 revo2vy-,ywkn0nq6u , tr:f;na8y lluo l8lutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of kywy;v8uln- uolf6alr ,80 :n,n2yq t0.80 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 revo83c 9aqc00 9j iq u-ztb4dxi(wdmc,iilutions as the car reduces its speed uniformly from 95km/h to 45km/h The t-(x4i98ci9 cbd cqqtiju00 iz ma 3,wdires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a rulaxjx e :g:t4 453swbike puts all her weight on each pedal when climbing a hill. xj4 u5xr34 selgawt: :The 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 je ;i tbz-7e5hdoor 74 cm wide. What is the magnitud -bt e57;jiezhe of the 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 about the axle of the wvp0 t7 cu355 odxkv1hyheel shown in Fig. 8–39. Assume that a fr kx3o71h0 c5dvtvyu5piction 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 ) eb(khu/n d5o2g* wxwri,7krrod which pivots as shown in Fig. 8–40. Initially the rod is held in the,xbek rhn5k(2wir* gu7d/) ow horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightet qivuw.xe4 /c*m). slning to a torque of 38c.xtsw*)/.mqu4ve il $m \cdot N$ 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 bo d n9i;)dgl9,slz -dthe axis )d,sl;zl9bdgndoi -9 of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm i n5;d b2+yqdhbp.8mrpn diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be;2r+b8 n .pmyqd 5bdhp ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end okyxzqc3v67jog,/1 2rjj ig9 if a thin, light rod is rotated in a horizontal circle of radig1z /39xi 2 oiq,7jvjrcg6kyjus 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 3.cn5vo- hgt 1t, 6tfom8ksvfa potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands afttsh 8o5kn1otc6gv v-f.m, 3 nd the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects fq6h jk 2t3zv*shown in Fig. 8–43 abou z*3jqt6fh2v kt
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose totacbf nodi(d (o+r9y)z,l 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 cylindrulrz(y/e-9 trical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. Iu /e9r(y-trlz f the satellite 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 cy t4)esz hd4h(65c cpzwlinder with a radius of 8.50 cm and a mass of 0.580 kg. cwzhs 46)hzd5 c4 t(peCalculate
(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 sr1s vwgptgp/w1 6,,2tfqb q(it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and is 0a*j)u;hlvpbp + hhm hi n.1y7oqj84zable to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictioh)n u*h bh01zhqv7mj p 4;8y+jpial.onal torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotatingf 5wgrv(4z*t :uis,tkw ;jh)r at 10,300 rpm is shut off and is eventually brought uniformlur s(gi zt;th4)k5,r w:vw f*jy to rest by a frictional 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 ball a4cxf kdy:rf4a 5hj3k8r3 2l jst 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 th3 oe2v)ke kqn31hj aw1e action of the forearm, which rotates aboute naj k3wh 21koe3q1)v the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, aswyf ne 7+:k/ o/b(xvrf shown in Fig. 8–46v n7fy( be:/owf+/xk r.
(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 kwoysczk x*/1p3u g.idc/) s3 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 hanmza4(esieym.,;, ( s2b e djm30yayummer from rest within four full turns (revolutions) and release; 3(s, de2eumn.am4 ayy,b0 eszymji( s 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 hasn1zvwk:p( qx0 a moment of inertia 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 o:w)yu u,; vap*vh9aik 8r/k taf 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 withoujm7jj,ckuqmec2b4b 1gb w:4 - t slipping down a lane at u24c q7-j:4jwcm mb1egj bbk,3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the k1v vem,a6/i pd40a)l tbdo7xEarth with respect to the Sun as the sum of two terov0)p k641vata/ d 7edlb,ximms,
(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 radiu6q-(sxyb8mjw s of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate86-sw(byqjm x 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 k v2 xhzhth-,)zic;c, vg starts from rest and rolls without sl-z; z,xith c,c 2h)vhvipping 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 jhfs7by+fdpp7tw)0 /eu1m ih q*( y ku5dl03x tip. It starts to fall and its lower end does not slip. What will be)0 iu*l7t+1q 0/ sdp(y73 kwhjyd ubxfep5hmf the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the enz/qgk3y ):fnrn ucs2 uk051msd of a thin string in a circle of radius 1.10 m at an angular sskzuynmngfc/ s1k:q )r32 0u5 peed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grin;nzhqn:r /7zf lkw:+s7 vi*a )3icceoding wheel of radius 18 cnf no7:*3r) zaiq:;hsi/ ecc7kz w+lvm 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 (+d btyv. ;eprplatform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal positip(y rb.e;td+ von, 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 re9uytvrf/ or;f bt *r,/duce her moment of inertia by a factor of about 3.5 when changing from the straight position to the rr o,bt/u/t ;rffv*y9 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 incre xf9t4/b i:3z rlh5ipcase her spin rotation rate from an initial rate of 1.0 rhx3 94tpilrb5cif: /z ev 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 p2v(9vj*so; 9pxt (joix5e kxaxis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg a2x p(j9sx(k*ip5 xe9; votjvo nd diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum oar qv )ws()9vrf 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 that +llh, 4iu5oe Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onb d;clf 3tvs8p;fifv cf :p7(1to an identical di cfps pivf7dfctf 1;;8b:lv3 (sk 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 x iae9ecj6160tg p8+xn 7tu l1nro 4cw2.4 $rev/s$ 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 rot ba)x/-oeh*x* .ujrwvating merry-go-round platform of radius 3u-whv.xb)*ao je* x /r.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angulhlx ne*yu);lk p-;h .r:kx8f rar velocity of 0.8 :8x.- *k; k; hrhlyxeu)pflnr $rad/s$ 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 pd)+ ;wk.yw jiabout half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass w j.d) p;+ikwycarries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds 4(z7hxzq6- hi 0g ceg+orgec* in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass u.6uxaz4kmo9pn; s 5q$ 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 without.t7jjkq3b3 uiyxj2.m frj tb373 2ijqm yx.uj.kiction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at n x3m6oiyw/4h5 r/rpj the edge of a 6.5-m diameter merry-go-round turntable that is mounted on o nx 6yrrj4m 3/pw5h/ifrictionless 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 ground with the 4ulwoe . :u4ciend 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 lengtoil. e :4uc4uwh of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the 3jay4zz.f 2r 7ox.kcd+ rtb q4Earth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Morda+7.4 yr zjc.k4 qfb3otx2z on as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest a)qs n b. 06d+(hqafeyyt a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a unifq(au g et3 m0wrdzt(i/)p:t/h orm 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 thudp3)0h z/ egt/q( mi: attr(we motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg /vhur/5 g)gik and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if ggh)k/v5/ riuit is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, h yy7j5h bx p6wkbdn/,4ow is the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were ri) u7pgu8ool 1 skhy83otating 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 the8)p7u 1k syghou38oil 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 lbbsp t 68)ji /gf,fod3ow-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 k8dot j6,sgffpi/ b3 )bg, 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 anb0hcdo12b tf v*i x 0 8kyqf,gntp:c:0 $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3.g4aq vj8ba69 s3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignor) or9l2ws(xwhc*9 i h68 ubilpe 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 the rod. A2 ploux wc)i6(hrh89l ws*bi9ssume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the fl ba.-erz5d awgwb,mhp 9a. b43jju/:yoor, and we want to exert a horizontal force F at its axle so that it will climb a step a9w -, bz:h4b/wm.udpe.5rgya3ba jajgainst which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flalu :cze yi* +it,+mi1 jhmhb*2t road is making a turn with a radius b:iyecjm z1l++ih tm h2,i**uThe forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into alax t5td/8uh8lv h 44j2n gfk; *kvos( sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after*/x4;ulhd8 vtg8k 24st5kj(vn lofha 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 h)*vq:/er .elymt7u cand her arms as thin rods, making reasonable estimates for the dimensions. Then calculate)h* ety7um:cl er./qv 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 cylindrical platedus v:h;;yti bwiuq*h42);j ps, of masspq)d :j2;yuvwbii4t ;u *shh ; $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fig. 8–5m)(:*, 2cd4/r znbrdjzed:-zs atgiygv4 ;y f8. What is the minczd(r*d4tjg24:zsv);a/f:gr ide,bny- myz imum 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 do not assumelvi9 r/ h f-fd0e.3f.z++g: yvdlbwky $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reducesv6dtqr2xck q ;y:t5n 8 its speed uniformly from 90km/h to 6y t2kq d8xtr;6:cvn5q0km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s