A bicycle odometer (which measures distanc7xbpt by, f)()bn;) -mz tapwqtm1te7 e traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bictt-bq)mmbxp )a,w17te;)pt f 7ny(bz ycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does ar/*8t:s/ve p5gm my,m zb,gcv point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangenyzbg/m,sr5 ,/ 8ve* tgpcv mm:tial acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single val nns 1)r7:tdfajrp)g.ue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger fmzex0c av 01pqds* 1x8orce? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head tha8cs i 6ge2rnb,e1n 3wmn when your arms are stretn,b nc 816imgw3e2serched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sr* wimr f(oy5d cjhu-5 +*2ijcprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear s*+dyrjo i( c-2fchui5 mw*rj5 procket 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 v+ j - k3)bwkby(ddl(ktn+)sgto run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotatsbb k+jv(-)+ kwldtdn)k3 y g(ional dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, jwd+ya oalnbjz*g2 0o)d27 6hnarrow beam?

If the net force on a system is zero, is the net torque also zero? If6k**xp:tpj )j86tfyba y rp8r the net torque on aj*f tr p68)j8byyxtr*ak6 pp: system is zero, is the net force zero?

Two inclines have the same height but make different angles with wqxw-8slnk ysgui8:1s+y2 g6the horizontal. The same steel ball is rolled down each incline. On which incline will the speyk8 +lq1ns-2 xgwyi 6: sgsu8wed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously sta/ma98kz pbd gk/8k( igrt 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 incline first? Which has the greater speed there? Which has the greater total k89a8/ z m (ibgpkkkgd/inetic energy at the bottom?

A sphere and a cylinder have the same radius and t1kvyxei+go 2*s)18 . dqnloqjw 08 pwehe same mass. They start from rest at the top of an incline. Whicds q io.ovx1n 1j)28wy*8wk0gqe pe+lh 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 greater rotational KE?

We claim that momentum an gkd)8b .lse4dd angular momentum are conserved. Yet most moving or rotb )4 lge8dk.sdating objects eventually slow down and stop. Explain.

If there were a great migration of people toward thlfhia2 9yn d0d(xw t39e Earth’s equator, how would this affect the length of the dyl9hnwt(a xfd0i 3 2d9ay?

Can the diver of Fig. 8–29 do a somersault without having av q9c og; a jl;q;c (sb::sxc-(lh9zjlny initial rotation when she ll9-;zs l((vas o9;lh :xgb j:cqq;c cjeaves the board?

The moment of inertia of a rotating solid disk about an axis sx r:;:-zojoxthrough its center of rsj:xo:- o;zx 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-kgltvd2np 264 qk mass in each outstretched hand. If you suddenly drop the masses, will yovp qd62l4t2knur angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same0a;wtr b 17+6xhvn5 8gzfeo6gxidp 7e mass. However, one is hollow and the other is solid. Describe an experiment to determine which is which.6xh6r1 zno 8w 07vtf bg5ea+g;epix7d

The angular velocity of a wheel rotating ony6-ubwr9 vnp3m zr;j / a horizontal axle points west. In what dir;/m 3-y69 urwzprbvj nection is the linear velocity 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 freely rotating turntab8 p)hwto l,ll*le. What happens if yoh*lwlp 8)lt ,ou walk toward the center?

A shortstop may leap into the air to catcho b,p xfoa-ezsnftq-xff51/l) 8y ,y: a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. Ifszf yab1nx- f y,top)-oe f q,5/f:x8l 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 o-a0viccsl6) 3i ybv-uf 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 op-6ivi0scl3 y)b c uav-erate to keep the helicopter stable.

  Express the following angles in radians: (a)l m4ubq4a0r/c 3ad+o; 8 ssugk0*ihln 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 coincidence. Calculate, usinc5 1z45 au vnct.al 3)r3nmabxg the information inside the Front Cover, the angular diameters (in radianlr4vna5m uc3nt3 zax) ab1. 5cs) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direc ftykh2 yh8jq.9c hgz4g641m x0oj; fhted at the Moon, 380,000 km from Earth. The beam diverges at an a;g o .k 8gyhyc6fxh1hm q424f9t0jhjzngle $\theta$ (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 ra)/g nqvh 6ybh7,gmb- tte of 6500 rpm. When the motor is turned off during operation, the bladnvb7g,/)6 htqm gb yh-es slow to rest in 3.0 s. What 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 the ball makes fs02/mulgw u*15.0 revolutmusl u*fw0 2/gions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter t:8 m ;,.-s biztfzltxjravels 8.0 km. How many revolutions do the wheels make?txz8b l; sj-t.izf:,m    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcu,xi2 :cy: qyz0j ggmz;late its angular velocity 0iy;z : :qgz,yxg2 mcjin $rad/s$ $\omega$ =    $rad/sec$
(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 comgr;- j;1zxm jcplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m -1;; cxgjj rmzfrom 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 it.i7aw* ueigi* s orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speedcjgol6a */ ab,wv4nkp9 gym1dqhx 0;rwp-9p9 of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from :ow*sbd,r c9vk3 1wzl(s a)nxthe axis of rotation is to experience an acceleration of 100,0ka x(nsvz)*1wclb3r, ds ow9: 00 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fra4;g o qxjy0k; h/eef8om 130 rpm to 280 rp8j/ 0 ;e fx;qeho4kgyam 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 radiusyft 0r w:j*s 6xhpi8(g $R_1$ 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 pnby (9 m0v orn-h;qf-* eo9vtthe Apollo 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 apfn*(v y or9oeq --tvh;90mn b 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 fr*9;cqeaz; shh om rest to 15,000 rpm in 220 s. Through how many revolutions di; h*es h9azc;qd it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to zebe +3vepcf4g 5tn881200 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 flyia;q 1l4ujfmac7(vlgv )8a .bx:u5 wes ng highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 jvge1lu:m8f 4a.qava) (s5xcu;bw l7 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 acceler u1chj*e n1 sm,qyx,3)ld,vixates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in thije v3xsx,*y,q u1cl1) dhn,ims time?    m

  A cooling fan is turned off when it is running at 850rev/min It tr ,w 3ws; j5asmaw0x 0a*0mnfturns 1500 revolutions before it comsnmjma,0sfx; tw 03*awr5 aw0es to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed unif4szie p e)zcqlchx9eb+: +b 1(.ncmc*ormly from 95km/h to 45km/h The tires sc z 9 cizbmc+):lxbc( 41epeehnq*+.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 ewf3mf65zg+ q xhpuv.gg/1,1q ts o4r the car reduces its speed uniformly fromm.o+ xhe zquvq3g g/fs4,5t1p16g wr f 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

  A 55-kg person riding a bike puts all her weight on each pedal when 82cl/:m3pl;rwi3 c c+ohsqnf climbing a hill. The pedals rotate in a +:wfq 8 ;llsi23np3cr /ch mcocircle 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 is bks4x r9/wcla56 dfnr h5p 8(othe magnitude of the dr rcwp( 5o68la/sn49f k5hxbtorque 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 of thrtco//cu47u z3 z/phqe wheel shown in Fig. 8–39. Assume that a frictipzroqut3cz /7//4uh con torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are at uvu t732bgzhxb7f,7vtached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of t77uvhxv 3zb2u ft ,bg7he net torque on this system.

  The bolts on the cylinder head of.oos bgwzb*+;15dw yn) q d.rh an engine require tightening to a torque of 38 ww)o5qb.y1h d;s+o r zn*d b.g$m \cdot N$ 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 3x7o nj3zl4p3 m ii:l/ n(o l9,yafuvbinertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its cu3o3zl9jy f4 (,xmilinvnp : ab7/3l oenter.    $kg \cdot m^2$

  Calculate the moment of inertc az x17oduym1jp,u22 ia of 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 cazam1op y 2,uxdcj27u 1n be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end 9abxx* wz5;0uau, gegof a thin, light rod is rotated in a horizontal cu,uw*05exa9g azg x b;ircle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potv,go7ph ta d(*ter’s wheel rotating at constant angular speed (Fig. 8–42). Tod v(,p*t a7hghe 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 point obqx220nu8i +go mzve2o*ew, e8jw7gq bjects shown in Fig. 8–43 ng8q2x 0me8ez 2be gw*o7ijw,vuoq 2+about
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whd0i*a *n5b l l::qpz,ovj3iqwose 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 spinnin ov vq,,g(izx qo53ur-r ca03eolv 20mg at the correct rate, engineers fire four tavo0 vo2,q0 guz-a3e5l riqv,3r mc(oxngential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm andmnz8j2x - )m2mk o8uba a mass of 0.580 kg. Calculau 2mz8 jk8obmx-nm )2ate
(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, accexvwu:o17n v h5lerating 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 smalw 5 o3ir hlwzs/3cd 6si8fb16ul 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 children (each with a mass of 25 kg) sit opposite each other on the edge. Calculateuii/ lwzf8336 co5r6w hb1ssd the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventual r1/02uy)hgxo63zl;ihnjcr( mw8 gnr ly brought uniformly to rest by a fn;hr )6m /8ziuj( gg2o0wyc1l3 h rxrnrictional 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 accel krt6+ f7d))6xokujaajlqcziw, z+7+ erates 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 solely by the actcu1pkl j; ixdc,89vp*ion of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest t9j8i kx, dc1l;v*cppuo 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 cons+0,qz-bnmfm r( n (knyidered a long thin rod, as shown in Fig. 8–4- nyfkzmn( n(rm +,bq06.
(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 oeo6-n ed2fexf:cj-r)f 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 ff 2m dldwate 02 uq1ypk3 vv2fr(5((pdrom rest within four full turns (revolutions) and releases it at a speed u dd5 (r1f2 2wep(t(l0 23kyvmdpvafqof 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 inerup 3xzb j3x /d5ty-* lrckvdwa:6*e1ktia 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 tn m1jo e,zi49morque 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 radiusjb bb(n/qlzf.c*r7 l; 9.0 cm rolls without slipping down a lane at (n;b rqj/zb cb.f7l*l 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respeb:rj8ifjuf 43 ct to the Sun as the sum of two termur8 3bjj:f4ifs,
(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 m8bell+tj+zge64 cots2) p n4ow much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is ajc4ep4n )l +sot+gmz6be28 tl solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg stas of*wcvu6*+u rts from rest and rolls without slipping down+vousc *u*6wf a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fagmwff 6 ufuh 8 eo-e1-zc7-fo*ll and its lower end does not slip. What will be the speed of the upper end of the pole just before it fhueu -zocw-8eg6fm* 7ff-1 ohits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular m8rlh87- m rm5bq5fzgwomentum of a 0.210-kg ball rotating on the end of a thin string in a circlhwfblm8 5 zrq75r-m g8e of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum ofj7*6: oozrx co a 2.8-kg uniform cylindrical grinding wheel of radi z6 o7:jxcr*oous 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 p2btq0 ffpqt8vkhq9 6: latform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. q6qhf 29vb8p tfk0q: t8–48, 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.6kts :.ah.jnv 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 1jtn .sh.:k6va .5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate frg,x.yl z.8ilqm l:-zn om an initial rate of 1.0 rev every 2.0 s to a:-.l. 8iqxgm nlyl z,z 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 through its centh( n em.)ayx41 k djw5nwwn6:jer at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and d6: wym. njx54( whkew1ndj)naiameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a :s.ja3.mc9glk0 p2vdvfos* kfigure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Eart8l do :ys1oiqv:t;, pih
(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 q6xx5oj u98yt;3b h qu.frgo9 moment of inertia I is dropped onto an identical disk rotating at angular speedqx outx 99u f q8.3ryoghj;b65 $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2+g56ehwf g0ra6ek/oe1 q7 we j.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 roc4axqzk- 4-v. da jbv27yo (ry75vdcxtating merry-go-round platform of radius 3.0 m and moment ovq zd.ja7- xv 5cocbyy4 x(dra-k7v24f inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an albu *6p6tn5t -gkq t:,tuswv 4mp 22ugngular velocity of 0.p-lk,q2 ugt *w btm5:26 6tu v4unstgp8 $rad/s$ 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 collapsei:hl7e6191z a k1st7ahp wmha s into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s nwham pea 1i119szl7:at6h h k7ew 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 windsphuire3q3 (qh) ky36r 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 : r-eua,fnrz4br2*k4om y o:g$ 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, initiagc 6c/arnk ;2o h:yoo*lly at rest, that can rotate freely without fricc;hnk ay/o rg:26*o oction. 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 edg qo0+9w8u9*cknmajp le of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1700qojkcul8 + m*0 nwa9p9 $kg \cdot m^2$ 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 th2 vmx7lihl0 7ke 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 far d0l 7hv k2li7mxoes the spool’s center of mass move?

  The Moon orbits the Earth su3 v/p;k9dc fb nr4tsw-xna)kkt; t0d+ch that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, trf9rn+k4cp/ )w;ntb;ks0t x-v td3 dakeat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rrt/ wa khh .*r2qa7)naate 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 u1/d )dmvzzh1*bw y5z iniform cylinder of radius 0.20 m acquires a rotational rate of1 5mhizz)bdy1 v/ dwz* from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg andmh7: pe3 wj. ar2u1(t -yom.vd/jojpv diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use cons1op-.7( jvtjh 2/ ayov dj.w3murmp:eervation 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 speed of the rear wheel ozvj feepc1c1;(. nd8($\omega_R$) 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 sizlcx)d6srn8ixy ls3 :1e 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 its mass in the process, what would its ri6 dc1x8:sxln lrs)3 yotation 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 ush(- ge,x *pz49ab5i9hceyxt8zve 4 fze 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.pevy9hz9e- 4aifx4b *ez85hx ,z(gtc 50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates av*0 nxk1zfr7v n $H_2O$ 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 s2v u3tiz2.jo rxxy ip8f x9,4qurface 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 freely (i.e., we igcjc y7ul+t7hg7 y u*b7nore 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 aclcy jc *u777utb7h yg+celeration 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. It is standing vertically on the floor, ah ax va(16e7okgdloi6 7a8 fl5nd we want to exert a horizontal fode 7i6ok aov5lhxa 8(1a7g 6flrce F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn with q(os ,zi(*crq a radius The forces acting on ti(( qrs qozc*,he 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 s: ggee4p74 r(l figh3bling of length 1.5 m and begins 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 plg(37hfg44ee :brigtorque come from?    $m \cdot N$

  Model a figure skater’yaiy+ v ,f82ru +on8eqs 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 ar+q,no8yye +8v ai u2rfms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plate,7el.mq:/hku jys- ots, of ma,kh-t/y7s m.: lequjoss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 m+ 1tcfvr7xdc7 k fgy(kr5 u/:rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest pointfk7vyg m+ fccrr/ ukt (:175dx of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume ytvwq,,30/ fhaao4dm$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifln.l0bfj3zmwlef89c8-1ki s ormly from 90km/h to 60km/h The tires have a diameter of 0.90 9 0fj kl8zl.wein8 1s3cfb-mlm. (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