A bicycle odometer (which measures distance traveled) is attached near py .k3*.ous 4gxsazff n.-d t1the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle . 4dk3.xsu pn.*sgz- yft ao1fwith 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a pmt td/y:6 bue5ifrv -ebh5.( foint on the rim have 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 linear am: tirbb dfyf/5e.( eh5v tu-6cceleration change?

Could a nonrigid body be described by a single value of the nikw3(+ unod; angular velocity $\omega$ Explain.

Can a small force ever exert a greateruzhwt+qfc3. 3xhii-77gl qc7 torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hn+xyrq0e u: 4eands behind your head than when your arms are stretchedue0rey+x4n :q out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has sevqotqtm3-pf dqc*v tj -t:q8 06m.f0ceen sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a smal 3mt o6*-tq -e0 t:0d8cvtfcqpqjf qm.l 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 being able tonusk2aj;6c xx(8zfjd l 7( 5ll 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 didua5jn2x s8lk(fzlx;jl 76 (cstribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35jod;vc:z i 6t6( vb5h r3fito5) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If dqqq li3hf y)u4yfjt+*: 2 v(gthe net torque on a system is zero, is the net :liq 2jf3y(t yvhqfqg+ d*4)u force zero?

Two inclines have the same height but make different angles with the ;meziw1mu-5hj+sw 6 vhorizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ballhusz1 +m;me-j wwi5v6 at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down a kjl0( zmz vsacg (5t3:rhp-7mn incline. One sphere has twichj-sr50z3g:zpm l (( at v7mkce the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mas5 ewjurlmt1i6k*(n c0 s. 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 tkr10 icjm w6*e5nlu(the greater rotational KE?

We claim that momentum and angular momentu( qa9:jldmh9 vq b;zqu1t o;f5m are conserved. Yet most moving or rotating objects eventually slow jz9dtm v(9la:qo; hbqf;u 1q5down and stop. Explain.

If there were a great migram6pd q(ya 8 u6yl8waz+tfe)e /tion of people toward the Earth’s equator, how would this affect the length om8yzaup6+ /(wft8 6)ldea eyqf the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation p.7 ; /znqxwn jvb;(yywhen she lea7vnw;xnz; jp(. q/byyves the board?

The moment of inertia of a rotating solid disk about an axis through1cqy;h:*da v8i gity-ij w0 7y its center of ma*vy: hg7 awyd iy0c8ii;q-1 jtss is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass *gw zxs2.et aw3miyff4t +2w /in each outstretched hand. If you sud egfx*fy34w +/ w2tz.2 iwsmatdenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howevervbozmbn e 5*zl .yj-03 k8us4p, one is hollow and the other is -p.l5b v8 3 ebns kz4y0ojumz*solid. Describe an experiment to determine which is which.

The angular velocity of ,04 e jwzo..djp2pfh ta wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a poinzp0 2dhwetp. oj,4.jft on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Whaechlz6 wtq/n e*+6gb :t happens if you walk t zbl/*tce enghqw66: +oward the center?

A shortstop may leap into the air to catch a ball and throw it quio70l ct(fl q8xqy8k -kckly. As he throws the ball, the upper part of his body rotates. If y7lq ql c0yo 8tkkf8(-xou look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of anguld( 3(ehgfe*olq4nw e 9ar momentum, discuss why a helicopter must have moree(h*e (q4g9flw e 3ond than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following,s)j )v6wyc bo angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, using-c 6t.nlmgu b5 the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, asn65 -cg.ul bmt seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam divergessaaz9 2sr3 il 2 ddlvtvo5,1/b at a2l5s /t 9a2 vabod sdvz1ri,3ln 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 r ziin 7f knu-*d+kbts90a,+jat a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3n s i7zk*jbau 9f+0r,k+di tn-.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 2y5ppkd,01cz lhgt 0 /5pe aic5t +1btkb zb9e3.5 m to another child. If the ball makes 15.0 revolutions, what i02p5y b5 t0k9ctce,tg zelb5 d+1k1 /bihzpap s its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revrz30d r/+tsc)bg 3gtpolutions do the wheels makeztdbr0 /g)t3gs +3prc?    $rev$

  (a) A grinding wheel 0.35 m in ;d5 vi0vnt 3o5q7kojs diameter rotates at 2500 rpm. Calculate its angulo0k5 von7vtd5si 3j;qar 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 rvh)fc vg-pzjt,6g vg0)-j .j vevolution in 4.0 s (Fig. 8–38). (a) What is the line tg0.)-c-jj)v6vpfgj, vzhg var speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the s0/txxrkf6k1 2ztke u .p; c)dSun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear s r1u806hu nsytpeed 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 rotv89s3j3 pjrzv1ny 0-jyv rj/ mate if a particle 7.0 cm from the axis of rotation is to experience an acceleratinr1jj yrvszm-j 3 v3jy8p/v90 on of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly aboutm7w(hl k.x)5 lw- gcvp its center from 130 rpm to 280 rpm in 4.0 s. Determwvh)pl.-lx cm g75k w(ine
(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 radius9oi 77 9h v r5na:ehngjlny1j4 $R_1$ 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, astronaub5cqm d-4p j *zbq9;:ufs*tnjw ,f y1kts aboard the Apollo spacecraft put themselves into a slow rotation 15-sp9n4 fjmc* t jbu* d,q;fkbqzw:yto distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. 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 s68)wrd gjcxomg:7 uy dv3t28uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turnrtgdog88v)3:j7 y26d xw csum in this time?    $rev$

  An automobile engine slows down f ms*,z5v 8mobmrom 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 h, )kr6 7-,irfdgre)jjp xzy h+ighspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions befjdkx e)g hp-,, zr f+yrr)6ji7ore 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 rpm ;xr8 hk:n) l49zf invuin 6.5 s. How far will a point on the edge of the wheel have trlvnurnhk;)8x4f i:9 zaveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/mg2j (xnwrek.i9 q;,wn in It turns 1500 revolutions before itn, i.(qxj ;k2wewr 9ng comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutionsh3ld w .pn)v/x as the car reduces its speed uniformly from 95km/h to 45km/h The tires havewxl . n)hpv/d3 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 unif wbg/0 ye r49s+4jcggyormly from 95+ 9se/wgy gjb4y4rg0 ckm/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 ridingjsibz;g (h9r 1v ggz4* a bike puts all her weight on each pedal when climbing a hill. Thhg rg1z*4j i bg;9(zsve pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. Whatpe smn-h k9:i2 is the magnitude of the torqkmns: -p2hei9 ue 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 torquwd 2ecm hgdr3h9l3)( ne about the axle of the wheel shown in Fig. 8–39. Assume that a friction torr ecl wg3dhm(3 )2dh9nque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attac9rjoz4oc0q7 5: rmzvk hed to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod9:cmvr zz0q5rj4o ok7 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 tightening to a torque of 62+nj w 4l *b.ogyhpn. gouux2386nxl2.+b*2uhjg4 un o opgyw. $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the j6yw0hon / +9sqnajn8axis of rotation is throu/qj0owjahy9 n 6n8n +sgh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in3rg)u vyg z1.m diameter. The rim and tire have a z).ggvru1 my3 combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of aak z1z1l0hux 6 thin, light rod is rotated in a horizontal circle oza10 z1kxhul 6f 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 at ckd3p ; rtppo;4onstant angular speed (Fig. 8–42). The friction force between her hands; 3r;kdpto p4p 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 ob0 8cc)4yoz1j*avp3 )tu9 bns:w4u vj5 csjxbujects shown in Fig. 8–43 aboub5 ox)3w tcna4jy4j8uzjs b1u:p vvsu0c9)c*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 oxygen atoms wqq+rt7+okl4 d hose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at 4x68x2 xje8 cnjolb1lthe 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 jbexjx16lc884 nlx 2oeach 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 mass ofbmt -i3t c1md4 0.58t -t3bmmi 1dc40 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, acceleratinghh +r5( zjgn(wyq3(lk 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 ta./tpeut53khqa 9s;mnngentially on a small hand-driven merry-go-round and is able to accelerateau thq9t/ ne.5k3;mps it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rph*c. (xml)okvqkdid.n, hb:2m is shut off and is eventually brought uniformly to rest by a frictional torque ol, ni(d cvdq:)k2b hkmx ..oh*f 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 ac1i1lmjfx:n7+ -6y z-xq cu- se9fb wsscelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, whifxcs435edk)o 9efi1jup3. h gvl il- ,ch rotates about the elbow joint under the action ol3i5v -gsi)x4ljohkf.3 e 9ce1f dup, f the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be conc8 rauw4w4. *mt0vv icsidered a long thin rod, as shown in Fig. 8–46mv08uv ra.t 4i4ww *cc.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two mass o 404t 9kg+kleozh1ipes, $m_1$ 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 withink3nvlr rn7 n1. four full turns (revolutions) and releases it at a speed of 2nknn .vrr3 l718 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a momentrsjg7qjab- tlv 4dp4p7.e.r + 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 developt7+o7ir 3 gzmos a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg aq,- kg+ omuu7v+ cdyu* *1zftlnd radius 9.0 cm rolls without slipping down a lank 1l,g+f uq*+m z t7u-yc*ovdue at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energx37a rp ,lten. 2qzs(uy of the Earth with respect to the Sun as the sum of two terep7usz rt(2xn3 l,.aq ms,
(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 radiu8x db30e/j/4rlm t,g7ovxae bs of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder./debv,aj x/gr378t bl0e o 4xm    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts fro0qexe,q1vze (aj th.+ m rest and rolls without slipping do01ae, evzqqt+h ex. j(wn 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 ngutx(5)ks .6fh lk(plgb h1u 9tgn6)t0obm ,vertically on its tip. It 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 gro 9tl6 g hp (n6ougl,g)5s0t1bh)m.n kb utxkf(und? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of *5+ydyr- dip tw0cpkus+ 4s/ha 0.210-kg ball rotating on the end of a thin string in a circle of radiyhi0r+5y* wpsdut/- sdp +c4kus 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momfl wk(n8bk7sgs* y40lentum of a 2.8-kg uniform cylindrical grinding wheel of radiu s4kbwlkg87 y0lnf (s*s 18 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 rotatin6+cx2rj;s ;hn;hh2 ( v tghszqg at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rc zhj s ;vxh(2h sq26hgtn;;+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 i2 su5ed5h yv5wf2xsa;x-2 kxof inertia by a factor of about xi2e2yah 55kx ss2vf- udw;x 53.5 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 her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rots ndgq7i dr / ;:aihcp4)aul*9t)7wwpw.mu8tation rate from an initial rate of 1.0 rev evear7tm)w:utui9 4hs ln;wp .qd*7 i8 d)pc/agw ry 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 rotatig*5ag 6kze0nnac95y png around 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 rotating wheel. What is the frequency of the whe5zgpk 05aan6 nce y9*gel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning 8 z(m 3dw g3.yfyv qb er(evq;asi4.l8at 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 Ears6b e;ip6r3.ve u. :(njxeckh 4xl n2cth
(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 e 12 1i*85kgg ku, 5pfjsfopplof moment of inertia I is dropped onto an identical disk isp*1g 5,82gopu pk1efflj5k rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 -anvv09ig+m2qqr7qb $rev/s$ 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 stana1m;c6d7kboe*pj*w+ 1w cwuwds at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920 6m awc*w;1bw+ o 7edk1w juc*p$kg \cdot m^2$ 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 n+dvw7n8 -nxo rotating freely with an angular velocity of 0. dnvo78wn+ nx-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 coll,xr kad1 6;)z kdvuv7bapses 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 massvkurv)x bad1 dz, 76k; 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 excesn 78nneejce -0s 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 (;/6dslcpu ; 5ury jr fa:b5ws$ 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, initiald.mv4;ti ndh(ly at rest, that can rotate freely without friction. The moment of iner;in(.t m4hdd vtia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5- hp)sl7/yn z:wm diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment )l wz7/h p:nsyof 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 o 2 zs/ r52x)(hjphcapmn 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 u2acp5/)(2sxhhrj mzpnwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such thez+cw*e)w1 a gat the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbitwcw)*+ge1ae zing the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate 6fburydo.3m / 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 twjum / ,r7m ;fad+tjeu:spc ke4n(2 .khe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from resr:e kpf2jumwan+js 4d ,tk.uc/ e;(m 7t 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 ts4km l5ja7* )nx:igc fwo 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 diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m5f4xnj)l asg*k m7:c i-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 abu5)r5qlhu 0b-+e ihj:8 7rg6hwywxhngular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our +dbnz3.ze *cn1+t 2to zpj4pv Sun, but with mass 8.0 times as great, were rotating 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 sphere at all times, an2bz4zo+ep3c1 *n.nt+j dz tvpd that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution twu2 9q)e*4xkvra8ek automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uakke*ve9x 8r4)t wu q2ses 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-m,yjr)emu /p7t a c+k $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3.d/tqtnqs3pk7ai (d3ht cd* 2-3 $m/s$ 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 az-h i 6f, 8 u.rhv9ywaow-/qofhg2c1ond 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 horizontallyfh,o9vr -fw1y.2c owq-h/8u6 oighaz and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. 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 the floor, and we w;,o5px.sr ch aw0u g;uant to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has heigha;u;ucwgp h .05oxr, st h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn we, qw0ae)hjf:46t.rj mj2 dqw(4sd fdith a radius The forces acting j.,jqd):w4 fsewrh4t(0jqmdae26fd 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 and begins whirl foii ,k-yf:n5v+i)rl u 5s/dying the rock in a 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 feff 5 dl 5v ,ri)+iyknso/y-:uiat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her a5k gj-a),/vl pvhzok59a :u dnrms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched ar n5) ojdvp5ka9:gzkua l,-/hvms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two o(1vr r+esf) x-sb0 c zi(lzvya-c*v1cylindrical plates, of masssc lif(zx11r+)b -*s v(vaer ycz- o0v $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 at :aa n c+85e1q+llxdxlong the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marbllxtlq1e: a+a+n8c5 xde 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 assume 3zolds d0f7lmi;)5m w$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unif:upuyr x-ac x6b 8c.n;ormly fromcbp6;ray:8- xun.xuc 90km/h to 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