A bicycle odometer (we l) o3)s7 0w /9nvnwpdp(zhzbhich measures distance traveled) is attached near the wheel )w9d h( vpb/ np 0nzw3eolzs)7hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a poingyy2f4uvkry/ 4 8/ay pt on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniyy4y8gr42kuya/fv p /formly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angula7pmo 7fj3(l r1h el,ddr velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Expyd)2r ;qacfx+1r co*qlain.

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-u 23+(qs,g0,j; rnxf yd zcgpjjp with your hands behind your head than when your x3gcr j,p g+;0jsqzjf,2y (ndarms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at7z,1.aifgaa7 : -au mrskr+fx the pedal cranks. In which gear is it harfkrum fa+ .7i 1xaa,:s7zg-arder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs withvw5x ,fry/2 rse9lvg9 flesh and muscle concentrated high, close twe,vy95f 9lvrg x/2sro 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, n huosxgfe/:i ji(3+fc(fs, u5arrow beam?

If the net force on a system is zero, is :r294r2 gxeqajxp p2 jthe net torque also zero? If the net torque on a system is zero, is the net:pq2jjr4a2 g xxp re92 force zero?

Two inclines have the same height but make different angles witl82ztjjb 0;3)t)zmjkm6oq dxh the horizontal. The same steel ball is rolled down each incline. On which2; 0b t)moldx3 zj)t kmjq6z8j incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously stpnui ;d. e6+0syuo: jfart rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incli+dp 6ufu y. o;n:0ejsine first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinde j3k-/ syv.ddnr have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the ysvj- k3d./ndgreater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most c1 7ey:/wn5r;t dnpqh moving or rotating objects eventualdq:c/pnte5 17wn yr h;ly slow down and stop. Explain.

If there were a great mit7c:c7hhwn mu+z1 c,m gration of people toward the Earth’s equator, how would this affect th c +nc mc7zm7h,h:t1wue length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation+t.w-bu6f(*-u.xwmn;mvk rcjsg g+ r when she leaves thwrmfg-g .*ur6nm jc + vwt k.b;(s-x+ue board?

The moment of inertia of a rotating solid disk about an axis 8a/4oo hds;x c0q p9tgthrough its center of mass ic g p8h9sqo4/;tdax0os $\frac{1}{2}WR^2$ (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 eac,*v+tbq-gtl0 :b qp 3nlyn.lza3 tgu+h outstretched hand. If you suddenly drop the masses, will your angular ve:3a3ql ngvq +0,ltnb -ug.ptytl*+ zblocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow a-blamy 054 pzdnd the other is solid.ay z0db-p4m5 l Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horitvxxs : /f kpbm*t125jrv;p6m zontal 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 accelerk52*;/:x61s f mvb jvtpxprmt ation 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 freesy0r1k z9x ;1sd4 qbyhly rotating turntable. What happens if you walk tsdry1kz9h4 ;qy1s bx0 oward the center?

A shortstop may leap into the air to catch a ball pi3zu e/ pr3mrlng6u6,h5ub5and throw it quickly. As he throws the ball, the upper part of his body rotate br 3 l5/m5npu3ure pg,hz6i6us. 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 l nt- b hnenh*)v.mmocyv0*s2+aw of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to v0n*o)cnvbsnm*h tyhm-e +. 2keep the helicopter stable.

  Express the following angles in radians: (4xt b .j+5t,6:jhovnlmxlu 0u 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 bech:o*ci zawrutp 2 71m+ause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radiaz:1mtwr *2+cahpoi u 7ns) 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 beam divergel5 e,1o5 obbqs(2sz sgs atb s2,b(l1qez5ogs 5os an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a en(u 9g 0ox1,,mqsdl ed(agm+rate of 6500 rpm. 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 slow down?g0 ndqm+o1 maseldg ,xu,e(9(    $rad/s^2$

  A child rolls a ball on a level floor 3.;zyo8p k, hzpc)o8 jd,5 m to another child. If the ball makes 15.0 revolutio8 cp ,)o;hdjz py8zk,ons, what is its diameter?    m

  A bicycle with tires 68 cm in diameter r)om) ves ora yo5t: /0db52()oenrthtravels 8.0 km. How many revolutions do the w 2(ore)r5o05e yva :r)d)t ostbh om/nheels make?    $rev$

  (a) A grinding wheel 0.35 m in di8 c/owfz xm qxgq-5vzu y0xq7ay)6;0v ameter rotates at 2500 rpm. Calculate its angular avwxoqc8-m qzyzx y0fgx/v0)u6 ;5 q7 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 49a fu pkl ;6pe9n;pu,w.0 s (Fig. 8–38). (a) What is te p6u;al pkuf;9n w,p9he linear 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 aroundl w,goegrh6 l0w3 7w+u the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pointpw-n 7caog k*8gavm.;
(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 alel- bk)u 6g qxgs.l:p8ww*n1 centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acgx k-e l.)l:gqn 18u6*wb splwceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from a9oxs2zyy h +dvc.+ 5m130 rpm to 280 rpm 5dxams2h+. vcy yz9o+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 radiur);n j,z/m exxs $R_1$ 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 spacecraft put themselves cxfr;gvz x/:k+jrl2 . into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, . rzx 2vck rl/f;jx:+gthey 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. Through houkg-va6a/un l:ih2 2nw many revolutions did it turn in thn hkvi 26l-au n2g/ua:is time?    $rev$

  An automobile engine slows down from 4500 itq; d3,1glql 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 inxitst3 7h whj7,0exd8 a whirling “human centrifuge,” which takes 1.0 min to turn ths 78wthxx7d3e t0i hj,rough 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 accelerc1nzm1 v9)4+5uj +aekzqab5 xb .mwzkates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled 9abkj1 na+b4mzcx .55 vm +1zu)zekqw in this time?    m

  A cooling fan is turned off when it is running at 850rev/mi z1,v;eijozh;n It turns 1500 revolutions before it comes to a sto1e ;,ozijvz ;hp.
(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 revolut3k.q )0wfcy9/ku 3i eru l7trlions as the car reduces its speed uniformly fryk0wue q.r /9)c7 lrlift3 u3kom 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed umly(i 1 n 0.2r8oazompniformly from 95km/h to 45km/h The tires have a diameter of 0.80 (z0n pioyomrl28 1m. am.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each uw txeic3)13rpedal when climbing a hill. The p uc1 txe3wi)3redals 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 doortbh-f)naw ;m , 74 cm wide. What is the magnitude of the torque if the force is exer,ftwb );-nmhated
(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 insyeelt, r2ox.k (;hl jqm. m1 eenh..2 Fig. 8–39. Assume that a friction torque ofx, e.h oe.lr.syhjk;emnl22e1.m ( tq 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, arsvf5gztts*rwvzoe+k c. 56 ,+ e attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held iceg.6 5tzkrvw *,vo+s5fs tz+n the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an e+ccrl,s( p-i0wb8 jq dngine require tightening to a torque of 38 w+, sc i8q-j0r cdlpb($m \cdot N$ 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.c jaqa5;7lg4c 8-kg sphere of radius 0.648 m when the axis of ro5 ;cglcq 7aaj4tation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bi4 2dvd jpg d;6x0y0yjfxogl -+cycle wheel 66.7 cm in diameter. The rim and tirexgfx2j0gd o4d;j-vypl6dy0+ 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 thin, light rod is rotated dzz+ vhtf ,52win a horizontalz 2,5+zdwfv ht circle 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 bx t.tm+as d-*oowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The . mo t-ax*std+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 of the array of poi vz: t.h+1s5yyyz bgem*;b -nint objects shown in Fig. 8–43 abb*z y:z; syg.v -bhim1n e5ty+out
(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 ob 0( pfmrppnn58h:z9 anj- f8ff 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 s5 g)6pt tq a)7.f+z:.yzp mc*p jetlsipinning at the correct rate, engineers fire four tangential rockeeyz lt cjigp*at.tpm)qz76.+f s)p:5ts 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 with a radius of 8.50 cm andbmt br 7k;-ro) a mass of 0.;7 b)kt-mrrob 580 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, accelerating it frxyq1qjc3-ljro 77 0myom 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 smal. d(9jlww-omj l 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 has a mass of 760 kg, and two chw mdjl(-jw9.o ildren (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 rp0hfn;ws4--er-r 2ygqoqk) mg m is shut off and is eventually brought uniformly to rqen rsrq-k;0- w 4-)ym gfgh2oest 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 ac6qvzhoivifx h5)4 - *mcelerates 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 thro/f24 u .yf;v t vxvxuv1a2ij4wwn solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muswjx/ fv12i.;ayu2 uvfx4t vv4cle, 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 thin rod, asirpee)3+:*us4e zpq t shown in Fig. 8–46.zr 34es+)p:eqi e utp*
(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 ccr8 gtbc por)5-aw6e;onsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates thsp,0dn7 1v0jw e0u ,9jyas xnye hammer from rest within four full turns (revolutions) a u ,yejx7 w1ad0 9p,ynvsns0j0nd 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 o2idd6* eci zku+-x n6gf 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 t kljd7c9;uf u.orque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rollbc 4x-:ooc f)as without slipping down a lanoc)4ocxab f-: e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the +ns/3a x nx:jwEarth with respect to the Sun as the sum of tw nxax/ j3:sn+wo 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 arr0ub)gg r2 /end a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation r )ugr0rrgb/2e ate 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 and roll ;a69ijenejh. ;pf /gks without slipping down a 30.0 ep i9naj ;hg/f;.e6 kj$^{\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 verticmpult+:qs 38z ally on its tip. It starts to fall and its lower end does 8+ lspqmt:3 uznot slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotati -xn:*hwx4b fq5 dgs5+g(asrl ng on the end of a thin string *sg- 4g( f+w nd xrlxah5:b5qsin a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momente4*o+q r7 dfh 7jm4huoum of a 2.8-kg uniform cylindrical grinding wheel of +47m o* 4hj frodq7uheradius 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 rotatis;7bh brcoapu r99.c- ng at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation c79 - b; orau9sb.rhcpdecreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment ol7yee *w9z n3if 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 sn*y3w7iz le e9peed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.0ueqr 61akjl ,a12,3upu -dabh rev every 2.0 s to a final rateq-,l kburau,a a6e1u ph3j21d 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 rotatig8oxakm/f4 ;pbf mc4,z;s k .ung around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, appro;g xzk8a o/bmf 4u4k;s.fm ,pcximately 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 figure skater sp p1hk+ jx+b7m vurl5q:rn,a;sinning 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 momentu,kvjjf w kj ku l6ta:(v3ua3q1p7kl+ 0m 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 9 iwh g6r2fzl6 6vjc3fr8.ystidentical disk rotatjv3 8l 66ft9c6hsyr f. rwg2ziing at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns 0:b9o v+xb q4a) auwwvhb+mq15nvr; p 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 the center of a rotqg4xa/+kn*1jcm a uedv g h3(.ating merry-go-round platformcgq *g(m1 3+ad avxku j/h4en. 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 rotatinu+4kxn 2t ko2+ * ud*6 2 wfx,fufnjdrx9y(pirg freely with an angular velocity of 0.*(of f xrtiu46u2d2 w, +yn*+nkj dx kf9u2xrp8 $rad/s$ 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 itsgkn2o 1nvv2z-ia( r-k 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 would the Sun’s new rotation rate be?(round to the nearest integ-2kgkzn nv2( oa-r1iver)$\approx$    $rad/s$ (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 1d n/.deb ocrqv(*pg3+20 $km/h$ 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 pn2kzv zxon 4429zn *i$ 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 rotatepszf0 9fao 2i5vj6vc3 freely without friction. The moment of inertia of thefo9j6vzv 20p fc3sia5 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 diam0zsbh3 jsd8h5 eter merry-go-round turntable that is mounted on frictionless bearings and has a moment of iz 0 jh8hbs3s5dnertia 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 9q 3-xftj1ioe r;wlc,with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How fariw1t-;rjl,fc9exoq 3 does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. /9kc;y6vv q 0solg)f n u 2knpc+c)8atDeterminen;fkty c/6kn+2 oc0qapu9 v)8 vls )gc 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 Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at cxhybn. h18snpl8m6. 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 offmy2hqkby 6o:9( /o3 3t xont2a elay- 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. Calculate the torque delivered by the mo36yy:-l3bmo o khn f(x29q y2ata/ etotor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical qp) o9u,f/sop disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0pf9 qspo /,)uo050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speiky+jop/5 +w:oyy-lyed 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 sizo yn2r-ko bje hqk80,4e of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of i-2o0 bhjk oye 8nqr,k4ts 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 user zdeq0 ,t:uq, energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a :0ut eqzrq,d,uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustraeni l2 r*3hc36f1 yatv5eap/ates an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling oe1o+nf 9knb3n n a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivm3eny a+o v5nrmg+g3bs t*f 7;8m q5mtot 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 masf5ont grm q+t73538m*b gyem+ vna;mss of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R.c:;z89 a k avliza,3dh It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so th z9:l 3aazck;a8dvi,hat 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 road is making a turn wlnr )vk2z9 c*v9) eqi+opy.n cith a radius The forces acting on the cyclist and cycle are the nn9vpnc*+.2r))q z kyciov9 l eormal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and beguz pq*+ l76a)ropqv9jk)ft c *ins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque coqpf9lqjp) cor+)z uk7**tav 6me from?    $m \cdot N$

  Model a figure skate1) rx lnh6a;lor’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 ;a6lnor xlh1 )arms held tightly against her body.   

  You are designing a clutch asse3n e ag,bdu/d5-buoki1xr38e mbly which consists of two cylindrical plates, of38ud1iee uba- o ,xnr5/3gb dk 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 along the looped 8vel jwkb 10y dm6u7xk:p xdy m/-q7d5rough track of Fig. 8–58. What is the-u p5d7078 k bwj6de qmmx kx:dyv1/yl minimum 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 doec (bkquomrnk58je :s )fn8;, not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly fjek)sjmx;y(2m(8vi ew*e . fmr*esik e2x.w (;y( mvm8 mj)ejfom 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s