A bicycle odometer (which measures distance traveled) is att wk3omydll)4zv f+g14ached near the wheel hub and is df yzllv) domg34w14+kesigned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at 1 lx,1)nrz:7 g5f9zmy nf hjnpconstant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity incre1nyf gzmhxnrf,5lj7:z1 9np)ases 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 describ2 sm hikldh9d1gi4z81ta8x3k) h g0cq ed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger 8 einsk29)d2: xna) 8eyvedlv 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 difficanvu/u 2uo m09ult to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A/9u n2muoa vu0 diagram may help you to answer this.

A 21-speed bicycle has seven jj,7 w( aw5mqrsprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to p(r ,jw7am5wq jedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on b2s4.6 cncg 28 qc6ylvpaq4 gdteing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this dipc2ca486d6tgsyqlq2 vc4n. gstribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrojy r8x(r 0+;z gshcx)c24dpd yw beam?

If the net force on a syse27c5o. ypvuz x 0y5xmtem is zero, is the net torque also zero? If the net torque on a system is zero, is the net force zpx5 yy7u5eovcxm 2 .z0ero?

Two inclines have the2.rd f6 g/ -w:rcw6butw g4ubr same height 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 b/tdg46r6 :buw-f b2.cr ru wgwe greater? Explain.

Two solid spheres simultaneously start rolling (ju3ej sq*j h8jb(( tnax :7sd4from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline firstjejj(* snu4ahs83b tdq(x:j 7? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius anavyj9uw *jj:iu,y q;0d 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 kinetic9jwjuj* y0 :a;y,i vuq energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yetl,vbj lo00 w;s most moving or rotating objects ev,0;w lsvo jbl0entually slow down and stop. Explain.

If there were a great migration siu 1ou18rchi3)r3rkof people toward the Earth’s equator, how would this affect the length of the 1u1c8iro33ihu rksr ) day?

Can the diver of Fig. rx4w:vl;b:ta 8–29 do a somersault without having any initial rotation when w; :t:la4x brvshe leaves the board?

The moment of inertia of a rotating solid disk about ,t *t0p pj-zepmkjc9) s(9hqu an axis through its center of mat*u)k0jz(h p9q,9 pm-tjescp ss 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 al (66a6bnbahv9pc d( t:lhmy- 2-kg mass in each outstretched hand. If you suddenly drop the masses, will yo6hm (vlhn6dbplcy -a(9tba6: ur angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass4blximv )tv1z8ie)sy;iexp6bw(. k8r 68ki c. However, one is hollow and the owkmr i8ic8b4xvt)8)1 ix6le ;.ivsyp keb 6(zther is solid. Describe an experiment to determine which is which.

The angular velocityi 4 x98j 3w)qtmq)zmq4kghfvc-v/; h r of a wheel rotating 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 speed increasing or decreashmitm9 w;gq hqjk4 r3vx-zf)4)vq 8/c ing?

Suppose you are standing on the edge of a large freely rotating tureb82ozn9.)h +y rxkl nntable. What happens if you walkr9yk.o+ hbze )2x l8nn toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he th yp xwapr1/vk(9n, xou(1sqb,rows the ball, the upper part of his body rotates. If you look quickly you will notice that his hips ansq( ,n/x kprvu pw 19,(y1baxod legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservatw*:e ;ciwi /kiion of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to/ wicw*:;keii keep the helicopter stable.

  Express the following angles in ra3 xv*xxw fx6.n 3,fyagvztq3 ,dians: (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 a9aaab3j mipp :y.u35yn amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.uabpa3ij9 yamy p:53 .
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beamxh2ov5t7t* kk9 xrc*2nuzt * 9puq3on diverges at an2*thn*o *9 v3t q2or5 tkcku9nxpzxu7 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 rpm. When the motor 0 )q 4 yw5qabex;yk:slis turned off during operation, the blades slow to rest 5qqxly 0k 4 a)sy;:bwein 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 tovh3.q12t0 zsxp qq4zn another child. If the ball makes 15.0 revoluthq x q20nzvt.3ps1qz4ions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter6 ynp q0t,fj0ppkq, o- travels 8.0 km. How many revolutions do tfpk,py,j qto6-0 0nq phe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2(jd am 2hl7xn t 91c6csdzgmvx3328rf, sjft1500 rpm. Calculate its angular velocity i rvnc82 f,jh 73 sx113s6mlfmj9z adxgc2( ttdn $rad/s$ $\omega$ =    $rad/sec$
(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 completewz 6x6 hr5+sa+n+y o1drv+tzm(im xa 8 revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from m ntvxs1+zi8m5a waz++6oy6+xdh rr(the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Sun :da0,d,jg/ xqxnsr: ev.j8kz    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point2kg lys2 om 5hf8t43qt
(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 7evpm7eaxrc (0u /.t)k.0 cm from the axis of rotation is t( mrpaxt)/0v.uee 7kco experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about itsrya s z-t,w5g6o)xl+p center from 130 rpm to 280 rpm iny +x az65,r)pltwso g- 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius uj5g0f ciol,w 0n5q7 dvdu3 ;m$R_1$ 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 themselvef 6a.y mdo(cq-s 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 oyc(a 6mdfq. -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 unifbs,spqmr .z*2o64cay s)qs;formly from rest to 15,000 rpm in 220 s. Through how man; o*fbyc,mpr4s ) s6.2sqaqsz y revolutions did it turn in this time?    $rev$

  An automobile engine slows down from k(cz7 id4 q,gb4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying hig59 6uradj sg,xhspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 compgs 6u5darjx ,9lete 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 unifk/tt)lh3 kf*b ormly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheelkt3bf/t k)h* l have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/o,6 kfc 6gdbatriv/5 *min It turns 1500 revolutions before it comes to a sto*dkia6frg,v/ 6oc5 t bp.
(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 kj0f;8 ,+ tz2atvb+glej 8q ykthe car reduces its speed uniformly frtjkjvq8l 8a0ye fz,tkg2 +b;+om 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 (r 8obgyk 58q)alr b,ppp7r 4o9vlk+ lcar reduces its speed uniformly fromp7 r8(klg+b, a rbvoklpqr9ol58y ) 4p 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

  A 55-kg person riding a bike puts all her weight o.,z,bcytx u+a +tw j*zn each pedal when climbing a hill. The pedals rotate in a cxtw +t,z* aj+b.u,cyz ircle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 8 w-:2cale d8lah 5qgscm wide. What is the magnitude of tdg8laqw c e52a-h8 s:lhe 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 wheecgg31 0)nd oawl shown in Fig. 8–39. Assume that a friction torqu01gn o3gwda) ce of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass zrkm +hx78ip(m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net toi (7+zhmp8xrk rque on this system.

  The bolts on the cylinder head of an engine require tightez* fezzz gaqxb:xrhl 6d v ))*/x/24zrning to a torque of 38rf/gz:zb/ 62) vzz*xxq d eh z)4lr*ax $m \cdot N$ 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 inert gb 5( znz8+tjfe:g9a-cxyqq5fon-9 tia of a 10.8-kg sphere of radius 0.648 m when the axis of rot:gz(t5gy f +z 9o8-b nq95entcqaxj-fation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheelct8 j(c,xr 9rm 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The tm9(c,x r8jcrmass 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 r) a0r wg01f xbpo7h :iqlf(1imotated in a horizontal circle of radius 1.2 m. fx ig(b:17rfp h00 )omwqi1 laCalculate
(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 vsxmj 6d6 r+0(7uaos zconstant angular speed (Fig. 8–42). The friction force 7sdsoj+6a0mvxz6( ur between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig. id11bv2on5.frb i lg.zqid398–43 abo12. nr3.dzqd59 v bibl oigfi1ut
(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 oxyge .rmdq fw3ya)m+owjfs +(dh,pp7 2bx;n 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 spinnido 4hdm /;+1fy r0am/bx drdf)ng at the correct 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 force of eachra)f 4 /y+1md0 fmdrx/oh; bdd 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 unifsv :lo0) k*qwa.2fgc korm cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Cc qo *0k :)vlakfs.2gwalculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest to 3ml gc:,:gth.++;o 4ble oa sfh $rev/s$ 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 n4vgxx9n*p0k,. e h//t9qev svrky8 yhand-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 v/exgh.xy tk*9 ,v80nskvy/9 p nr4qe 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 :0 pagyr 6pobsryb):5 fa )bo3rpm is shut off and is eventually brought uniformly to rest by a fri:6)ba05fo ry o3ay)b psr: bgpctional 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 at h r,;)qmwrf*w7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the agk cep9/6cuv 2u(n7xxction of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. Th7k 2cxuupc vex( 6g/n9e 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, as shk*glvuk 4qx ).own in Fig. 8–4k*ugvxlk q.) 46.
(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 oix ty.w x9 3ddu)fi7(df 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 ha/x 2ndj5m3cot mmer from rest within four full turns (revolutmxt/ 35 jcnod2ions) and 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 hao m-n,ndg8(dmoer:(d1f3 g20h w mdlls 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 of 2ggcy- akt)1q4o98,w :f; rga o -vumsa80 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius bd f +w9t +jh-03sbox(y /idgu9.0 cm rolls without slipping down a lanijbugd/(t30+hxwof9 y-b +d se at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the :yapv /yxo1vt3ifn; 3Earth with respect to the Sun as the sum of two terpx n1ov;/3 :afv3 iyytms,
(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. How much net workubl11pv+ f 6lqj/,tl4jv u el, iej4 l16t/ l,,vuu b v+pql1ljfs required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and hy4rq/c+ zt os(n o1 p0e7qpb,mass 1.80 kg starts from rest and rolls without slip7p +or(p4c/0q 1 yzeh,tbso qnping 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 verticag2a * ktjl(ixu rc*p)7lly on its tip. It starts to fall and its loj)2txrl*c kp i*au(g 7wer end does not slip. What will be 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 0hrww:/ 9whqm. 8a1ilpcyy t79.210-kg ball rotating on the end of a thin string in a circle of radiust /p91 .ihwraqy7 hm:ylcww9 8 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angu+yedf 9 (j1 hbk09p,kz .csqf2)xr wpslar momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 1srk +dbf)pwf yx91,(9zpk0.2hj e cqs8 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 rotating at kc5: yr,/zk oua rate of 1.3rev/s If he raisesr,yz 5 k:o/kuc his 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 reduce her moment oq:qh( *xupz)2 fw; glpf inertia by a factor of about 3.5 whenuzhq:w p ;2q(*x)gl fp changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from anvep1.6;l6*pdm4 b1k6 csu v asu9 kqhf initial rate of 1.0 rev every 2.0 s to ap v41vm6;.ku6pb 9he 1lk6sqas* fcdu 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 through itsvjk u):j h*t6(ig0r3;luxcx z center at a frequency of 1.5rev/s Tvhuz0jj;* gc k63u )rxi(l: xthe 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 the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figuah-vfd9*oe6xk4e. o kf 7mkf2sw+ 6nc 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 of the lpb4 )uht.)4x8s tz wnr3ihsk7v.z:kEarth
(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 onto an ideqnh164fdrc x7d. *da6m eufq5ntical disk rotating at dc f7q6x5a rd*efu.qd6 1m4hnangular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns arugo-p y16 t)rt 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-rbm.4 5zs:f1 c2s xwpwnound platform of radius 3.0 m and mom wsw1nc52:bm4pz.fxs ent 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 free6tzk q(fr i/tl/e5j -kly with an angular velocity of 0.k5q6-klrz(j f/tt/ie 8 $rad/s$ 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 clnm vsk*pt;k ;.hzf:( ollapses into a white dwarf, losing about half its mass in the process, and winding up with a rm :*tlshk;n(vk fpz .;adius 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 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 8zi5f2bk hcdtybbq u/x6;f64 winds 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 bw4num40x- c*yre+vi $ 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, thaw4 : u +lfj tmi0)j5u48cptarqt can rotate freely without friction. The mom airjp5 j4m+lt4) ucfwqu:80t ent of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 8.9f0zdpe ,tzisx,3xqp wen6 6.5-m diameter merry-go-round turntable that is mounted on frictionless x p,ezdfwq .p0z6 98int3sx e,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 i72zoz5hv mt2on the ground 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 theizz 2to5 2mhv7 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 Earth such that the same side always faces th);ya00 z lg;r xxheonc u2i:u,e Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter c;yeh:zl)cixro;0 gn uaxu 0,2ase, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a cdo6fp6mm0 ;l 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 unir kpj ke0uco,u:/ wmhw 4m;ron8*t2/hform cylinder of radius 0.20 m acquires a rotational rate h/roktn * e2uwm0/p8,;uwo rj m:hkc 4of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of twp+p8zb 6w 6py 4st2kzoo 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 diametto2zspk 648wbzyp6 p +er 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rea1fx+.b)dt.x bg zv ldj. u3,zw)chp f7r 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 rotq2j.kozv4;uq hcgq;t n* -0i vating 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 uniform sphqo.gzv *-q h kn;c 4;2tvqi0ujere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution autom *cfkkv0d9+jdxlv rz.e j aq5;43s v/zobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a tot 9 lzd j/v5kd4exk3 0+ c.z*fjs;qrvaval 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 an 5pd c06qx8uau$H_2O$ 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 wi ,kgjw-dv9ph; *to /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 freel;ft0r2yc;rlxhv g g +8y (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.08yhxlg ;fr;rt2+v gc At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius jc66 zw qiv1 fm/k(s(dR. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climjkfs(6 dvwq (1ciz m/6b 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 roaelqxwouo9 31pg 1tvu53gbno-dfl0 2l1 n bv .4d is making a turn with a radius The forces acting on the cyclist a5vlx 3olbu9uv10- d3. p41bf1 q2g owgtln neond cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and beginn4nm1w o7 zybzz-dr ) dm3ko6(s whirling the rock in a nearly horizontal circle above his head, accelerating it fr4ooydk bn6z(zw3rm) zn1m 7-dom rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, makisys0dp3*;2vs:) d qqii9cdzsng 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 againstd*;s i zy:qiq3 2ssvp d)09dsc her body.   

  You are designing a clutch tg*j df x,gz9vs6;y:jassembly which consists of two cylindrical plates, of ma;, yf:v 6jdxz gg9tjs*ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ha,3m2b4ue 1 in: e6nbn+evlkr rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the h 2 3kml6u:bine1ebn+n4,v eahighest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not v)k 4/2asfos.dzod0;s1d uanassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces iu*a8di (8mku:rdyo +eb ) 8gjtts speed uniformly from 90km/h to 60km/h Theudrg8 d8oj)8bit*k( :u+ mye a 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