A bicycle odometer (which measures distance traveled)h2fmp ;gzzis u:6 v39s is attached near the wheel hub and is designed for 27-inch whev :pzg39fushmz2i; s6els. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim ha; rxa 9 qr7we7g,y6mxvve radial and/or tangential 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 lxg x;v9qr7 rwy, e76aminear acceleration change?

Could a nonrigid body bcamd(um 4:xzk9,mj9y ;kd-l v e described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forcema/i x;iftf4wk bwcd ( 317x6q? 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 tod 484v8l)ulys ol* xmk do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A 4o4ld )*vksy 8xuml8ldiagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three m+qs:z*s)uq lat the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harde *sq)l:qsu+mzr to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have wn 75a m-nmlth 4t,r8rslender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, expl mha,nmr 75n4-wtrl 8tain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) ckj u 8emp21/7uc3cgl+rsqln 7 arry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero wgm9vgiiz+6)g5.nfx? If the net torque on a system is zero, is thgg iwvxi5)6+n. mgz9f e net force zero?

Two inclines have the same height but make differ1qu-/g:7qd w. 3q srkzxrx1ry ent angles with the horizontal. The same steel ball is rolled down each incline. On which :gr-7 q/1x. du3rq1sykrqz xw incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from res sew2*k,m1x9t c)y egvt) 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? Which has th t2x 1mv9we*k,c)seyge greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start fro9,c4 3 ahhs).k mwxd:1csvc bpm rest at the top of an incline. Which reaches the bottom first? Which has the greater sskpcdx :s )193 cchhb4,vaw.mpeed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conservnkp4yx(t*y wh lf4 *2yed. Yet most moving or rotating objects eventually slow d4xl *n4fw t(2*yypykh own and stop. Explain.

If there were a great migration of people toward the Earthd z0g1;zh1 mcivw8p-v)w lkgv3pj 61zk28xzr’s equator, how would this-g w3 dg18z8m6z v;pzcx1 izv1k2 jpw0lvrh )k affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation y )mb;cv0). sa hwzs)xwhen she leaves the boardzy c0vs).a xbws))h;m?

The moment of inertia of a rotating solid disk about an a7tsfgav6 f 79hxis through its center of ma7hsag f6 v9t7fss 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 wa2rdpc m+eq+6. c+,o5tyfn na 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the ntra+,p.+ydc5fnemw q2 oc+6same? Explain.

Two spheres look identical and have the same mass. However, one is hol9 c9f8:kpj0n 5jtkk uflow and the other is solid. Describe an experiment to determine which is whic ukfn8j 5k: c9pt09fkjh.

The angular velocity of a wheel9y *uqub8-jxw 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 acxwu*-yubq 98jceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a lar cuk9m5 58jqf80imisvge freely rotating turntable. What happens if you wal8f0ju k q8i9v5mmc i5sk toward the center?

A shortstop may leap into e0(+9ofmq)n* 8ej fs3jyam hgthe air to catch 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 hips and legs rotatensghoe*fa38jjm+y9 0f( qm e) in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a helpts flk l kp5156n6gl*icopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate tlp kl*p6fl5 1gk6snt5 o keep the helicopter stable.

  Express the following angles in radibsn.2 6sao+7 aqyp6 uuans: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculatrbqd*1h stlz9kv:cymkn ; ly 0x 2f,44e, using the inltyy0n;dq4,hvckb2 9kz:fsm * 4rxl1formation inside the Front Cover, the angular diameters (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 beam divrihru, -d50i verges at 5r,- irih0vduan 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 is tu6.vgsb y p5 a v-s8qjtp9ry07rrned off during operation, the blades slow to rest in .6 yasj89yvp5qs-v rg0b7 t pr3.0 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 anot*bk/ vuf (w.o+d1qa.xc 9blexher child. If the ball makes 15.0 revolutionsu/c+.xe xb(k 9q.b dafwo1l*v , what is its diameter?    m

  A bicycle with tires 68 cm in;pz ; h+44*9.yb, l x* uvyfsvyimeocc diameter travels 8.0 km. How many revolutions do the wi ; .9+e*ybus; lh4xzvo*4fc, vc pymyheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates aty7 wfcajfyf(bf,2w- . 2500 rpm. Calculate its angularw,c j.y7bf2ywf(f -fa velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–i1rgir4e4s 5 q38). (a) What is the linear speed of a child seated 1.2 m from the centii grs4qe14r5er?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Eut wtiiv0 :):karth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed hmen ni(nmp): n/xy39 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 2 2sj;vi mfi7i:/ecnw particle 7.0 cm from the axis of rotation is to experiencj 7 i;ves22/im:wfnice an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheea- 8nz b*0z wu2spkx/iklly-:l accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. lpwi* zbn :ka -k-syx8u 0lz/2Determine
(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 radiushvb)d i (9;kma $R_1$ 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, zp pzv-3jpgzew 1anh/ve2 m( *vus5-*astronauts aboard the Apollo spacecraft 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. The vp1(z3 vuz wvhee5jgz2p - -/*pnms a*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 rpm in 220 s. Thl (,n+ ph9xt ixltbos-9zsaut0 rx9oi6081ev rough how many revolutions did it turn in this timn91s a(v s90x 8piett0 9ulib+ x6-to zrlox,he?    $rev$

  An automobile engine slows down from 4500 rpm s/e3 othn.ga g5, 9hxig-d7z nto 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 du,qgvrpp63 , of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its finald ,3vqug ,prp6 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 rpm in m h,clq/ 9:v,lz;h8+vxns azs6.5 s. How far will a point on the edge of the wheel have trav, ha8/ q,vl;+lh 9 s:mxcnzsvzeled in this time?    m

  A cooling fan is tur4c9 n5uyr2dvoned off when it is running at 850rev/min It turns 1500 revolutions before it cor5yov9 u dn24cmes 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 speed uniforn1lpby w hrb0-x(44po mly from 95km/h to 45km/h The tires o0pw41-b (bphn r4x lyhave 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 speed uniformly bl) :qoa668todz -hl) 4.sy egc)k flofrom 95km/h to 45km/h sla4h l. )q)of8go:zoey l6b c k)t6d-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 edj3,v se*icg)jh r5+g9 sa 1fon each pedal when climbing a hill. The pedals rotate in a c di* ghs+a 5cejf93sgr,evj)1ircle 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 55nypevvhto:5m f 84w).2m l1fy N on the end of a door 74 cm wide. What is the magnitude of the torq.8lmoe5v p4 1h)fy ftwm2:y vnue 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 to9x8zvj (ha gd z8akr lro3:-i7rque about the axle of the wheel shown in Fig. 8–39. Assume that a fri9r -kr3jh gzdoza :8x(ai l7v8ction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass 4e5h9pcal jf0 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 th4h0jl 59c peafen released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a /twebzg)3gf * torque of 38 eb)/g t3g* zfw$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of/ng5uc 4du2ufi* fg x ,uy3wu198crec radius 0.648 m when the axis o *e rc4nf195g fuc8u cug/,u2wx i3udyf rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wqocridi*(: pirc9z(.heel 66.7 cm in diameter. The rim and tire have a combined mass of zqip c o((*r .r9cdi:i1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rodqaloyno 1hl5kf zq 31;74iz; i is rotated in a horizontal circle of ra5oq l4ohkyzin 1 zl7i a;q;f31dius 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 bth l3+w2z;jp w07y eugowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force betwu l3z ;02wht7+gpj eyween 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 ax 4dx fomc 58e7l )azq69bb2hof inertia of the array of point objects shown in Fig. 8–43 abof b dxqho97a4lxz c6 b82a5m)eut
(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 consi.b+ehd0q i +yrsts 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 c 7f(c od0h6jwiylindrical satellite spinning at the correct rate, engineers fire cow(j670i f hdfour 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 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 unifdt q;1msle1yo/2f tc(orm cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculatqt( ldo;t1ec2/sfy1 m e
(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, acq24k( ao2l u tr i8:vw7c(dog10drektcelerating 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 wuvb yfjm)2/4 small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. fv2)/4u jwbym Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brqb a5whj*p+ /gought uniformly to rest by a frictional torque 5a +/pb*jgqhwof 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 7 / m8v.ppbx0i z$m/s^2$ 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 action of the for23 slfuc;luv3kx*vokkx/) d( earm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball ivl32ku dc k* /3x(vux klso;f)s 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 ben (6jfouhq v.esn j.jw ij)e-wy l0*7, 3g8ptd considered a long thin rod, as shown in Fig. 8–46.. wi tjs-(g jjdon.h6wqfunyp*)je7 ,e80l3v
(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 og(xp2 zu8-6ivz 1rrddoo .y:v f 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 from rest within four fuf8lc80ho mq 2yll turns (revolutions) and fy0c2q8lh8o mreleases 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 ha0qkk:e4.bp 8/0, ij zlkjk8m oxrr/ew s 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 tokd9u82.*ang 7z6m3vdj w st ywrque 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 withoutg-f .x3,dci ova3 o ke,1dxdv+ slipping down a lane at .k-ddcoo, ,f1ai33v evgx d+x3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth withlt b1 beu: hzgbf2rw;h x;rw +88;9wgn respect to the Sun as the sum of two te1lbhn + g: r8;ew;g2h8b9fx;twbruw zrms,
(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 woqg4c 9igic86 -ie9vej rk is required to accelerate it from rest to a rotation rate of 1.00 revolutioivc-94ei i jgg689 eqcn per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and +d t:d7(jc ajomass 1.80 kg starts from rest and rolls without slipod:jd +(7 actjping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. Itajcnb: yq4w(ufv:, q6g1lzg + starts to fall and 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 conservation o1 (gnbzj ,+qwglacu4::q 6y vff energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin stm+d ccdb* 8p-fqq 3l+oring in a circle of radius 1.10 -qbc pld coq8 df+m+*3m 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 cylindricaomb4,8u002 xnsft qltl grinding wheel of radius 1828,flq b ont04mxu0 ts 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 a y19tx;ko vt45w ux sb:oal)a+rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation datya;o1xu5v)tw o bk: +l94xs ecreases 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 of inertia b5 6/d1czrvbtwy 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 tuck position, what is he15ct z6wr/d vbr angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can )m bj v+/,nkj6oubb/dfj d:c8increase her spin rotation rate from an initial rate of 1.0 rev every 28u fdb +6cjj /ovmnjb)/:d b,k.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 arq/ - ouv;d3wl:c2kv t z1:bnncound a vertical axis 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 of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotncq t:; co2z/bu n-vk13dlvw:ating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentc l g5f:;bjmk1 n4(k-+h)sseu) dfxeo um of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momenef4y 2n.vkx 3itum 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 identibj0t9tl- 46m ewp1onmw j:j9qcal disk rotating at angular speed :j lpm4tb1ew 9-wjt6o nmq0j9 $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at kc629hqc gnson 2a(y b+k53ie 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 fcj:(6d2 p wb,eapunu)ls(2 xkg stands at the center of a rotating merry-go-round platform of radiu :(6aw2,jexsuu(pdpfnbl) 2cs 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 rotatiq z;5y.24 wg7bcppmtsng freely with an angular velocity of 0.8b7 2 gm4wc5y.pptz;qs $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half it ;ywf8pge2dkqvn *o. 4s mass in the process, and winding up with a radius 1.0% of its exiy4pogw v.*8nqd;f2k esting 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 winds in y ,lzlx;fue, f4qp63 texcess 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 e8 51io14n4h4vqoz;dguh qa6 )ceucz$ 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 platola7)y ;tz tv3form, initially at rest, that can rotate freely without fric) 3;totvl 7yzation. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5c:jo)+z1x) q d8j/hsyp dzc (f-m diameter merry-go-round turntable that is mounted on frictionless bearings and hp/cfhsj 8y(zc zj) d+x)1 qo:das a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying, g/letft h)ir:;q:xqcg6ur3 on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope u hgl)xqir:cttug;:e3f, /6rq nwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side al *quvz; to4gxv x4jm3(ways faces the Earth. Determine the ratiou; 3z44to*g vjxxq(vm of the Moon’s spin angular momentum (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 from resu0+. 6q8rlsdyw0 qdmst 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 shapevwgor 5x+5 y9d st49d4 braa5y 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. Calculate the torque delivered by 5txs4d+agr59vw oy5 br9dy 4a the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylr;/f .esv ua1rindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches thra1r.fv; /euse 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 speeb,58d zr e/ofvne:g t4d 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 mak(i:ri3ql8tks e 2s/ 7lnpdesz,sw04ss 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If is /srs (lw8t nezlei7 d3 k0:qsikp2,4t 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-pollutuud c,mggzksa/4 5 6,gion automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, m/gc,a sd6u,kz45 u gguses 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 a /w vw-rkq u,2nv6k*(x:e tu5vti8vuw n $H_2O$ 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 horizxiax35 ; h3a5/p( zsfr5ryc afontal 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 massz,o1y-jniq66 rgbt 1g M and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the lineagyrqj,o1nzb66 -gt 1 ir 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 stan nbi,1l;ee,q6 eh 1ppyding vertically on the floor, and we want to exert a nh,eeq6e1,l 1bpp ;iy horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn withr1o5mtnt;ux tpc k 8y8t;q;5 sk t.ca. a radius The fotqs18 .c8kxot ;u;. cmp5ry5tk at;tnrces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m ah:s:o vc6o1ku nd begins whirling the rock in :shuco 1v o:k6a nearly horizontal circle above 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 cylinder and her arms as t)(v19vifcj t ghin rods, making reasonable estimates for th1ti 9cj fvv(g)e 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 whia* 5arr/9 lfd dpo*p:ech consists of two cylindrical plates, of 5ra rp*oe9f/dald* :p 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 radius r rolls along the looped rough track of Fig. 8–58srvm ,uhxq7f3;* x e5h9i l/dz. What is the minimum value of the vertical height h that tes/h* ;mld,xhix 57r9v3zufqhe 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 assum46uxh ek5ipzhw)7r5c e $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its sg yw:u4+ybyogh0r8s1 peed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accelera:y4b +o 80s1wyygu rghtion 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