A bicycle odometer (whichz20xlt,n ,yzi measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens z ityl20n ,xz,if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates aep s9g 8:pqf l/ qp6*5fwqt0ygt constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have r9 *tf6fps0pgyqlg8/ qw5p: e qadial 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 angu(pejy7 .*05qcxg w 57abc3,wmpys rwhlar velocity $\omega$ Explain.

Can a small force ever exert a greater torque tha r ,b)b/sayqw7n 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 difficu fcc42:3gm(ul )nxw kflt to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you tn u4 c: )l(32ckgmfwxfo answer this.

A 21-speed bicycle has seven sprockets at the rearf w041t24+zgf 5nn0ff w c5pdnc a)cuu wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocke)ztw0f fucfg c5un10f 4n5 2acp4d+wnt? 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 fasttza)smt r )y9/5wq0sc have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why cs )m t5qt/9 yraw0z)sthis distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, naody0r / 0amu+qrrow beam?

If the net force on a system is zero, is *nob ,yg hw,f br9jh,4the net torque also zero? If the net torque on a system is zero, is thefohr,,wb n, bjghy4*9 net force zero?

Two inclines have the 3v )dcvk5xd4*cubh 6i /oztua59 j)zpsame height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will theajx43u vcb zi)vdut )pz/ h 69o5d5*ck speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolw+o(x r9y 52p6hue ejkling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches 9u26j 5o+exrh( epwkythe 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 hav c9aw5*i (wng-ygs 7lpe 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 greaa-5 w*ns97l g igpw(ycter 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 movingv.zs d7c5yh* y or rotating objects eventually slow down* yvs7ch dy.5z and stop. Explain.

If there were a great migration of people toward the Earth’s equator, eohflsl6lmp- c t3(07 how would this affect the length oeh pc o3l(t70ml-sfl6f the day?

Can the diver of Fig. 8–29 do a somersault without having any 4miq(8k e9 -xgjx r+uminitial rotation when she leaves the egrukjxx 49-qm( m8i+ board?

The moment of inertia of a c/uajcf+ g ,k1ig-d0r rotating solid disk about an axis through its center u0igkrj/ g dc-,1a f+cof mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstre0)08ct 8a+b;xb d np-phtxih8lzjb9 dtched hax+pt0c)tzd0; -bh9bbn id 8ljxah p88nd. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. fr,m q:kj*8pj7m o-ytHowever, one is hollow and the other is solid. Describe an experiment to determine ,8jkojmt:pfy7 m * -rqwhich is which.

The angular velocity of a wheel rotating on a horizod,d e3htnx1;8f-+t q /ov cyebntal axle points west. In what direction is the linear velocity of a point on the top of;d+f tv-3yt/hc 8bxde,1eqon 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 f54 vot 99tc.engnzx 9yx ;f-hfreely rotating turntable. What happens if you walk toward the cee o4t.v z5xng cfxty-99nh;9 fnter?

A shortstop may leap into7n-bf4y5k*gvu+nm n t the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in tb fgun7y-*4 5ntnm+ vkhe opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a helizfro1k. s2m2 q6mx:e rcopter must have more than one rotor (or propeller). Discuss one or more ways tqkm1s2.zxorr2 f: m6ehe second propeller can operate to keep the helicopter stable.

  Express the following angles in radb c+i7gp1ob2sj/d4o tians: (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 coinci xfn* xs/i5t zqy:w/6i3qpr3odence. Calculate, using the information insidefpw int5i//yqo3sx*xrq3 6 z : the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam du7sj 3nyxbcp1u 6e.;t iverges atc7 sy;n3.xpe b1 tuj6u 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 z9y2 g. ,/mkrnl p(lbwrate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. Wha9n/bk2wl ,.(yr mlpgz t is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If axm af26( c -1l-svqnxthe ball makes 15.0 revolutions, what is its diameter-qsvaca mx1 6fl(2 -xn?    m

  A bicycle with tires 68 cm in diameter travels 8.5+hb : -*9 md yk;bkl8pnve:e8nckcz x0 km. How many revolutions do the wheelpkveclbkhk:ce -b :*8 5zy mx nn;+98ds make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angucldxuj(-)c d2kg-hc zsc: 0 /)qnww7alarjaxnk dz :cqcd )swc-07u(hc2/)g-lw 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 onef4ny7 m*xv8 2ki l3r7mtr/iwx complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of w8*m/lmkri 74rx2itnyf7v 3 xa 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 around the Sun cbp+kfjakz0 u09m2*t    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of aoi.;xpn t)iy ; 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 centrifu +.rvu2jc.qnv6izrw, ge rotate if a particle 7.0 cm from the axis of rotajvnw.ur,r6+ 2zqcv.ition is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center f(lxvy7m2 h54:i :dx nfhumwcqyw8z7h(*r l j 9rom 130 rpm to 280 rpm in 4.0 s. h8 yduhh7v :fl75*:xyw ( lizw42(m9rcmn xq jDetermine
(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 radiuu hmo mt*c2no)a9ls*,x0xkg )s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacl4ob)g :vot;g4j(e knecraft put themselves into a sl4eg;(:) t ooblv4n gjkow 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. 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 acceleratesa2h1mu *7xjhn uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn iajm2*7nhx1 uh n this time?    $rev$

  An automobile engine slows doh,zj d(cf -fniz0) 2mnwn from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of fl+6e .clv+n3b ocsam8 dg4pcdmjs4 t0;ying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to t8p4l4m tmdb so6a dg.vjs 30 ++;nccceurn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformht*u /:(jragogp -y s9ly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have tra*tgrg/asp:- yojuh(9 veled in this time?    m

  A cooling fan is turned off when it is runnisc-pa/qg4a u ;c ,u8cnng at 850rev/min It turns 1500 revolutions before it comu uc8cqsa-gca/4n,;p es 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 makej7 +o3.fpu on.aup1fsufk9c/qd g*n6 65 revolutions as the car reduces its speed uniformly fromua9s1u 3/knup.g+ oo 6d7qcff* j fpn. 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 revol8zccl3- z4e.62vhpsulm ;cfyutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 mz4ml; zsue 3vcclyc -hp.86f2 .
(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 w q5,-dw7i-md nj/tfl6p 2y g2jomzi0jeight on each pedal when climbing a hill. The pedals rotate in a circle of radius,jom2 /j6y5pifi- tqwdjg 2zdn0 l- 7m 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 awy 1y5cgr37 fs door 74 cm wide. What is the magnitude of the torque if the forcfcs3 y 75gw1rye 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 aborahsfi; x/y o14t*o.out the axle of the wheel shown in Fig. 8–39. Assume thayrsito/ha x; 4f.o 1*ot a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless ro//d7ic,g0 shl ztie6z* .zvzy d which pivots as shown in Fig. 8–40. Initially the rod is heldztzi7clz*ze/y 0,.gh v id/ 6s in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head :xt12z qnlm t7 x5,wu8*d ued jj:t5xzof an engine require tightening to a torque of 38 t:zt:*5,njxw j7dqu5lx1 8mud tzx2 e $m \cdot N$ 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 radczx 17*eh2xdz0yg)l gjd1 *inius 0.648 m when the axis of rotation is xehx ncjl) i*dzdygz210* 71g through its center.    $kg \cdot m^2$

  Calculate the moment of inertia ofx6 +a bt/tji524a hsez a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can 24 /xt s+ij65ahtzeabbe ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod 4h2es:kiaymo) +wb9r vu)sd0 is rotated in a horizontal circle of radius 1.2 m. Calcur vwas0):)4seomyd u9hik+b2late
(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 potterzwc 87j t:rg8b’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is :8r jtzw8 b7gc1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objec kes5o 6lbjgh+d :w5(sts shown in Fig. 8–43 abeb+w5lsk56j gs:hod( 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 of two oxygen aty;vv sa*(pf /lxr,btj (8qw ,cfj.am1i8y*fp oms 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 5p-+1p tgt 6ls) jbas;s:kuibat the correct rate, engineers fire four tangential rockets as showu i b kgs):a 1jttp;5-sp+l6bsn 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 raqbrj ut20zs4+ dius of 8.50 cm and a mass of 0.580 kg. j4qbrs2 t+0z uCalculate
(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 wdc, 9:2vby4 rb0qvxw bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go;n(:y(u pv/z v yq+psh-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleratio; :/ zy(+spphq(nvuy vn, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is s1 mjk)l4;7g3llv mhhti 0w hk/hut off and is eventually brought uniformly to rest by a frictional7wg4m 3vjt;/h 0mh)hkl k 1lil torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball at 7 byk31j l af5soqnxu, v:5o eu j5/0es-$m/s^2$ 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 solely by the action osx c2vkfx,pg6oor )02f the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. o,g6v)0 pcfso2k2rxx 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 y 3tr1a2ci eu r;9ot3abe considered a long thin rod, as shown in Fig. 8–46.yet i23 ur a;9ta3rco1
(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 if2 bt;cqk)v*m 7o3:pxkk cv.two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four fw nh:d 3 x*hn-e9s.rx bi.:vgbull turns (revolutions) and releases it at a spb w. : 9r xvhgh:3xid.-e*nnbseed 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 inerti:j xa*s xji3moy.8*zma 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 28 cw k4myp+/e,i0 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mar1tzdy4 t64pu b/nb pae0u*h1ss 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.3 p bpu abune16r*h1/z 40tdyt4$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic ener/p np :g-u4ekpgy of the Earth with respect to the Sun as the sum of two te/eppk4np: u g-rms,
(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 and5v(;aqps) p vq a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.0v ;sq5ap()pqv0 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 auce3p)ybd. -a nd rolls without slipping dd. u b)a3y-pceown 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 balacx*y 6c6) 6r,1gwpcwynwj d)j nced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of th)w,w n6y61d )6pwyx*j rcg jcce 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 the end of a thin srhi i-/(*mhk7e h, hictring in a circle of radius 1.10 m at an ,7 *iherihh ki/hc-m( 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 wheel 3-8m afyygng0 1 lkr,eof rady a gfn8mgye lr,-31k0ius 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, 0r.r6 mbqcw2fcl 4cv. pe 4oa1on a platform that is rotating at a rate of 1.3rev/s If he raises his arms t 0rfq6c eopm42a. wc4lv1.r cbo a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fif zox9 *o (,-zx/ull4v v;jqtwg. 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 isl, -v vx*/xu9qozwtl; joz4f( her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her ycuy m.;5twg2nu91 ddspin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final ram 1ny.2 yu5dcdw ;u9gtte 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 throughfbc8,wnr ns+ 27r78 ttera8 vs 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 c tevrsnrsar8 7, wf7b2t8+n8shaped 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 skat/ kw q3 4vn hdmcy duebhl* k-8*m;2/iggspl26er spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the v ce(17gtb/gb 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 identin uqbn6t8hdu2b .pppf2+f 0auf.u4 (zcalu.+hfpfp.82 pqznut2d460bun(fu ba disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a3 )fqyjo *tm9j0am bc-4jtvi8t 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 rotating merry-go-round plat e2- nx(tf tlsolac,27form of radius 3.0 m and moml2o,- f2(a7e tl stxncent 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 rotatingtvq l .y4 dnm0fud)k+; freely with an angular velocity of 0. m.tdn)l;vukd +y q40f8 $rad/s$ 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 int 11;-bftarc klo a white dwarf, losing about half its mass in the process, k t;l rf-1c1baand 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 integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in exc m+q61 uwc+w hi-smsd,ess 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 - jfw4..f sxvfbg6ic p6y/v- e$ 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;/hu 69pnox -pzho,sc ) whq+v rest, that can rotate freely without friction. The moment of inertia o6nhvp)-h ,+xowz;s/uqph c o9 f 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 m:cdsy,b;4 qfwerry-go-round turntablec,4qbys :;fwd that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope r j0fde w-ue/*rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? He0/e fju d-r*wow far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always f 3dhd2s( hfl,taces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earththf( d32l,h sd.)    $\times10^{ -6}$

  A cyclist accelerates from rest at ava4ys0k-n 9/hf-t ama 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 cylinder; 8mz2x iw52wf1sw neu8mn:e g of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s intexgm2e;:z 1wnw 5smw 82ue f8inrval 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 0.050p(se5r dffo:4 c2 yyv6 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to car2f:6o yv 45cydfes(plculate 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 p2p.1hjjg,..cvrs xsr q,,rn angular 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 ofzy8/ x12hzppg)j bg- n our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carrip z12g npx/8)y-gh jzbes off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is fci viq .z4m* hovfp/u,6p1 jn/or it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mas/6jz.um p/ p,f cnvi4*vhi o1qs 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 illustrates638c apzsfm5 m 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 on a fceo9wsg ;*(jdx(an)horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot f x2krbejdj(uj iw8610m0j3e :xe v(i breely (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 forcee 8 1v2jjjx 3 mdibek6(web:x0ij (ur0 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. It is standing vertically on the floor, and we8 zxn 3r4* dxzds e9)5vsrxc3p want to exert a horizontal force F at its axle so that it will climb a step against whic rrn334de czxxsd9 zvsp )8*x5h 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 z-k-bejsisv 8j3 pg09with a radius The forces acting on ts8-s ive9kg 03p jb-jzhe 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 length 1.5 m and beaivk 2m )x;z.hgins whirli v)axz2;m .kihng 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 come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and herim lnnmv*j (3- arms as thin rods, making reasonable estimates for the dimensi-njm3im v n*(lons. Then calculate the ratio 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 whicq /jdvhcxe8+;7t yyc0 h consists of two cylindrical plates, of masyt 8c70 vy;dqh+ecj /xs $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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-a9hsn8ael8-srr):v 8u 3x ybvxw*kfc mbg4 9 rough track of Fig. 8–58. What is the minimum value of the vertical heignh3cv-fb ly)r 9e4r9ms xxaasg -*v8 :88b uwkht 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 assumeec(/ r/wtphe; b wnx ,6qeu87ag d+d2f $r\ll R$ h =    (R-r)

  The tires of a car make 89yw,/ mk hyn c .5xyplory *9t2gt(eg;5 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleratio .9*rpghxyy,/ n emo;l2(gywyc tk9t5n 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