A bicycle odometer (which measures distance traveled) is attached near the w h-9gyva v7h b 5ca1tcjz5+c mx/2pd6oy: 4yspheel hub and is designed for 27-inch wheels. What h1m: hpo56xacvj tz2/ c4d5s7+yp ahv c 9-bygyappens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a poincv3j ;. k6x7gj r1rgwxt 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 acceleration cc gj xx7v .r6rwg1k3;jhange?

Could a nonrigid body be described by a single vc133cmf3 a*- mdyuughy*kia; alue of the angular velocity $\omega$ Explain.

Can a small force ever exert a sjr b*79aaa/u.d0mj e greater 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 7 )y7 4lvp4nlhjpw7yvhands behind your head than when your arms are stretched out in front of you? A diagram may help you to ans j774l)yv4pp7y wnvhlwer this.

A 21-speed bicycle has seven sprockets at the rear wheel andmrze txpu(;b;l (1gq8w- s9vo 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 9;vl8;ptrw(uxszo bm1eq(- g it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender hk3.2om5nw l/ ;k ll5e0lp/ddm0c,ufx +nphvlower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why th2/dul kn;0m,c/ 5hhp 3.0d5fl nmpoel+vxwkl is distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) c4fwk (ot-a)rz di7xi :arry a long, narrow beam?

If the net force on a system is zero, is the netjy+kr)u3go zs2p 7l8 snwvu 6 g5pb8l5 torque also zero? If the net torque on a sys)+5lls8p r8zw5gv bg6s jk o7u3uyp2n tem is zero, is the net force zero?

Two inclines have the same heighp .1(j3 j1 h 2-jcyog0ju qlubswfv/v)t but make different angles with the horizontal. The same steel ball is rolled d)vw3 u(v0 jus l /jgh1f2j yj.po1bc-qown each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest)7d o; gp 3kabkw5 xj0ot+5j cq)0m+ofw down an incline. One sphere has twice the radius and twice thj+;wogx75k 0akp) 0dwtfo5cj mo3bq+ e mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass.vy hp/i:l-0z x They start from rest at the top of an incline. Which reaches the bottom -yipzxvl: /0h first? 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 m. vm4s bl l47zq74wqkaost moving or rotating objects eventually slmkqalz.4qw7v l s74b 4ow down and stop. Explain.

If there were a great mi)xc+ fbqkq+/ 7wruh8k gration of people toward the Earth’s equator, how would this affect the length of tfk+)r ux7wqc bh+/kq8he day?

Can the diver of Fig. 8–29 do a somersault wiqsrqc7 g;ryvli3idn.m 858g(thout having any initial rotation when she leaves the boar sng. qvi38drq7igl m;r8cy 5(d?

The moment of inertia of a rotating solid disk about an axisv8y 7epm292lxer9bf e through its center of mass ib2rpyemf7 2le98xe9 v s $\frac{1}{2}WR^2$ (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 b+tee7.u q lf9 t(ldc-2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decr-7+tbc.9 l (lf ueqedtease, or stay the same? Explain.

Two spheres look identical sbrv)736+ 1 ej0ss )pxe tczdjand have the same mass. However, one is hollow and the other is solid. Describe an experiment to dest+xbp0vzc )ejs13 rjd 6e)s 7termine which is which.

The angular velocity of a wheelzl3ghp ,8uwk3.vpvfuxl,4 q- rotating on a horizontal axle points west. In what direction is the linear vell hxpwp4lf. g,z8q3, uv3uv-kocity 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 f w)2. s1cb-umyfpi.zireely rotating turntable. What happens if you walk toward thecyu2 sm-ifw1bi.z)p. center?

A shortstop may leap into thi ya pv(d2/z0b3 :3rcdswud:oe air to catch a ball and throw it quickly. As he throws the ball, the upper part of hisday p2( izw:bcr v/d0sd:3ou3 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 +tziw((( zt9ul,mp j6kbz7vp of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller).6u w(7p, jpk (mtvt9zlzzi+(b Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following a xu(9zl- 2 b;ceg +hlrd1yx)ntngles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidencu1zyxng8h3 2aw vcz, 2e. Calculate, using the information inside the Front Cover, the angvz w22ny3gc , ux81ahzular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beaiv p.cr8:ek1d; ( ipvfm diverges at an anglcvd 1(p:.p;i8fir vke e $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned ocu v1/s7ai 9g0(*zojvnduf, f ff during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down? o fa uc7i*,n/dz u1v9vsj0(fg    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. I1:s q/9aa rta3onv7fdf the ball makes 15.0 revolutions, what is its diaqas3dtao a 1nf7v/r9:meter?    m

  A bicycle with tires 68 cm in diameter travels 8.uonbsmd5 /a , gcb:5e(iedhh7k jp(41 0 km. How many revolutions do the (m7g pns5du4ob icheb/ ,j5de (h1a: kwheels make?    $rev$

  (a) A grinding wheel 0.35 m in dfzlf 8,ml6bhy, 5e,vw iameter rotates at 2500 rpm. Calculate its angul,5 mhl6,fbwe8l ,z yfvar velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–387bv)ehuizobc, 8vc- 7). (a) What is the linearhz 7-)bvc,ebcu 7i8vo speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit 4fyevxof9xc m2 ra,*u1jdbfy 8 6r1x/around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poi x+gc9dio (ybi2vq.(5i noi( znt
(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) mustx7dth,r5uz:l :ka kd6 a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 10rzdlua, t ::kd6k7h5x0,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abourx ;c4pq2 xug+*pl *ewd,ix0 bt its center from 130 rpm to 280 rp;q pcpxer0b lgix2,dw + u*x*4m 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 radiudt d wi2v ftd;85x+.wrs $R_1$ 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 aboard the Apollo u54+w9nj jo1rxz/kf co(u8 ht spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. Thfhkjuc w (18o/u9 nrz+jx54to e 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 15,065v3 s4c mt pgun-lap9l v(0la00 rpm in 220 s. Through how many rev5p 3ll6p va0-lu4n s(9vgmta colutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1 w,,x,rvlfgy 4200 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 jetfk /4 .v*jyjqvvb xc(7s in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reakf/(cv4j j7vqv*bx y. ching 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 accelm h9;s,z 2z8caq.vlhxerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the whee.9v2 sh8 lzmch,;ax qzl have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/ 4ir0k- b5q6fdb+fe nyepr) l/min It turns 1500 revolutions before it co-ef/i )k 4e5 p+y0n qld6frbbrmes 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 reduc2,giwo g oe;btk t,a *lig83p.es its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 8tog;big*po2, lkw,.e3 gi at 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 uniformlyc -4bbic*( qu, 7iwtls from 95 ,-li *4qbcbw7iu (tcskm/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike pn:d7q*iko)rvkw 4o*dj(0g hwuts all her weight on each pedal when climbing a hill. The pedals rotate in a circle kwh0ndikr jv*) (*qg:ow7o4d of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What t9h u8x b,x3dq(p3nnp is the maghbq(xtn8 pd3p 9 u,nx3nitude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle qch,c *n-bba+of the wheel shown in Fig. 8–39. Assume that a friction torque * cbq-ab+c, nhof 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 d-z1tk9fr)hr of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnfz19dt-r )h rkitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine requingn2v v qs/f2: +.,unu civ;wjre tightening to a torque of 38 i:unn,f;2+cw2sqvn.v vj ug/ $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0ehy6 uk/yfik+ 1a- k.o.648 m when the axis of rotation is through its c u ok6i.ha+yy /k1ekf-enter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7n* 4nufm v8tj48nx7vu7) lfcb cm in diameter. The rim andfbuv*t7 4v 7n84fcnx8l munj) tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a t3do0 0abe;1-)j7hlyyjg iyzchin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculat)0 b0gj olc ye;zi7d -1yj3ahye
(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 constanstxqbahv +)qo9b8: he54 dvm/41 z qlzt angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N totaq9qs4vt 5qh4)z 8+ /av1:hzloxe mbdbl.
(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 in Fai,) 0g4dxthr ig. 8–43 about0 ta,dx)hig r4
(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 cquo17tdc z2(2:baf b jonsists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct ratlibxro*,. ,kp8src- d e, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required r ,prkd*o8c-b. xli ,ssteady 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 une u0d:e x o91qrx,l ei.*)iup /yv2yctiform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculatei 2)urqxyuo /e*pxc,.9 led 0 i1eyvt:
(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, accelerating itfs k( *v10 gkp dpma ;v4vi:qup04psw. 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-driven merr0 .c2mat;k bvyy-go-round and is able to accelerate it from rest to a frequency of 15 rpy t 0cma2;bv.km in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 1lt 7w r s(ntn+qv+++ib0,300 rpm is shut off and is eventually brought uniformly to rest +s7l++wqrinbn( t+vt 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 accele,pbgb8bz4 f*fgri0z6rates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown v 3wxu+0lrj (xsolely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45uwxr( l3v +0jx. 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 thin rod,8+ v.qh;y jcj 0ubzj7y as shown in Fig. 8–46.cy8 7vh zyu bjjj;0+.q
(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 masf1 o yeo/eywk0:wrb*rn8sy 6 wdd v12/ses, $m_1$ 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 withinn5isi5( 0lplumj3aju4::w b h four full turns (revolutions) and releases it j3n0 b4(m: wai5lpi ujuh:s5lat 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 er)h onv*el4v1upd 9 ;a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 23: v om37oahw;;d nwqe80 $m \cdot N$ 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 slq+cjsc41ak 8e.9zlia ipping down a lane+j9.k1 i elaq8csa4c z at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic en5n:lo f;c /a kc9hdyb9/g (dw-5kttvmergy of the Earth with respect to the Sun as the sum of twotvw5(nb kc5: //dfh9y;- am c9ldgkot 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. H;jv hr:yfl: 8fow much net work is required to accelerate it frjvfy8lf:h;: rom rest 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 ag; 8 dj0h:a2*b0qz kr jd030k putwvkxnd rolls without slipping down a 30.;kjd3drku02 bj0:k twg z qahp00xv 8*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.;t udouttt/8 b6bs8+z It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use cons 8 d6+;ttbbut t8uzo/servation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on then*4b r6 qxqk2a end of a thin string in a circle of radius 1.10 m at a 2n q*x6abrq4kn angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding +jeitv+jdvf2*n hiti59s; e-wheel of radius 18 cm when ro+9iji ;ij+5 fht* ed-tven2svtating 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 rh-7k cs*u f6shate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8s6hc 7-f u*khs–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.zy:wrp d d1;7l 8–29) can 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 2.0 rotations in 1.5 s when in the tuck position, what is her angular spr71wzl y;dp :deed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial ra;4 ms no.xrf5kte of 1.0 rev every 2.0m ;.s45xnofrk s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis throu jcbawvm *6to2.) -ocggh its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. ac 6m.*v)- jwo 2ctbgoWhat is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinningp +) ccacz/ ma(ng.+ic 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 momewbwjm.dw2u+j 3njl6v y 3)xy,ntum 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 byx97ih. jm5wy xa0 3oof moment of inertia I is dropped onto an identical disk rotating at angular spbh 0xoya37x.mwj9yi 5eed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsb p. rnkc*v:fa) :6kzk at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at ty4/iibfx5 -sv *qssar 1 --bkohe center of a rotating merry-go-round platform of rad xv45bk1bf- sisa-q ri-*/ osyius 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 merm4)tn) gte ln x.8-usery-go-round is rotating freely with an angular velocity of 0.4m snut- gnx) )8tle.e8 $rad/s$ 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 axgnzo s,gq..pu.:m c:unw4 u - white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its exix,umnc zou-.pg wg.sun: .q4:sting 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 winz,g 9 jut.:xtxkk6c1 xds 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 5rv ;r-o2ln b ksp/zj /9+t 3qry-lsfo$ 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 restn :n6st59wkagzfd1q 4prd 5k9 , that can rotate freely without friction. 9nddwsk 5 tz4 pak6ngrfq195 :The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-gar:u t0f,vaj( o-round turntable that is mounted on frictionless bearings and has a moment of ir fta0:,juav(nertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope ly3u 75 qe(wa*eus5bsyying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spoo bw755ys( suqa*ue3ey l? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always v c,mv/r;ih-is (f.gnn43mh q dro2u8faces the Earth. Determine the ratio of the Moon’s spin angular momend;rn34m/q, 2govuhmc- fv .sni8rh(i tum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates( umy -x wjim;*gp0tp1+i0zbibo0f4w from 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 the9q (lcur ,or9 z,3q4l ghq8bfe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calf(o9ucqqgh,l bqr43 e,r z 9l8culate 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)nv 0je0gb.g y 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 li )bvg 0gn.yje0near speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the reao)wd:tvl :g. (pkeoyb6.+qz ur 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 Sunfx xq98g0lo r.h 5zowa )a1am7, 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, anl8 aw fahq1g)5x. xomor09 7azd that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it toc*ff,h s wxz u; .obf+6rzr2*g use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and masscb ,6s*f2wr;hzgo f+ uf*zr.x 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates a gekl zb/.:0ncn $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3.3 q )l 7om/rsyzfy-ttwwi 5 :zq+(1;itj$m/s$ 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 frictiop3b. j(vk9 sqd: o;*vg m;*xpbyvfc5nn) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment ofgc*n :(v. ;dsx mp pb5*k vjo9fyvb3;q release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. cc(ucxod j o-f ox; *vbci9h6(26o8ntIt is standing vertically on the floor, and we want to exert a horizontah tb2j8odcc;o o ox*u v9ix((fc-c6n 6l force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling 6jj/6zwo1e / :83lb0 liplaofr y9rdi10m svlwith speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are 61d w/ jpr:ml/j ri6yile 0sl8f93ovoa1bz0l 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 +z2cqn:y38iy zo j2wk into a sling of length 1.5 m and begins whirling the rock in a nearly horizontaz2io+z wjqyk:y8c2n3l 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 r 87bq:*celq;, wwuu mas a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratiqwbr ue:c lu,m8qw *;7o of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylinn46h6r7 anx 2 d 9qo4ollupdii:8+cnwyg:y1p drical plates, of1 i8 6x74 noohply2:4i+a96u ypcndql n:wrdg 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 rollk;q+gj gh2 4e 1dufwqdi6rd4-s along the looped rough track of Fig. 8–58. What is the minimum value of the vertical heightd+1r4; i wqkg dj2 hu-6efdqg4 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 assumecxjol1i s++zkc32 ,a s $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly z3rg sckn i:95from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration orig s:nck5 9z3f 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