A bicycle odometer (which measures distance traif 62mwq -ai(tveled) is attached near the wheel hub a taw (im2-f6qind 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 ange cd cl.*+e1ykular velocity. Does a point on the rim have radial and/or tangent dc lk1+*.cyeeial acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value z g5sy lqmr;ve+y wsm(.z51r)of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? 3zzj7qioz r4)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 y;6qg0imbz3v6vp 3c oto do a sit-up with your hands behind your head than when viyo 6bv zg3q3;m6 c0pyour 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 j.(8weq/ndfx+(hal i qx),vu and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprock( 8qulvi na, q.)dhjw+(x ef/xet? 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 /kw :5y,pei emslender lower legs with flesh and muscle concentey5: pkmei,/ wrated high, close to the 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 long,qx0*ipe pvv *qp1n-c; narrow beam?

If the net force on a s7 lgc 96(sde a(t5e1 k xcanaes/u+rf/ystem is zero, is the net torque also zero? If the net torque on dk/e(( e/c5fr+ salscu na9 txa1e76ga system is zero, is the net force zero?

Two inclines have the same height but make different angles with the hrn6,qo kpxl(m it. 8:i/) +pikx1s fdyorizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the (k.ik +m)f/sdi6qry t8,opnp:i 1lxx bottom be greater? Explain.

Two solid spheres simultanei qxdchk *kf4)( /2bmpv, l8v2ahthy:ously 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 greater speed there?*v t,2h:ql/(acf) i4p2 h 8myhkvkxbd Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the s6. upsgznc4f1 ame 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? Which has the greater rotational KEzpnu1cs4fg6 . ?

We claim that momentum and angular momentum are conserved. Yet most movin qm54ux (p4 *4)b*vk zrnxuk2d4p ynihg or rotating objects eventually slow down and stopz4)42* *h54nkuxp rivd (nxmyu4kp q b. Explain.

If there were a great migration of people toward thgfk4 0yp1 dtnsgs88u/ fdu( r0e Earth’s equator, how would this affect the length of tyugdg(sp0k8/r s nt0du18f 4f he day?

Can the diver of Fig. 8–29 do a somersault without haviu dj(g )s7e4fqng any initial rotation when je7ud (g 4fsq)she leaves the board?

The moment of inertia of a rotating solinmjc 0 xq)l1ek6uk):xgs5qv ,d disk about an axis through its center )kg x, nek1mjx5l)q06cs:q uvof 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 2w -wthf5d i8u4-kg mass in each outstretched hand. If you suddenly drop 5-t4uhdwfiw8 the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hol,yq0wl k. m5kx5*vgh slow and the other is solid. Describe an experiment to de0lgwv ,q5xmkyks h*5 .termine which is which.

The angular velocity of a wheel r:c *( q t1x tmk2lfew-lre4;zdotating 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 top of the wheel. Is the angular spt(:lek1;x-r c4emdqz*w t2lfeed increasing or decreasing?

Suppose you are standing on the edgeb2xt /15f d1n 1ngk3xe*npxy 0eeq*cv of a large freely rotating turntable. What happens if you walk toward the 3x*nb5* xexvee/dt0g n y1n2k qcf1p1center?

A shortstop may leap into the air to catch a ball and throw it se1m4mp1. 1q32s adg u:laamxquickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fim1xuse42s1gq 1m. adal p3:am g. 8–36). Explain.

On the basis of the law of conservationlk 4b d 6zv5fxd8do,n2 of angular momentum, discuss why a helicopter must have more than one rotor (or pd2v k d5zxbnf6,4d8o lropeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians:g*i+ *i0fwbx:py:o1u 8x llda (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 amazi5tagyc yk;q71 r 6+i2lvwn 8kxng coincidence. Calculate, using the information inside the Front Cover, the angular dwv 7n5a2yg+ty k x6k1;lrcqi8iameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beamksz( qmc 6g6e3 diverges at ane3 6(sqmz kg6c 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 rata xyttatvl d/+q vh:ves+c+k;)v)/+ me of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What ic x /at kvsl/ a)v;eh+t+:qvv+ +ytm)ds the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a leve72bc9 ;kqz52 n yluauf4fxvr-l floor 3.5 m to another child. If the ball makes 15.0 revolutions, yf;n 9 rfkul4z v2caxu q7-2b5what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How mfdo7x-d gezby+w ; rbau 9q-k2ecz0)c10/z ahany revolutions do the wh7c+--9 ed0cx b;kdr1 aa eufgzhz z)wyq/b02oeels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate 0 fi.av njy//qhp 6y5wits angular velocit.yi 5 np /ywqvf6ja/0hy 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)04qy,nes og): *w toxr. (a) What is the linear speed of a chr:4 0oqyxgo,e ts)wn*ild 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)g :k.qavi2id40py)vf t3; wwi) jxp-m in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe/,*lw8gqlqx55 e (auix hwbm1ed 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 do b1ki3u) 9a t9zf5gj)mft j37.0 cm from the axis of rotation is to expe9i31 ogjkfj53ttfz)audm)b 9 rience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm8:e -j uaij8aq to 280 rpm in 4.0 s. Determineaju 8-i:q8ej a
(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 radiu7.e6;qhpv6n +rji)iz 4z)jt7s xxdscs $R_1$ 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 spacecrq7/ ,b*llem tekyy8 -saft 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 d ylmeys8q*el-, 7/tkburing a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 1583 c/nxnx sux5jl: wa/,000 rpm in 220 s. Through how many revocnx8x:/xw lj3s/5au nlutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcuu7g)r hk.jj4 f**tt folate
(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 d uidjv dg rceko)6tlbu3a;/3 *t0(y5jets in a whirling “human centrifuge,” which takes 1.0 min to turn through j(/0t53 dtu b v6)cgio; ula*dd3rkey20 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 tobtq9kmk ;60+y f/v2z ap1vbrzq54 awe 360 rpm in 6.5 s. How far will zwfmbr 4k1/ek v bv9ya06a2qq 5p+ tz;a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is repgk:7:3k j7fhrr; gbunning at 850rev/min It turns 1500 revolutions befop 7;gr:kgfr:b 3j7he kre it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speye 7u9) f*a)cctei(e hed uniformly fromaee*tc 9c7f ei) h()uy 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 speqt )k/i 4.g.r u azh9:tlbp:zwed uniformly from 95km/hg r./abt: 4 u):ztqlzi9.wkhp 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

  A 55-kg person riding a bike puts all he)m 96f 9(8cjch9u/cbhq esh sgr weight on each pedal when climbing a hill. The pedals rotate in a circle of radg(8 f6/ hc9 mujsh9ceqbsc)9hius 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 aqks/cykr22v- da +93wzq5ynwide. What is the magnitude of the torque if the force is exertesa yw3c /5q n9d2k-z+yq2ravkd
(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 jq/rhr(iob xi0 xw*b// 2jvx,of the wheel shown in Fig. 8–39. Assume that a friction torque of 0 birrxx,qj2j//0h/(i o v*b xw.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 rog*dl-9q+d)+ wht -vvvm/a q epd which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal pos q dvhv/tp-)9a+dqle-v+w* m gition and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of 7 c/ka/uw huayq o*3xb8gdw4/ an engine require tightening to a torque of 38/o h7d4yu a38/wc* b wkxguaq/ $m \cdot N$ 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 10.8lfm k 01w)pe(r-kg sphere of radius 0.648 m when the axis of rotation is th )1rpfewl0mk(rough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in dibr4, skv:b uv-5eo6wa(caaq .ameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ica -r :av6bwevb (5s4uqka .o,gnored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizon/7o51lmtn he powh52w tal circle of radius 1.2 m. Calculamo17 o 5lwthe2 h5p/nwte
(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’xck:nb z q;fb elo 63u e/-v8ddpfo;sewx(( (5s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N ; z65pl(d/nbdxox 83co(kfe;sweue -v (f :q btotal.
(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 arrs)9: zycge 7hf(8 o cd.rgunc;:v8 hox5zyjm* ay of point objects shown in Fig. 8–43 ab:98osc hrzegxdy58y) .gu jof(c*:nmhz vc; 7out
(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 consist 3j w1rt96kptys of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correctm rs4 ks8bkcy 7vu77i7f 0z4sr rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady fobm7r7zf 4 s7sis7r k8vy40cukrce of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radtc+a*nai rs 250g/oqt+b(8ew gl: pdhius of 8.50 cm and a mass of 0.580 kg. Calculabq5 s 2tai+w0d(nre*g lc8pht:+goa /te
(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, acceleratingz df3x;-fg w*c) th1 mb1fk+jj 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 merv s0(xd.k-,bjp 3 qc4v,tga yory-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 each other on the edge. Calculate the torque required to produce the accek .ogc vy0j a,d34,vqp( sb-xtleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventxf 1-iv 2q qlr:9uui.yually brought uniformly to rest by a frictional torque of 1x :1r9u.liqyv-uq2 if.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 accews j.bm+4s:q5lusa a *lerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown 03d zw a,4x. hub6k*2txphjn csolely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball. nah k z,hct 2upb3jx6d*0x4w 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 bla:6dqytzc xqv1,k;tzhb 2rb8 5 de can be considered a long thin rod, as shown in Fig. 8–46.b8bx;z:,5z 2rt1cvdt k q qh6y
(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 conlnu)p by3l: 9dsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer ft//l 7diz,3b) e*m0 gspmjqsj rom rest within four full turns (revolutions) ,gimslz)3 jme*dqs 0 /b/pj 7tand releases 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 momenq3j//i hy fdwt.v7mn 7t 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 develohn7l-zf w d9r /jo75hlps 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 w 57jkgoal(4dv ithout slipping down a lane at 3.3 5 v4ajog d7l(k$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to 6l hmmu;0(7 qw n qcd*jd+dq0nthe Sun as the sum of twolu*0w6mmj+c ; qq0dn(h qd7dn 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 o)q- dmb rdtfh3 .ox54a mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a3mro4 ofd.xd) hb-qt5 solid cylinder.    J

  A sphere of radius 20) sqj1oux c38 2hj(jfy)bh4jt.0 cm and mass 1.80 kg starts from rest and rolls without slipping down a 30.0 xyj uhj)1tf(h 2j3bc)4j8 osq$^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall anz9y e3iq;w(n td 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 9zye wqnit3;( conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the en.d+cdv p ,)ywr(w8yh qj2zcn .d of a thin string in a circle of radius 1.10 m at an(r. hqp n8ccwy .v, dyw+zd2)j angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grrf kk(uusl+em9l6en9u8u i ; g1r 7cx8inding wheel of radius 18 cm luiu rx(r7+96k;ckuls9ne8e18 mfg uwhen 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 pns9ezct1n/v2*.uknp 8 5 k 8thix .ywelatform that is rotating at a rate of 1.3rev/s If he raises hin n zushx2.tp8k*etvw5 iy8c /19e.kns arms to a horizontal position, 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 red8t3 bllp k4uw6uce her moment of inertia by a factor of about 3.5 when chlt63uw8 4pb klanging 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 int:3*b s3 j/. sbsvmoe+e:ddpyu4n)jpcrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s*ndby:v/. )mo4judpsesjp3 b3 : se+t 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 :pw5f9 5 k6asgpkalf-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 ofl65ps5:a pwfak9 f-kg clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figucpx2 *)kiqk(csn 3d e,re 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 tivb/;p4wj ube06* nm of the 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 drop qy.x jlz2*ze/ped onto an identical disk rotating at angular spe/ .zqyjzxle*2ed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns dr2z bz4kpdkw)95ld .f tp8x8 at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-roul 4px43wd) m69rzf hkxnd platform of radius 3.0 m and moment of inertia 949mhkpwr 63xdx4 l f)z20 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocmky9.ixk ez,d /qf;o(ity of 0.8 dox q,(e .y9z /kfmk;i$rad/s$ 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 dw xc0kac3r+,x j,s*x6o362s2agzrm 8an xtv xsarf, losing 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 rotation r36rx *g,xkm s o3ssaj2ta06xacrc+v n2 ,x8xzate 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 excesspp70n;dkqn/ ved 0x5: kzt+ xy 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 x 7b7iji8daf)$ 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 withoutuw 1ngb:8vfq+ friction. The moment vqu+ 1g w:8nfbof inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diamet.;zk4 ,ilv3xf an5t iter merry-go-round turntable that is mounted on frictio4n,i zaxfv i t;l5.k3tnless 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 groufqbe 3qox-- :fnd 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 behin-3 obq :qex-ffd the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the 97.e zuvy96r 2kp/,thum xx 7(izr i5leji:lkEarth 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 Moon7v : mk2er xel.p (79izz ,y9xi6j5kulh/u rit as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest;p*c g*t u.pbn at 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 (pc1lzf7 foo u x;w47pthe 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. Calculat;( oux1wpl7f4 po c7zfe the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made osehn .w(.192ikiqtq jg6yo+w; ddm, o f two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 o gnq w(ij.kh,diyqew;+ 1 to6msd2 9.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 it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is thei-jgg wj,r ;kcwe3og:3 fn-a: 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 rotatmk ns :x2 l,:2u iebkcd+j:1zwing at a speed of 1.0 revolution every 12k+x 2l d: e:i1mbw2usk, zjnc: 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 uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-dros 42c je70,edmv0kpollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cy0moe cd 4je r0sd,7v2klindrical 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 illustra8dr, t kcsy8ex bu0va*un,9,vtes 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 horizonub9jlo p he903tal 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 pivot freely (i.e., we ignore frizoom(,kpx /) mction) 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 rmm,o k oz)/(pxod. 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 szj5* 3ivag. gmtanding vertically on the floor, and we want to exert a horizontal force F at its axle53.jgg*mavz i 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 rc.+ mtbev5-t=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 for5c+mr -vt.t bece $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 lengt 9y ema*n)wkf;h 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating k y)*; af9menwit from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solz0a;ofw j /-tuid cylinder and her arms as thin rods, making -ja ; zoft/0wureasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates, f(vs -*scllo +ofo lfcs-l *+v(s mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and ra86b:1c (2s0 r qhyuynfrjj3 dodius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that thej(rcyuhd s0 6 :yj8n2 fb3q1or 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 as pa2i+o;ngcp:- l( 7usbnu c+ysume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the7b psry vt63b2 uy*9sj car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. ybs pju39r2s7y6*bvt (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