A bicycle odometer (which measures distan;1 evoa7tj d7stfcw,4z:hq+ht3ifa 0 ce traveled) is attached near the wheel hub and is designed for 27-inc7 3:4j ,ztaavdwq +si ott;0hfc7efh1h wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a ,-98 i kbn+ue.n lih5n- jqplcpoint on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or j5n+ilb-chqnpk, u e-in8l .9 tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a sinurw r u.rl3d7n;a-,m4pons.hb l1 kk 5gle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forcelb6;s(w)il6u dp lmd b1s5:dd? 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 3zxxy *kdv1e3than when you3 x*z d1xv3kyer 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):ojbfdx) c4d pedal cranks. In which gear is it harder to pedal, a small rear 4b))dc:ox fj dsprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with flel5f-rnu7e 7 iei/vqq1sh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this d7/ul-efv 5nq1ier7i q istribution of mass is advantageous.

Why do tightrope walker2-h9jq)h n zvu h+nysv/ a q2+*gy:p93nrwyk bs (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the nb0 hbm)w r qd4e*+j7poet torqu7ewphmb 4 qdb0+jr*o )e on a system is zero, is the net force zero?

Two inclines have the same heighqweb j-4 lrts:q5) ,nnt but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the 5:j ,qbe s -lnnqtr)w4speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest),uwwjm b44 kwg /ub4i, down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bo4bk4i4,uw wuw b,mgj/ttom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder 3;obqpn gzszka 7ske15-u89 lhave the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has tl; au71s5k38kn9 zso gzbqep -he greater rotational KE?

We claim that momentum and angular momentum are conser(t 6akpvhi6 5b:uof9 gved. Yet most moving or rotating objects eventually slow down and stop. Exp vpf69:ugt5kao 6 h(iblain.

If there were a great migration of people toward the Earth’s equator, how wouldm/i o9;.xmkf;p kctfq zj*o9+ this affect the length of .fj;oq z9k+m;o9 *f/ tpcmikxthe day?

Can the diver of Fig. 8–29 do a somersault without (e9 wy) cq(l: mfurt-chaving any initial rotation when she leac ur fem)(-:(ylqtcw9ves the board?

The moment of inertia of a rotating solid disk pdz 0 0b (.yzugcdh2f/about an axis through its center of mass is y/ b2 0phu .(0dgzcfzd$\frac{1}{2}WR^2$ (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 av5mi 0uik 5q-x 2-kg mass in each outstretched hand. If you suddenly drop the m05qv5-u iixkmasses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, q0/uuicf( u/ *l; hb; fzff:xcone is hollow and the other is solid. Describe an experiment to determine w hulz;c(cfq 0/ff:*/ix; fuubhich is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In whl.q6* 2 +mr radinif5jz b/ny:at 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 ann2 rm+rf6.qd*za5 jiblyi/: n gular speed increasing or decreasing?

Suppose you are standing on the edge of a large 0 vrs4m*fi 4.algw jx/freely rotating turntable. What happens if you wa w4/v .m4jriaf lxg0*slk toward the center?

A shortstop may leap into the air to catcf.5pb0 iaa dmlhsl3x r8dk+g5hr/:3uh a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his himha8 d5 b sx3lpa/fd 3r50:+hkg.urilps and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, d- s ahx 2/6v(h,dk5cpa;2ijzz7hg:sr koy;yliscuss why a helicopter must have more than one rotor (or propeller). Discuss one or mo5h;y:2sz2s c 6ga7 xl,jok(/i vh-ydkzha;pr re ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 303/ v8-e, w/fusvilusj/smv :h $^{\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 becaddvi 1i,ksrlmydndj f11),( 1use of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of thdd ,ivmks 1rnjyl1fd),1di1 ( e Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon 7iy+ vlk,b/z:u/r (clg7iji v, 380,000 km from Earth. The beam diverges at an angly7i:i/c 7jr/gvv(+zu ilbl, ke $\theta$ (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 l,evi3sv) *f *t7bs5px tepg9 of 6500 rpm. When the motor is turned off durin 7 xee,9s3glttbips5vv *f)*pg operation, 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 th9 otlz*xq6c2u ,zx dt1e ball makes 15.0 revolutions, what is x c d2,*x9l1tuzo q6tzits diameter?    m

  A bicycle with tires 68 cm in diamet4enrk:c *0xu . 6ztirrer travels 8.0 km. How many revolutions do the wheels make?z4k rn r0c:.uri *xe6t    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 r8k/kt(lh2d ,wftjd j)pm. Calculate its angular velocity in k(tt/j8w2 d,hkjdl)f $rad/s$ $\omega$ =    $rad/sec$
(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 co,ka4i; dsn mv/c j.x/zmplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1/mks.n;i/ 4 xzjva, cd.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/avl6dc- i:,zt8plbpd i j2w6 of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speedm),(krm c )zdp of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axis ofig(x n izul n9,p*s;d9 rotation is to experience an acceleration dnl(*;upin ,x i9sgz9 of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to f/l4l9e. n7e dsa-m. bzuq0eh/pe6c q280 rpds luq e407e.-z9c q/ h enapl/bf6m.em in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius zha wk6 py4: p n5a+ wq,lxpf:iu.u4n3$R_1$ 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 spac.p5t9mogu5c9pq* (lbss )xbrieg(- wecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. Th)9x. l*9oi(-5ct b gpus(b empwg sr5qe 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 15,000 r i:2c bjygks,42jca emv+0e6ypm in 220 s. Through how many revolutions did it turn in this time?vj 4asi:,0kym ecyec + 2gjb62    $rev$

  An automobile engine slows dow-cacuh u 0 -aps9,twv(n from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying;4yk u;l yz18u ,ypdhr highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions befohpy8 u4;z 1; kl,yrdyure reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 r9 jg0 g3v ilhoyx xf7 )0by-xh;a0z:nspm in 6.5 s. How far will a point on the edge of the wheel have traveled in;hjg0yf)sb0goai: 3hyx xxn -970vlz this time?    m

  A cooling fan is turned off when h,g :m)(b5vo4ne:hl n/gpgec it is running at 850rev/min It turns 1500 revolutions before it cog n:b:gc5 / (,ve mgonhe4ph)lmes 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 gy 4wlqhu7f9.h+5h dkxarpo-*j0sc 2revolutions as the car reduces its speed uniformly from 95km/h to 45km/h Th0qa gjskfo2r- hpy+x .hwd*c45h7ul 9e 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 sp0a- a t*x5qqjcg)o+ ikeed uniformly froqic0)-k axtjgqa 5+o *m 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 on eai6r)q ,qceho6ch pedal when climbing a hill. The pedals rotate in),6ioech rqq6 a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is the mao:+ re1.ciu /wcd/qp zgnitude of the torqu1r : dpc+o/ci we/zu.qe 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 a3lypa4n9x 29.ftdp )ojn 8 exe+sd0gcxle of the wheel shown in Fig. 8–39. Assume that a frictdla pfy0o9)gs.8ex2n +px e j34cdnt9ion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mas.jx)fcp 4 d 7e(e oxnguaq3*9rs m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially .3xeue pxj*7o9 gcfda4)qr( nthe rod is held in the 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 tgbt mamgh.*9:ightening to a torque of 38gg.:9*mm btah $m \cdot N$ 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 sp-tb /;qv ;t01d:/h msxyvnycthere of radius 0.648 m when the axis of rotation is-m0d yt /v cbx:h;;1 nvtqts/y through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm i637ren q. b7l9su5cxt (ula bon diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (whyea5nu3cr6lo7( ux 7 9sltq. bb?).    $kg \cdot m^2$

  A small 650-gram ball on the *vsdrt( ylh*x(*5fvct 5pf t,end of a thin, light rod is rotated in a horizontal circle of, vth*x*vl(yf 5ttdr*(5p s cf radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating az+77 grg7ue 4crutsw5t constant angular speed (Fig. 8–42). The friction fo5c rt74 suez 7ugg+wr7rce 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 o5 uh pmzo:vk .o9 + xcmllil)7s*1;ruaf point objects shown in Fig. 8–43 aboua)+up1l:7o5ormz s hx cv*l9umki . l;t
(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 oxy dd ot *:8rmbjx9wt+w3gen 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 cylindricvune-we7 /.ig t.rgo *al satellite spinning at the correct rate, engineers fire four tangentig.e7i.r -g vw*unot/eal 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 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 radius of 8.50 cm and a ma6h ;ffhkctk w91xl r89ss of 0.9fkw;h 1x f89rtkchl 6580 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from restbzpq xx10fz;5yy8 7 hm 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-r wo/l h9zojgxb; y1u-)ound 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 thew uol9jzx / 1y-;)hbgo acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rp2(3ys )sue- xxc8e ctzm is shut off and is eventually brought uniformly to rest by a frictioezx uecc8)2st x 3-sy(nal 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 balba 5d/dxqz hw;x u)/i*3; fiobl 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 t( ewcr6ia(x5 ohrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerateda(5x6o(e ric w 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 shsw 2 0gs5kbvqe;cb(v4own in Fig. 8–46.c kbv2gbq s0 ;5s4(wev
(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 nm6 o7 q79zfaytwo 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 th q9(c fz c*la,e-c5nkde hammer from rest within four full turns (revolutions) and releases it at feq5k*ndc-9(acz c,l 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 hr 8 39(gq e3c2niyxgy1cat4v ias 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 o7kx,9b68 z ryurwa2z wf 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3vrx *lc;7obmc6uer,a9;w9 lzb8y viu*xf:h3 kg and radius 9.0 cm rolls without slipping down a lane at 3va cc:9bumel;u6x 3il,ofyr 9 b *xvr8zhw;7*.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum /)v;2nu/rz 7cqt sdt wl8b5xjz.r(tf of two d f8ctw; 5zr v/7t) subqjrz.(tnl/x2 terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How muc1awv(;zo en9 u)hd5qyrh -sn*h net work is required to acceen5nqz1 o)(s h -d9uvhwya*;rlerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withou 9c bdfq or7rf6uco7/.t slif/6do9 crqcfo7u7.rb pping 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 balnx8(lvgnoaj59c/d 8z5t5q hq9pih; ianced vertically on its tip. It starts to fall and its lower end does not s(i98tqghvoc5pinxd; 5l/98 qza nj h5lip. 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 0.210-kg rfypk3 zu. /:gball rotating on the end of a thin string in .r/ z3:upfy kga circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding f.h3,9awkgt bqq1+3 ncc 1 qomwheel of radius 18 cm when rotating at 1500 rpm? tchof+nqmkq.g1 w c13q,a93b   $kg \cdot m^2$
(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 tw7sqow0rw*+te cmp11 hat is rotating at a rate of 1.3rev/s If he raises his arms to a horcmtpw w sow071*1eq+ rizontal 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 rg6jvm9 fik* 0ueduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tu*96iukvg f0m jck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate 8n qsy cq-ils;h9g1 o)of 1.0 revcsgs-y;9lh q8nioq1 ) every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a freqj1 g8 low+c8s xsu07ypuency of 1.5rev/s Theuc +8pgj w1 07l8oyssx 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 figure skb - q hy)e05q341tvnmq l6cxlwater 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 m1ma ju7rp5b6womentum 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 dropped onto an ng,h0p(r* qzdt8x o .w5arwb8 identical disk rotat0(n . zw*apr,q r 8bgtx5wod8hing at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsm6 yynswb 0d71s,av9s 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 rotatins9f8 ss7 k23 npani78mg6piilg merry-go-round platforp8k6pi7 sln a 3 8sg2sniifm97m of radius 3.0 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 wi z mn;qwh lz,;7+ilzw:th an angular velocity of 0.8wzz lh;mz, wlq:+;7in $rad/s$ 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 eventuaz +c4 l 1l1qz3lg4oohnlly collapses into a white dwarf, 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 rotatiool1lno+cq43lzgh 1z 4n 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 invol h(cohw x bi;y94l wz7sin7,t8ve 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 bi,i:e 5c*a0x i*xwg w$ 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 freelr,tczn1 u) k 7q:yks4v;* d.)mgzfvjdy without friction. The moment of inertia of the person plus the pcuvv*:k, dmr1zk ).t jz dn7f;)q4ygslatform 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 standgr9qkm)t02: u xhk*2-gvrazns at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of vkz2rk a g* tqgu9xh0n-2r:)m 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 end of the rope lying oncf c(3,69jian rf0osu the topjo6 n0f9 scr u,fca3i( edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Eaezyg-;dq:ru/ubr 6 pbw.g6 k:rth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentumu6;z:bpqbgrwygk6: .re / ud- (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates 16wkosq+ky/l swz i :p-m.5z ifrom rest 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 the shape of a uniform cylinder of radii bkdd nnpia9f+ :pz26ht; i(p(3d0srus 0.20 m acquires a rotational rate of from rest over a 6.0-s in(tns dpp;(a:rn6ih +20fipdi93 kz dbterval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mzo 9cr/a +d5n2nzzq q8ass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculanq82z z+9qanc/ozrd5 te 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 rear icmqu hgf46 u.17tms7 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 m/s so: o3xgdx1vf(9t mass 8.0 times as great, were rotating at a speed osdosog3:(x9 ft x1/vm f 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 sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobileozbc1d1r9 rqe (hclt230 f pe: 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 cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be r9qder(c 1fc: 0 zh pb32eol1table 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 illustrabx rp6tw b0/2d.4zj*ex5 udlqtes 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 horizont.d v 4b1hk1tsual 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 freeli n me08+1pmux jc0po1y (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At thejc0um n0i+p8pm1 1ox e moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on thruj j;bd2o1s (e floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fi ;2u(osjdbj1r g. 8–55). The step has height h, where h

  A bicyclist traveling (g)d durzv0 h p4ns2 6b1p9lcswith speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the normal n)v pzus9b4g0p d6 c1s(rd2 hlforce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 ro:s:esfi; jp7w ck into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above:fp;se7 wsj:i his head, accelerating it 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 solid cylinde6rs i +q gxv8-k)2 dp/wj3rsvlr and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinni 6rp2-+jwx i3v/sks8v qd l)grng skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylinyqrg1 z7hzd5w f/nt c6gq8vv1t w :d87drical plates, of massrh1 v ftdq 16tz7cgyzn/gqv85 ww:8d 7 $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 , 7nn( qgg)qglrough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble mustng,lq )q (gng7 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 73eo: 4qyqev sassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly fr 9f+ny9f zo1p5e *d4nk5( vwl gbigo0dom 90km/h to49g5dz+e no9y0w*( bkdln5ovgffi p 1 60km/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