A bicycle odometer (which measures distance trace,ns-tvfc- 0veled) is attached near the wheel n- s vtcfe,0c-hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on th /3nkcl4kz:;/tgjr ro e 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 eit3rr:j okc;/l/4kg nzther component of linear acceleration change?

Could a nonrigid body be described by a singlxg a)v h lnjr/q +d d/..*s-n9rtnms)se value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a lpe gg.c q7hs -4hi5i(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 swxz ),k,tc +lwusr )m1it-up with your hands behind your head than when your arms are stretched out in fron,)+s1l,)kmztcx wu r wt of you? A diagram may help you to answer this.

A 21-speed bicycle has sev6j60 o 4-0oeacnagbw ten sprockets at the rear wheel and three at 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 harder to pedal, a small front sprocket or a ntawa046ocjeog b0-6 large front sprocket? Why?

Mammals that depend on being able to run fast e2 74lo 1 rsth9-nsmw twns;o1have 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 distribution of mass is adve;-2ws 4 st1o7 mo1h tw9snrnlantageous.

Why do tightrope walkers fnhpj3 s376w 1tyxpf9 (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zr 8og.4sgj:fuero, is the net torque also zero? If the net torque on a system is zero, is the net . fjsr8g4ug o:force zero?

Two inclines have the same height but make different angles with thz0(d,cynxqx6ml0 8pwja gb: wn/5ze9 e horizontal. The same steel ball is rolled down b0xlj:56 eyp,nc q wg z0dn/(a8wz9 xmeach incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. qp5ur iw 3,t 2.gl6t*4mx(yerb)c jmrOne sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the inclm6(xqe r4 t. bwrcju5 l3my)rt i2*pg,ine 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 massyj egr:9uz3 ww8dy.e;. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greaterd9gjyz 8ee. wrwyu:;3 speed 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 conserved. .*yr 2yelnlqcs *g*t;n.9ax qz 2nl4;jdpr5a Yet most moving or rotating olaan stcz92q. q*;yd* gre ;4nnyprl2*5.l xjbjects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s g fl1z5lqs2 w5wjfj8+ly )o/f equator, how would this a1 +5jlj )g8q/owf5z2f fw lylsffect the length of the day?

Can the diver of Fig. 8–29 do 6p9 a/ lfiqug;a somersault without having any initial rotation when shfiqla9;/ug p6 e leaves the board?

The moment of inertia of a rotating solid disk about an axis thrmma9ki76s dbgz z0 -/yough its center of ma96 azb7mi k-/ 0ydmgszss 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 inbwa 0 ,nea-b)d each outstretched hand. If you suddenly drop the masses, will yo)nbbd,ea-wa 0 ur angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and hav+py*u g7m a*fl: bqqh9e the same mass. However, one is hollow and the other is solid. qyufg*7+h9p lm aqb*: Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle p+2t;x p7ir znmoints west. In what direction is the linepn z;xmi7t 2r+ar velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large frez8w9+)r cww5hl4b n he)lc /yeh7k9jwely rotating turntable. What happens if you wah)ykc 9)j+/h zcrelw9whwb4ne 5wl78lk toward the center?

A shortstop may leap inem.)c-m-r bt h4bi5wato the air to catch a ball and throw it quickly. As he throws the ball, the ae .th-)mm4iw5 rb cb-upper part of his body rotates. If you 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 angular momentyz :.yfc)fb h1um, discuss why a helicopter must have more than one rotor (yyf1ch.:b )z for propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following anp2 fuq,v mv52+(zq vgwgles 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 ofj q2 3lja2*ybg an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diametglbq3y2 j2j*aers (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed v8nzm* /+h qspat the Moon, 380,000 km from Earth. The beam diverges at an azm /hv8sn*pq +ngle $\theta$ (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 65fift3893 ot p,k:lrrs 00 rpm. When the motor is turned off during operation, the blades slow t :,tfkr3p3 it8los9rfo rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on+ 9 cuu+idijg, a level floor 3.5 m to another child. If the ball makes 15.0 revolutioniu +u 9gi+c,jds, what is its diameter?    m

  A bicycle with tires 6pq- i,g5n 7a7;ksrke r8 cm in diameter travels 8.0 km. How many revolutions do the wheels make i7s,qr7k 5g-paer;kn?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2v2 ah 1udwzd08500 rpm. Calculate its angular velochwd8d a z1uv20ity 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 rev;38gxcqfj gfi n5/8 mmolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the fq8mc / fx5gi;n83gj mcenter?    $m/s$
(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 em8)uu td:r2bv, 7 n0emrmq:iorbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of zr02 mqilgcq x1kc6.)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 parti f0 bcw7m r7/z9+yc4nwxuahmn3o0 p /fcle 7.0 cm from the axis of rotation is to experience an acceleration of 100,03 7bcucx mw/h04m0f /7frpaow9+nnzy 00 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformot a+hcrkn*cm4 - b1;vly about its center from 130 rpm to 280 b;+ knm4vhoc *-c 1tarrpm 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 ubf:p /)zp l9ud*yr 7a w+-kin$R_1$ 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 ab ddsg e16f+9mhnv++p eoard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the stae+6+g1hpd+de nsvm9frt 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 uniformly from rest to 1i ;t++38xyueqnfp yf4,9slyz5,000 rpm in 220 s. Through how many revolution++efnyqs38py, xyil f;z49t u s did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm i8adj9rh7sk4) k r kjy6n 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 stre.io.hvdpt2w ,l0l qv6sses of flying highspeed jets in a whirling “human centrifuge,” which takes 1o , tvp0 ..whv2li6ldq.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.wgdb)vy u r.z 1w-u;e93hr.ax5 s. How far will ;9 v.xwur3.yebgzu1wd ah-) ra point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turnedb4mr v6-wqn7vghjs3 +px j5j./cgey +ovr4b- off when it is running at 850rev/min It turns 1500 revolutions rv-q v674po wb+/ebnmcg 5s+.j r vx-hj4gjy3before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the ca*6h q x7fym*klr reduces its speed uniformly from 95kfq*h6xklm* 7y m/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 ul+p/jd0 jp u8revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The8lup0up d/j+j 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 ezj*u 5o9itf 80/he4 hf9rgexeach pedal when climbing a hill. The pedals rotate in a circle of radiu*tz/5fhg8 f ehe0 9ejuiro49x s 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 olti z55cubslx.nb 8t*i04b.mf a door 74 cm wide. What is the magnitude of the torque if the force isc* 5x. bbisl 4nl.tmu5t8i0zb 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 torq)h26 u-(hdg+sra mnft ue about the axle of the wheel shown in Fig. 8–39. Assume that a frictioan hs6(-dtf2mrh)g+ un torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends ofdw 8 )dl; xy)dkj3,x,cavqeg 4 a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horijd d ;kly,gxcx3 )v )weqd4a,8zontal 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 torquel* 0j4faxrm k/ of 38 a/mkx f*4jl0r$m \cdot N$ 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 43 j7+qgts0jmxq- ixn sphere of radius 0.648 m when the axis of rotation is througqxmjqs g ix7j4 3n0+-th its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 c et3j3g9:gwn 6ebd va8m in diameter. The rim and tire have a combined masa3g69dbj :e8v twneg3s of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on u s+otji9x.iwv04 8fithe end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculatvwxjsf0 u. i+i8it94 oe
(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 constant angular qyztj5,v m4;)mm j) ecspeed (Fig. 8–42). The fr mq4jvtjmcm),y;ze )5iction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown bmnt z.wot97 .in Fig. 8–43 z.9w 7tb.o tnmabout
(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 whose total massd 45c ec)tcjieop*76 g is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the 4k tu;kfal3pn5tiq;c qe- 4 w4correct 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 rcqpu44nw -le35qik4t;fka ;t equired 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 cylinup7a(hmp:( y etze9t4der with a radius of 8.50 cm and a mass of 0.580 kg. Cz9p(:(tye4utea 7mphalculate
(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 swi*v:cgp:iyf( bngs a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-drivmxz9v (s -k:vl .gkmp.en merry-go-round and is able to : m k9svg.k.mlv-xz(p accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor roas7edqv1l .w51k5z7 )0 zuv 8brkmpy atating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictionalpq7bm vs8ua1 k.d5wkrv5z e17 0azyl) 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 ba *5mpc:0xn*+to 6eso (dkcaswwm3 u3rll 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 throw cfx76l)x:e z,qbfmt) n solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. Thexec6 xffbtmq7,))z: l 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 considere0cw2h o5:h kszd a long thin rod, as shown in Fig. 8–46.: hso5c0hzwk2
(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 90y: e1ylf* ctfmjbq/of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turns (z p.k37njx gg,revolutions) and releases it at a37zgnj., kx pg speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inakf yizel*+(m4)q,a nertia 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 torquz a54yo ; g6fecn.z9cte 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 witgz *4medm4csog9:0fkzp 8cr +hout slipping down a lane at 3.3:z 9d*ggezm f+4rs8o0 pk4cmc $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with rewqz)9 csslw 2q.s5+prspect to the Sun as the sum of tw9zwprs+sl)2cq 5qw s.o 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 maw) 13c79si *o fuo7gglcw3tswss of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cyli7* g 3cfw3olog1s us 9)7wtciwnder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts+(m :pwi)vr v::vicsr from rest and rolls without slipping down a s rp+:: ii( )vrv:wcvm30.0 $^{\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. Itwje:i 4td m8gj gd(0;i starts to fall and its lower end does not slip. Whatii:mdegdj0(j wt8; g 4 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 ball rotating ontgo+znu 3vz4r* ,z0hx the end of a thin string in a circle of radius 1.10 m at an angular speed of 1z+gr4,o3nzxv0 z *thu0.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grink5sxk;.d jq gn+ /(vdxding wheel of radius 18 cm when rotating ak/xxv.ndsq(j ;5g+d k t 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 iskp1oc53aa 7nqfu r7 hma(a u*. rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of ra 35qa17 no7fu(.*hrumackap otation 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 momdvyu-v.af*98wh f ;fz ent 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 tuck position, why* f w-fzvuh9;vf.a8 dat is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increa8 upwj 8tq44sxse her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate o4u jpw8q t84sxf 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 a9 6v1vu5hrrz; t2jdr6g0s fwqr-kt - p 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 wheel after the c1 t2g tfj0pv wuh- -q 6;zrrrkdrs569vlay sticks to it?    $rev/s$

  (a) What is the angular momentum of4cwm0dqwee8(6h f d6j z 3eu:r;yrra8 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 momentum of the Eaem c:yaa080jm o6bb 2hrth
(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-cz 9e a.iltz1 disk of moment of inertia I is dropped onto an identical disk rotating at az- eaz. 1i9tclngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 1dl540 skrg rhkmx+f(vg p3a *ex *a4u2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round plat9; ,c/m pznungform of radius 3.0 m and moment of inertia 920ugn9 z/mnp;c, $kg \cdot m^2$ 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-aiyq;,bimzcy+ x/6.n go-round is rotating freely with an angular velocity of 0.8.niqa /xb6m i+;zc,yy $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dw) zzadc)gy w2ufnve5 t5* u9d8arf, losing about half its mass in the process, and windingf*d5u8veuy)wz2c a5 d)gz n9t up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation 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 tkb* ,rpw(iin.vuytj:dl80t m:e l*/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 xs m5c::xaia*$ 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 platforiq+ kaw3.- wf xe -)sw,c8ubmsm, initially at rest, that can rotate freely without friction. The moment of inertia of the per,qx m3wf a- si w+wscb).-k8ueson plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round nmv (c7biddxi5/z;v /turntable tha cxmvd75zbn/;d v( /iit is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls o guc3rlmeb ij3txm( xqv9s.6 i0d6)y/n the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person withoutcyx30ev9blu6/ mj)msr3xi dit.g(q 6 slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earthjq8 he fxid78-ur: 6va. Determine the ratio of the Moon’s spin angular momentum (ab a6irvhq:-xf ue88 dj7out 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 rest at a rabzvdf n1x7:-fib9 sq0 te 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 rvznae3 -w7hc,.h7d fg adius 0.20 m acquires a rotational rate of from rest over a 6. f7d7evhw cna-.z3gh, 0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cypbohcs1v i1 f0 mpde7sf58a8dd ben ;+:hhg;2lindrical 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-m-long string, if it is releasedbeogm;dp v1h5 ;h f7pihe+ 8nf s:sb82ac10dd from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular s)1 f -y,h;l ;stjamqua,n a7axpeed 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 sizec 95gd.y *j)a9c3obb * craeju of our 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, and that the lost mass carries off no angular momentj cj3ug .b oacyrad9ec)b* 59*um.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobi8z/me we o9rdb20gz(hle 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 e/gwrd(9ze 802z bhom 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 illustravvj) zxd;;gfp-k )ik6w8v ,ma6fq6l rtes 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 eli9aeq(0pa w hy-b: g+p3:jkon a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frictio+b+2id/mm3qnu ui c;dn) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod,i2m/nd cbumu+d i3q +; as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertical o-ukrcse u/9;ly on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The 9;r-uko cu e/sstep has height h, where h

  A bicyclist traveling with speed46l hn syllw o2.q-nr- v=4.2m/s on a flat road is making a turn with a radius The forces acth.qyll-wrls4n- 62o n ing 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 into861r0bdkqms o, dk +7ag g olyb;z33oo a sling of length 1.5 m and begins whirling 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 0oqo68o+yl,g3;zg rs kkbd a7b md13oachieve 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 7wil7 6smkdy+wo2l ;8s(q th was thin rods, making reasonable estimates for the dimensions. Then calculate ;il+77 tk w28yw ld(sq somw6hthe ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clut 48cbv m pl:pumja 5ju,e3i82ich assembly which consists of two cylindrical plates, of masbem4i88upl5,3cp mi2v jj:a us $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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. g g vziukv0vgp0lcl1,0mfpe j q*;v9) va4h84 8–58. What is the minimuq4pli;1*ke9za vg m g) vv4 jv0hgvcf,u80lp0m value of the vertical height h that the 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 assuu1wq ply1ku96x,lh )vme $r\ll R$ h =    (R-r)

  The tires of a car m1ai7/0qixsen05 3aget ll 0gn ake 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have30nnae10lg g5itq lia x/0s7e 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