A bicycle odometer (which measures distance travgg)-ce:zj)zw pm k67r eled) is attached near the wheel hub and )-)gjmz:ecpgk7 6 zw ris 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 poa(aoloq*, 0v5aaz o (tint 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 cal az,(vqt o5*(aoa0oases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of th:nn .gi m- bn;,pmj5jje angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Expleh gqtpz//o ,me:+ f-aq77l ntain.

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

Why is it more difficult to do a sit-up with your hands behind your head tha0o/x96(wt r/ zf. 7gngdxphwpn when your arms are streto6n/p(09wzxpg t f.xwhg /7drched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets atei9t0t+ bz6 zd369 y1dcexzyayzql7 h)h,ip: 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 p6) dy9t xaph: qzzd6ehz ,byi1i0yzlct39 e7+edal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on be.q0yf bk+ju;bing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the f;ub qkjy. +0bbasis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkersc d5z2cn 0 5rtych-uv, (Fig. 8–35) carry a long, narrow beam?

If the net force on a system u 2+lu5 otna*orcrr9v1c ;, uf/jtf .w3q yzd-is zero, is the net torque also zero? If the net torque on a system is zero, is c l; rufv dtyf-92qu+aort.c,u *3rj5/n1w zothe net force zero?

Two inclines have the same height but make diffeav u7 g8htjvz/d6o,k7rent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of7 ,z6u8vjav / gothdk7 the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) dow*rbufc8g26 4.dei.bx:as.rz 5gp m h qn an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has thubxd. s.fq2.m gh5grrai:b 84zp6e* c e greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder haj / mpzyj( -q((hnnlc*ve the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom jjnp z -*c/yl((qnh( mfirst? 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 and angular momentum are conserved. Yet most mv:7gol*qz lt0qtpw 4+i.j ( gyoving or rotating objects eventuallj.tqvl w *7ol( +p:04gqzygit y slow down and stop. Explain.

If there were a great migr4/q,h-6,1 lghhudlc,feb jdk ation of people toward the Earth’s equator, how would this affect hfu , 4,bl1qchjlk,hg/ -edd 6the length of the day?

Can the diver of Fig. 8–29 do a somersauldb 2p v-t7w0ny+5lvptw3 lz-ht without having any initial rotation when she leaves the bw5 pwz tt0-d 3-2hy7l blvnp+voard?

The moment of inertia o+n0m(y yedk46t ad;kjf a rotating solid disk about an axis through its center of mass isj+ ;06 knke adytm(4dy $\frac{1}{2}WR^2$ (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 roti 66fx he/+iphtv 3 .doubu//uating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, o xth/ueo/ud.3 f/+iuv 6 bpih6r stay the same? Explain.

Two spheres look identical and have the same mass. However, one is r3z m1fcp5/ l* fo2ahwhollow and the other is solid. Describe an experiw* lf/mfac3p5zr 1o2h ment to determine which is which.

The angular velocity of a wheel rotating on a horizontakyb6,6u q)zcll axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleratioz 6b6,q)ucyk ln 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 e//grt)ay4*zf l 0i8a(smhw.8slk mw freely rotating turntable. What happens if you walk toward the cent )8sz8hlwlr*atf.kgea0 4mym(w/i/ s er?

A shortstop may leap into the air to catch a ball and throw it quicklu6 c -;.+ msf:--ojplcwqvf uxy. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips andf+6v l-fs- omcqu. ;p-cu:wxj legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momenv/x5j;ms9ym.o +tcuth :9 w .zxnbrg2tum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the secondy.jox9;ht+uvs :m5cxwnbtg m 29r ./ z propeller can operate to keep the helicopter stable.

  Express the following anglj438d 8 t ua25gt-rm nibp9ltqes 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 o+lkkhsz :o5gz8 b3 f7zf an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen b z3hk+z 7lsf:ogzk58 on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km frovs nx*,y6nha7t* k 1gjm Earth. The beam diverges at an anglea1 h 6txnng**7vy,kjs $\theta$ (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 td2ei2::d vp m6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the ang:2dde:vtim 2pular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level fl 0t j)a, bdpf 2:eie v:2a(vnr9bh2g9yjn1szooor 3.5 m to another child. If the ball makes 15 2gd:2p sbt1 ,yh2r aj) bezfe0:ivo(9avn j9n.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revol 2sqvgj sz)ridh3uy-65 .,exd utions do the wheels3deq5jg). xuis ,zhs rd2y 6-v make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculib ry7m(n4ukdn5rl 8;xo:a: figp c;6ate its angular yu k:r( 6il;cmb4fn7 oxrn8;i: p5a gdvelocity 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 rec nrg34* tg:zqvolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of zcng rg4*3:tq 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 angularemk2phn ai*( z jj;c5(wz(jp 8 velocity 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 scmys3h-pp(3/kdqa 26 yu9pxp peed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a partk -5rj)y kjbu25v, fyuicle 7.0 cm from the axis of rotation is to experience yu )-kfr2ykju5jvb5, an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly a+r1u/g bm5jgo-j 5a p llii0amw5 4ee))+zwaqbout its center from 130 rpm to 280 0m jlo+/a +iee)r1a -4izwpwl5 ug5mg)5j q abrpm 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 radiusf)xwvs fl xq9z+ 3qy3r8wka *2 $R_1$ 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, astxx2,f ; 3 7;mhgbimpmwronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’;m x2f;i3, mpxwmg 7hbs 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 2 -cdzho3j ;n2z20 s. Through how 3-hncojz;zd2 many revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm g5kfbk qh*; 9l7pvpl1to 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 ofrgg:y qa z9: 7yyyh8kfi/ 46a fam8k0t flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn throuay y7y0/ t azgqy ffikm 6:9r:488kgahgh 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 nv-3q qr)b6un 6.5 s. How far will a point on the edge of the wheel have trr 3)n6u-qbvqn aveled in this time?    m

  A cooling fan is turned off when itl0:8o r.w r un125jowkroi/at is running at 850rev/min It turns 1500 revolutions before it5i w.lort/kaw1ru:n r0oj28o 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 redq mg- n:g9ha z4+0fsmuuces its speed uniformly from 95km/h to 45mg:usz-0mh94g+q nfa km/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 revolutions as the car reduces its speed uniformlydw agcx4zo2cfa6* 8 dr*1lb8 y from 95km/h to 45km/h The tires have a diameter of 0.8 oy x4d81rf l8dbw6**gc2zaca0 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 bi7v8nu92etqz;y0 f j 6nv8hjkike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of juyefk nvh q8n28iv0;6 jzt97radius 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. Whatpqjx,6 sm zlck+1x: w4 is the magnitude of the torque if the force6c sx,zwpx+ml j1qk:4 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 of the wheel shown in Fig. 8–398xe4 cak hk7,g. Assume that a friction torque of 0 gkk 4e,xh7ac8.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, ac)aaw+qlg c 68re attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Ca6 )aacq+wg l8clculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening c.4 rxc.j3ys o qbgf-j;:h4 53bun79yvjxla ito a torque of 38 :3vj57 g-cbx 4b q4a;rlohu xy3c. 9fy.jjsin $m \cdot N$ 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 radius72 cochpm4 r5j /oaoagh7t7l/ 0.648 m when the axis of rotation is throught7/rah4j57 g/mpocoo l2c a7h its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle( +,he+/nmf 2ogzu ngu wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of theh+ 2(no,zgm ueugn/f+ hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end op9 s(w d1lpp2jf a thin, light rod is rotated in a horizontal circ(w9 lp2 pdjs1ple 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 sq)knb7d -+yl-jqyhw1m v w14a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force bqbj4 w7-yk1 + yw mvnldh1q)-setween 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 theccsw9z01o hz ) (o7flj array of point objects shown in Fig. 8–43 cz9s oo czlhj07 )f1(wabout
(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 mass is ;9sgmlfe/h b j(cr8g3 $5.3 \times10^{ -26}$ 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 ct8x8,k0l ,zp. lixxweorrect rate, engineers fire four tangential rockets as shown ie,l,0 .xlw8xxk8it p zn Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass d-unhl k)njpa*pmid;+) z;z5 of 0.580 k-*u ;zp+;aihdz5j) lmdn n kp)g. 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 ba1oihwxg v1m1 tm. ,*ro.s3 rzot, 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 ta47qnpuvi*r 7ks-m1 r4nuznp, ngentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and hasp*u-ur1 7,p4n4 nv mqkri7snz 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 rpm is shut off and is eventu c- xvwjbb6hgera z0:(y+1,lx/9 bgx lally brought uniformlya1bc (rjzxl ge0h+,l9vx-:yw/g xbb6 to rest by a frictional 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 acceleratei q5q*f5vov 9js 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 thrna, )8e-xte) fmvprb 8j5e,z4 tq 3vspown solely by the action of the forearm, which rotates atbep,e4p)vxv ,qt8am 5zjef 3sr)8 - nbout the elbow joint under the action of 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 considered a long thin38k mogqxvphuhfl8 3 t+ 8o*/mxie*v - rod, as shown in Fig. 8–43oq*/v+8-t oe8pmfmv h3 *u8 likx hxg6.
(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 masb6cw.4xcepp +8diov 2ses, $m_1$ 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 v,wx:,0hjss(*imrnuyl a/ y* from rest within four full turns (revolutions) andxur,y,v * wlsnj sh*( /0i:yam releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia of mcwl xuw7r mxuj6i) kr4d7 5b54jei/.$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develop.63 pg.6lg82iuibz tk+vmoxis 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 7.3p b3r4;v(pm e (k .yakw.77zojvs/hn y kg and radius 9.0 cm rolls without slipping down a lane at 3.3 4reysvyk.w7;(mj n7(p . vkbap 3hzo/ $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with reszcvm y9 /m)u+zpect to the Sun as the sum of twz+ c9vmz/u)myo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net wo)4:wx*v sdqei;;0 ypaexp bu-rk is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it ;spv a4ux0 ; pwbxe*:iy-de)qis a solid cylinder.    J

  A sphere of radius 20.0 cm .mty,+; mri w ;dfreln 2r)q:xand mass 1.80 kg starts from rest and rolls without slipping rem q;)mwxn,2r. fi y: t;r+lddown a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It start hunfnsnl c),e+jv td72lt0 p3+v+hy. s to fall and its lower end does not slip. What will be the speed of the u7 3tdch ut n+yvhp,lsv n+l n2j.+)e0fpper 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 on th72e2 dkpgtt:r5-twaxi3v*uu e end of a thin string in a circle of radius 1.10 m at an angulart etuv:2gi73 pkr5ux*a2 tw-d speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindk 9n,cykj +r4srical grinding wheel of rj9nksy r4 ,c+kadius 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 rotating at a ratw 5fbd2 2kk pxumy mom 80c0m.6w62ydee of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation depc o f0mme.26m6d0yu 2bk5d8ykm xww2 creases 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 h-cgdti/3qqnz8 , ( ntb:grca 4ob1b -ver moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position.vno-,qab8g nd qrtig1ct(4-3z b:/ c b 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 rotation 0jq. tv5tx:zmpy)q9g rate from an initial rate of 1.0 rev every 2.0 s to a final rate of )z:tpy 0vm95 qqjgx. t3 $rev/s$ 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 cy crr + wtyu;/ /ja0vxjw/oe.7enter 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 wheel after the clay sticks to it? w yuxe/7 ;j//.vtjoacyrw +r 0   $rev/s$

  (a) What is the angul.wn:dyae.4y. g/ mzn6pp)k(whbpg 6 lar momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular moment3* 41gjkyp3t et69udn/-zdansrdr0 rum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an ix;5wmq/xn 5j 0cef4wvcd, tj b90l6 qwdentical disk rotating at a;5qw/cb lw5wdjenx40mx0v96 fqc ,jtngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at ka0yx0r* z*u i:r 9avv2.4 $rev/s$ 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 kbs p89b,)j jr tuvd:l/z6ns1merry-go-round platform of9p1jk j 6st8)b ,d/nurzlsbv: radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an afv cqnme91qbq/w:- t9-r7w4zfcmnx ) ngular velocity of 0.8 fb947tqcxm wq 9n-wq1f- cmvz:)/ ern$rad/s$ 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 dwkbelhzd;wq- :b6y d55h0brj* arf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away j5q:rhbbd; * whe5z0ld-6 bkyno 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 it8m(6ib.ukk:uu2fhtk4w3(: d abpd dn 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 n2u1y 92sdkeh $ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotahy,xur l)ph;+ te freely without friction. The moment of inertia of the pers+ l,p;yhx)u rhon 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 merrye) fq 5 ti;exyn)dg*9d-go-round turntable that is mountedqe5)yg ;d f*x dtni)9e 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 on the * rfve8ef2 4( lukcma9ground 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 thu (vmc k 2e8*fafr94ele 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 theovkpyu3l 63bs( ghy 5+ 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 Moo3 5+kgshy3b( uv6yl pon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates f5g+s5)9 hpxqrqm c7hy j3op 2wrom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform h+ 4+) dffjw0r)k yl+7pn 7kw;t dfradbm en/0cylinder of radius 0.20 m acquires a rotationaydn+7f0f +l) wrwtk/d+ )7 ek0mdn4hpf ; jbral rate of from rest 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 two solid cylindrical disks, each of mass 0.050 kg and di9ab2c3bbls:spydc y / b+/tpy-+g+ f iameter 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 g+yla/y+p b3 +2c-psci9tsbd /: fbby 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angulaf g )lh( 3bn.-p1smrlvr 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 Sun, but with mass 8.0 times as great, wer*n4cbp jp /h1qe rotating at a speed of 1.0 revolution every 12 days. If it were to undergj4c1n */ppbqho gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution autommd og- dgo89g-m2kn .vobile 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 flywhe2ov-d nmgodg-g .mk89 el 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 anjle0 emz3k+i1 ,5jqey $H_2O$ 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 sq 9 -wm0wvbpq)-hph* apeed 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 alz-7 w0mpqdl w: a6q0nb:yhs*nd length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the 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. 8–21gylhp67m:wbn - dlwq z0qs0:*a.]

A wheel of mass M has rad-vl2 1 tt) gis6h5 6uvgumpx9qius R. It is standing vertically on the floor, and we want to exert a horizontal for6uhgiu-2mx5p )gq9sl t1 v tv6ce F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat roagiev(:y7v ip4w h-en7mu l9e *d is making a turn with a radius The forces ee4 i(hy gul7ne:*p-79w ivv macting 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 8h .4v.m(k/55 coaz9mvzf ka ts+onoglength 1.5 m and begins whirling the rock in a nearly .av5k.m th8coa smz+(v5/g4kzonf o 9horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rkvs9v6q5o jtk .9e rb u;5;zfeods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinningo.5vjfe ;;k5v9tbzk u9 sqe6r skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which chmoch txy:)6b(l /jv ;pp.5yb (zzn5zonsists of two cylindrical plates, of mas5z ynp5bz6tx/ yb;(.vj p:ozmhchl( )s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 v v7q+6z0wzsj/a t, rutrack of Fig. 8–58. What is the 7v +rswzjuz /0q,a6vtminimum 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 assume,bq0bb( hlqrf *z+jcl*9m9cp $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifp1t gmkle8f,:tm67pjj7, kg t ormly from 90km/h to 60km/h The tires have a diameter of 0kk1gtetm jtp,fm:6p8g , j7l7.90 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