A bicycle odometer (which measures distance traveled) is attached near the whe ik paqbp(x9z)pa/7,e el hub and is designed for 27-inch wheels. Whpa9 ap ),kpiz qb(x7e/at happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a pw49hfhfrz w p(n(0j5b8 m: psooint 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 accelerhnw wb4fsrf(p8mpoj59z (:h0 ation change?

Could a nonrigid body be described by a single y g 6 uw,nfk*wbo-6-umvalue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greatw54q1 nql8qoe1/bt uber 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 hands) jzir5m/9 kmc behind your head than when your arms are stretched out in front of you? A diagram may i) kmzmjrc9/5 help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at tn+4 (/pvqzh ael bm08bhe 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 smaavq(pbe m b /+z8hn4l0ll front sprocket or a large front sprocket? Why?

Mammals that depend on being able tob g*0 h dzli,g3zaa0q1 run fast have slender lower legs with flesh and muscle concentrated high,0,h0 b agz*1lgqai 3dz close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, ngg s-+chfq0ipes,5 0karrow beam?

If the net force on a system is zero, is the net torque also zero? If trycpn-v*0 + ohlya:n2he net torque on a system is zero, is the net foyy+cn *: plhnar20- vorce zero?

Two inclines have the same height but make different angles ph41pq j hw1k3rwo/ w+with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be p 4/q1 rhh3w1+kwopwj greater? Explain.

Two solid spheres simultaneously start rob0((o;rewgl6ew nw9a+h l e)leg5y. s lling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline 5rw;h e+ 6b eo9lgww(gl0a()es .nley first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder 6g2 bh qarpx.8v,idu) have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom firsipa2 dg. v )q86r,xhubt? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum azj ffb)x8z ab -cu yh+hg(:d+:nd angular momentum are conserved. Yet most moving or rotating objects eventually slowbhbxj f+gdz ::zcf-ya(u )8h+ down and stop. Explain.

If there were a great migratig* g+f6+jckyu on of people toward the Earth’s equator, how would thisgcku+6g +*fy j affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotati8ciu )1ab 5sxion wheni5 sic1a u8bx) she leaves the board?

The moment of inertia of a rotating solid disk about an aejgn86 lis47gxis through its center of mel6ng7ij s4g8 ass 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 in each 1lbjoeoe+c*hh (:2yzoutstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explaio* e1ec2( hzy l+oh:bjn.

Two spheres look identical and have b okd/5k*ps*p ln;,xm the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is which.; p*n,dblkx/ pos5*km

The angular velocity of a wheel rotating on a horizon7r ,yuy -gt,0lor 9jbbtal axle points west. In what direction is the linear velocity of a point on the top of thebj-70rly,y,9rg u ot b 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 8zcwen5 8 qv(9qq bo4nthe edge of a large freely rotating turntable. What hownezq8( c8b94qn v 5qappens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As h4my+r 0 59/b9aspsv nncl:ly ae throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate i4v+9baspls5 9:n0alyyr nm c/ n the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a o02 me x3 *gcft( :oxl0zik3afhelicopter must have more than one ro20fax: 3 0ooz xgcl(mftk*e 3itor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in 79vl su - mlq5ylgxr*/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 ;n 9)kx yxsy)vv fkb-:v fjdtg:3 ps-+of an amazing coincidence. Calculate, using the information inside the Front Cover, the-vy 9) t fvsk:kfpv;djgbyxn )+x:s -3 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, 380:5d0lpjv7 pq2ce jzt *ro64rqgm p5q,,000 km from Earth. The beam diverges at an angle ptq4, zpd 6 75rpgo 5:jr0*q2evmqjcl$\theta$ (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 rqcjh:i1 .d(fgpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slo.gi dqfch1: (jw down?    $rad/s^2$

  A child rolls a ball on a level floor if n(c 5s;uo 3d/1gvna3.5 m to another child. If the ball makes 15.0 revolutions, what is its diameter? c i a1vn ;5s(n/3ugfdo   m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How r4ntdtd.: tg ae0c3 df*2 dc)noc *-vemany revolutions do th0. n*cav 32 d detc)c:t4e-d*rfnogdte wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotay+ud ;8m1e7vpy1xtt4 uyl f9ates at 2500 rpm. Calculate its angular+v78l9 ttue 4p amxdf y;uy11y 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 completewp2gf . ,f7cqcu5a2eg revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seatu cqfg5,2c fgpe7a.2 wed 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 velocitmi .k(f6k0 vp av*d ;d ).u ziuepx1af8cli.c3y of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a p c texe4,83xtw x9i+oo/r om/hcj. ;jtoint
(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 +(9xihb)cai 3s ;fv33mhjqrd pb+ lx32 vv6nurotate if a particle 7.0 cm from the axis of rotation is toai +f v b 33xjdhbsx6 (r)+3lvqpn3imhvcu9;2 experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 138y8rwyscx,4zgb2rc s s:s9t,0 rpm to 280 rpm in 4.0 s. Determi4by xw2c 8gc8s :r9,tsrzy,s sne
(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 radiuuuu;o y;w /f*w(1bjxq s $R_1$ 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, astrona h( urnkqn n(qu9xrpga,)2j fel0m0)(uts aboard the Apollo spacecraft put themselves into a slow rotation to d0nerjmqln, r)x9 )uha((( fp ngku2q0istribute 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 uniformly from rest to 15,000 rpm in 220 s. Tl.6c;xvimqv *hrough how many revolutions did it turn in thi.vi*cv lm6q x;s time?    $rev$

  An automobile engine slow ,; 4zbion6+h c 1 wqsxiii54qrd1apc4s down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirlxhdia )yh 3c(cwuttf- d r) mqk4h/o6x8 (w1k9ing “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speh8x (k1k3fcq)i9rdhcxwy am t4t )-w6 ohu/d(ed.
(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 2ob- yre v77dz540 rpm to 360 rpm in 6.5 s. How far will a point on the eyz v7r7o -5edbdge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min 68ry1g os wu9ux9of8k It turns 1500 revolutiongk 98usuoxo 86 rwyf19s before 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 car reduces itjdf:,m el;d:-a0 qzqt4,srtlw * mam7s speed uniformly from 95km/h to 45km/h The ql:f:mtw*4m7sj e, rzdqmd, -;ltaa0tires 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 mu.go,f2c5 .czwd3g tgake 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diame,gu o53g gzcf2d.wtc .ter 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 h0cumk pcbbu d79v.;6l er weight on each pedal when climbing a hill. T.c6l7 9du p vkmu;0cbbhe 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 74u7g vne fo0)c7kaaz/8.dl;zt cm wide. What is the magnitude of the torque if the foo;d/uztazfce7.k 8 07)gvnla rce is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle ofj dk9)3zdfpn *mm91h, f7jjle the wheel shown in Fig. 8–39. Assume that a fricthn,p9j l )k d*f3j9e jzfm71dmion 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 of a massless ro:kks 27.jm/ vzsx4;ffs d1wdc d which pivots as shown in Fig. 8–40. Initially the rod is held in the hori/1csfxkz f2sd:jw d 7k.;s4v mzontal 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 3o48q/q ls)v a kfbzl/tightening to a torque of 38 qsk fqaol83/lv)bz/4 $m \cdot N$ 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 sphergaf 4y n/ul;r:e of radius 0.648 m when the axis of rotaru 4;/yf:ng lation is through its center.    $kg \cdot m^2$

  Calculate the moment of in:(ny powd/,jg ertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combinen/jg, owp:(y dd 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 a thin, light rod is rotated in a horiz)(x:xm yruc4a( bxc +wontal circ:x w c(amxryxub4()+ cle of 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 constant angularkk4 t0h bx ,iquo/alsr:05t-r speed (Fig. 8–42). Th0k0o l tqiu/kr axr45sb-, ht:e friction 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 ofwd w4v8q2kw7p g b*alj /2pjq6 the array of point objects shown in Fig. 8–43 2wpbljq46v28/*w p7jwk dg qa about
(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 ox/pb3lv:qw6 (bjtjlx) ygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the corre ; abbm9c9g63 +q*y j/qnudtsuct rate, engineers fire b n/qu*9qbt m;c d9s 3jguy+6afour 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 uniform cylinder l :e mkk w-9hv:wbu20jwith a radius of 8.50 cm and a mass of 0.5:j0w 2ukmwkhb:l -ev9 80 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, acceleratex2+ce/cuw 0k( hjd 1aing 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 han 4-g1;r mh8cj xrw6o *ko-nwhzd-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. Calculate the torque required to produce the acceleration816o-rrwmhn x*-jz;4 kch owg, 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 brou83n*niw: tfxm*6iv pfght uniformly to rest by a frictionanf 8wixt fnm3*6: v*ipl 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 accell 3t za y/m5v2of2pr3berates 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-kr-zv+e .q.xc c9k9 fzxg ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 .qvx+e 9x- zczcrk f9.$m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. nudh;; ifob:5as .z3sep6w8ft6qoes31b ,u w8–4qh36w, f e6;dbps e3st;ozfwu:n.ou bs8a51i6.
(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 consij*ct 6 0n1cbmdp0 yn(dsts 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 acceleratplt9:xa0f l)mh9m0s m 5;ert tes the hammer from rest within four full turns (revolutions) and releases it at a spe5xrt e:;lmfa09lpts0 )m 9h mted 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 inertia - d p4,kbv.8es(7gxeqos+ jyt3 *kv qtof $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 2805f s)mma.0vuy $m \cdot N$ 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 without slipping downz)pal uu5t3nks 0a/fq * ct*)f a lane pz0cu*k) ulq tatf*fn)3s/5a at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wiq uqf3 ov0vpn 0mx8nwhwx+(;)th respect to the Sun as the sum of tw( 0fn )o8m0;v+ vphu qnqwxx3wo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius ofeekd n9.986a yz1( r3jte dsackhin1 4 7.50 m. How much net work is required to accelerate it from r31tyri6a ke z.d1n n89ehjs a(kc49e dest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest zs,+ js9h-gljhpw27 m and rolls without s9 w+sgjpl2jhz -m7, hslipping 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 balancedn o)m a .,+;5ryk;rg9 :fvcrjxsyo:b+st. fj v 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 hi:fgnj a ;9;ostoksy.xf). :+,rb vr+rcj v5 ymts the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotatin;srx8uci rk0,*ae+if 79uu axg on the end of a thin string in a circle of radius 1.10 m at an cs*ai,i a;exfk9ru 0u7 xru8+ angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-k-vlgvm zp;;4;) wzs sig uniform cylindrical grinding wheel of radius 18 cm whenpv-;m v w;z4gs;) szli 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 platformm 3x-v xec.k:cn .;du7lrfj(x that is rotating at a rate of 1.3rev/s If he raises his arms x c jef(x-k;mv:..u cx3n7rdlto a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) cap- 0mt;ny9uk+zofgm e fjhlt 7g;s1;9n reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes kg99f gj+mt s-;mu; ; z1t70pnlfeyoh2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initialw*ww d,9zcl (d rate of 1.0 rev every 2.0 s to a final rate of zddc(*www 9l ,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 thro1quu0dwr97 n lugh its center at a frequency of 1.5rev/s The lw dnu7uqr109wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figyo9ftwp7 (3j z-hvdy +ure 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 momenklwhu aa +ve1p 8:m/(xtum 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 momef8foag t m7,idj xp-.4 6c;vsunt of inertia I is dropped onto an identical disk rotating at angul;jfpg,m -itsfx.a 67uo8vc d4ar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 26n,8wggi;:sl r0lia:ly s 3/alcq )ea.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 platr2jit31g3 ibeform 3g3 iertj12bi of radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotyetct*sx) p+, ating freely with an angular velocity of 0.p)x+sec t,y t*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 collapses into a white dwarf, losing about half itsuv8 h5*3bu1 ctno *4x(to bdrd mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what *(b ovx4 butdrhd18c*n 3otu5 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 excess of b8,gb ) l(0wvk56t+ sveaerevwqb(f 3f(vi0v 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 mj;s( a .f6 +vjko)cta$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that cph(b /b*x2t/a ,1rz- ng dyzscg3 iio,an rotate freely without friction. The moment of inertia of the person plus the platform is xizto1 g*ni(3/,-2 gpscy/dazbb ,r h $I_P$ 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 edgyp1r-ci04e0ivy3uf g e of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearing0u3yrf01- 4gpicvi eys 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 ropel4vzh pdxtk5 u/4py5p 5* xx1t rolls on 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 u5 1 d l4ytp5xv/p xph4z5xtk*L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always fazhrsb/ lo6m jo;*gfbb s10/,)u/8aa ngnzy/ yces 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 orbiting the El0 6/m n;oy/,zfaozb sgsu/y/n j1br hba8g) *arth.)    $\times10^{ -6}$

  A cyclist accelerates from res:doa :vjyu(3et at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of 9uh;jn lkt,7 aa uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over 9u,7j;ktlh an 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 two solid cylindrical disks, each of mass atmaf5-c( :jq 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 im :5-c(f atjaqs released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the 1.svnsc v5)b3) ctt 1ll)xkkj 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, b)gwtf/ge2jx+8 qb 6+n2i eru nut 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, losifqg)gntj8/b+w2 6ie+eu x2rn ng three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy storej 8 l+mqygsw8.d in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindricalj m g+lys88q.w 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 ann 6s/ cn81g:re4slt g8i1y vf3inp y*c $H_2O$ 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 aip /czsi(x34z1m: o5g-nnp a it 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 igs:f .2nxwg- ornore 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, anw srgxnfo:.2-d (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 fg. c -5q(gg8hqz(q eiradius R. It is standing vertically 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 step has heigh-qef5g8 gh(iqcgzq. (t h, where h

  A bicyclist traveling with speed v=4.2m/s on a 0dwj; d8mel 1jflat road is making a turn with a radius 01jdl8 ;d jewm The forces 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 0.5cg vlxr2+ -mb.0-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerati.l+xr 2 c-gbvmng 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 skatej 9z8wxd1 m*bee,wjl:*q57 al*zcecur’s body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with armwz7cl**jeu mwx5q*jl, bae8c z d19e :s held tightly against her body.   

  You are designing a clutch assemb4hpjuhab+ o51)hx-o 1lhri, zj/1 tzkly which consists of two cylindrical plates, of z hh4xu-+5bh ohrzijpk) l/1to1j1a, 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 alonm dg6(p-r(k mz*ol9xccz7 sd+g the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop withopc cz do6dzskg((7rl- *9+mmx ut leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, butelbaq8g asbfkgy4o .f (v*+2 tx. 9h,a do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduceh- t8 pf.4oy s9hrtgy9s its speed uniformly from 90km/h tohso 4 h8pgt.9fy9 -tyr 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