A bicycle odometer (which measures distance traveled) is attarulqc :y/3q /1v5g*lv atd y(bched near the wheel hub and is designed for 27-inch wheedc/( t qgry51vvb: 3/q*u lyalls. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at o29.y:zn2l y/ju yb edfmjp:pqz2a9 4 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 radial and/or tangential acceleration? For which cases would the magnitude of either coq 2mobj9lj:y9y fn/y4p zeu.dpa2: 2 zmponent of linear acceleration change?

Could a nonrigid body be described by aq x0 2ao4llsn- single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque tpkiy;o. 7o,ul han a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head than whu j0ljgid 11b+gi39kr en your arms are stretched out g0k r+93 1i idjbgju1lin front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven kj5c.z7 ;vf-+ zaotidxrs1 p+) s hh4gsprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In whichvhc 1o7-girk4 f +x.+;jdt)s5psz hza gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on )tcnn irha/w0 /l lp;; uc4ib*being able to run fast have slender lower legs with flesh and muscle concentrated high, close to in 4hb; /)/0c*wnlr;liut pca 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, narrow beam?c r o4 qp5)jl4ilr0p0i

If the net force on a system is zero, is the net torque also zerolzp u/d4/;yry,gg,/*i l oqull7 z .lf? If the net torque on a system is zero, is the nez4u./,/zyy/ilq g,f *rl g; dl lp7lout force zero?

Two inclines have the same he5 qx4o;vgd ef:ight but make different angles with the horizontal. The same steel ball is rolled down each incline. On which5gf d;x eq:4vo incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incl7o14ga;b by6nrw3vgz ine. One sphere has twice the radw1y637ra ogn4 gvz ;bbius 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 kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the sameo:wfuin2n7n/ :77bun ,pt vxy mass. They start from rest at the top of an incline. W 2pn 7iubnvyf7:,/7:tn xun owhich 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 and angular momentum are conserved. Yet most;+ migxtbkv)* moving or rotating objects eventually slow down and stop. Ei*kg; tv+bx m)xplain.

If there were a great mign9(*u9ks yo;mt4k ojsration of people toward the Earth’s equator, how would this affect *j9m4ot;osnk9 u ys k(the length of the day?

Can the diver of Fig. 8–29 do a somer (iy5wuby0d-g sault without having any initial rotation when she leaves the yd5gibw 0y- (uboard?

The moment of inertia of a rotating solid disk about an axis through its xur (fvai9s lh:+m.(zcenter of amvz.irf9hx+(l us :(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 outstret k4:xxygs)ncr.mg*ic 1u3 n )ached hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? s xm )grn3c:ni1xya4c.* k)gu Explain.

Two spheres look idenk/)ic( jii9pm tical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to ( jm9ic/) pkiidetermine which is which.

The angular velocity of a wheel rotating on a horizontal axl4o6g hdt,vf l6(ixcs a3in78l e 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 acceleration of this point at the top of the wheel. Is the angular speed increasing or decc g nsio 4lda7h6,3f lvx8it6(reasing?

Suppose you are standing on the edge of a large freely rotating turntable/;kb-7 hxc yhd x;xhqzp58v1ees7( hg . What happens if you walk toward the center?ex h;18 hvs;h5xd7 /ez gbx(cp7-ykqh

A shortstop may leap into the air to catch a ball and thrnhax z o5n.0g/tgo 3kus9d:r7ow it quickly. As he throws the ball, the upper part of his body rgo3 5z:ngo.7 uxktd90r nhas/otates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservavp6n sw)/iv ja7+y4c tzo+.cr tion 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 keep the helicoc7p /.t 6jw+r) zavi+oyn4svcpter stable.

  Express the following angles in radians: (a) 30 i4l4w,0u-wibyq8yp1mt v : 0o6 wzktv$^{\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 coinciel f534g3wa;nirlds :dence. Calculate, using the information inslwa4;s 53: gli 3rdnfeide 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. Thejw-6)am 6 :cmo rn(i6/fmarnff4;pcs beam diverges at an ar4 w c rm6sc/ op6-nnf;fmm() fia:ja6ngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpmbag3p*c( r3dyh.ga7 a. When the motor is turned off3*rh.gy(3 gc aaa7bdp during operation, the blades 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 ck;0ciw9 jjhm6hild. If the ball makes 15.0 revolutions, what is its diameter? khj w9;60jcmi    m

  A bicycle with tires 68 cm in diameter tr , x9 ;*+o,4 o5txpoqmsnqke lp6ksil-avels 8.0 km. How many revolutions do the wheels4l;nex9pt,*- slm 6,sq + pko k5iqoxo make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 25fq k-lk5 xd(/a00 rpm. Calculate its angular velocit/q5klxdf ka -(y in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s23*w0q s uvjw t/qf1kq (Fig. 8–38). (a) What is the linear speed of at3/2uw q1q qfv 0*sjwk 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 8zzo6)9nq4 dvq heul9sf3gh2) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point9: r7v q8uqfnw /kn 8tfsvg y)f,iy0,m
(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 centrifugjc ed73 5sy+3f5uvre;)h,tw gl paky/e rotate if a particle 7.0 cm from the axis of rotation is to expera5tl ve ke/f)hr 37yyugj3+dc5, swp; ience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its cenopnysy b4j6gr6a3 ,hl( u7m+yter from 130 rpm to 280 rpm in 4.br3 6+ny,yhsj7omlu4 yg6ap (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 radius ;eeegh, u u +yj b)fc;3xitp206,kuji $R_1$ 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 t *-x :ldw)mbkmhe Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenld b) klw*-mx:my. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from , hvagb/rhn4. rest to 15,000 rpm in 220 s. Through how many revolutions did it turn inhgrab4/. h, vn this time?    $rev$

  An automobile engine slows down haf. snb *0jz -i,l-mpfrom 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 flying highspeed jets in a whirling “hah*y.0gbk n8 a jd2;h+9cfpis uman centrifuge,” which takes 1.0 min to turn through 20 complete revolutions. h+f2ac *ki b;0 d9spjnay8hg 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 uniforml *y hbom* p 7z eazvt55ewi9yrej,39x*y from 240 rpm to 360 rpm in 6.5 s. How e95 m5zy*yw3pb* , hojatvzi *re79ex far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 revoluyi. l6lj:.gxutions before i.yl:6j i lu.xgt comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions ngv5kzc4u twiu8ot8n k dm+zc;z)85 qf1b2/g as the car reduces its speed uniformly from 95km/u/wdtu2qckvgi z+o 5fz85k1)8g8znn; 4tmc bh 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 g. s;p2ed/ aykcar reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m. ep d/.a 2;kysg
(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 bikeprzi;d.seko e+*i +)w puts all her weight on each pedal when climbing a hill. The pedal);k ie+ ewzso.id*+ rps 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 endqcg et dxs5fv,xaj) 7lc*+)0q of a door 74 cm wide. What is the magnitude of c gcd,*jeq f5s7 xtlqxv)a+) 0the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque1gd: 3xrqsb;cvag7i2s )o,az gi .r-l about the axle of the wheel shown in Fig. 8–39. Assume that a friction t3zida q7g;br :gsv,g1oc-2asil x.r )orque 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 atpps izq:1i,8 massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position ands:i 1tpqi8p, z then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of 38wag4j -tq pk -3f-x*yf*--jxt 34 kfqwayf-gp $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m rh c+m-2j o(cbm5tfy*/mou1 bwhen the am+*/- ub(b5 1 oorfcjmy2htc mxis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyw (j 1sqhji rgwe5 yn*:i-vhef 3/u+1 mepgj,*cle wheel 66.7 cm in diameter. The rim and tire have a combined mass3+gwj*u* qeii/ jje hwgym (hv5n r1f:,p-es 1 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 thi5oj )d7c60g b mwlba*4s3 ejizn, light rod is rotated in a horizontal circle of radius 1.2 m. Caloszm63dbci54ljb 0)*jw ge 7aculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rota ih ,x:pn)l4xeting at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N totalx:,ehlni4 x )p.
(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 pj1 sa(( z.vucbzp.lmbm*j m72 oint objects shown in Fig. 8–43 cm.(* ba jz1mu v7.pzls 2b(mjabout
(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 cons j b wsh:85oj1wecj7fb3 dzvrvr4 .+j5ists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrh i1bunv: 7b.mical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of h.m v iub7n:b13600 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.oa*r wz hxbs.mu(v.. lg.sk8 with a radius of 8.50 cm and a mass of 0.580 kg. C. k.8gsv a*.s.(xh burwlmoz .alculate
(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 from rerzz6aupda/ d23 rx:) u)bkq 2 )bkmry(st 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 h.x niauyg:0(qsx5 ,us rn+lt a8+i, agand-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 s8n rulaqu+,ayxx (g 0t.is:ni5+g a, children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm isg5qw7fn8xm7cy7 su.,ke at+j shut off and is eventually brought uniformly to rest by a frictional torque ofjc+yn 77a8 .gmw esq7 u5xkt,f 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 a:3g9q8wcol q:vltn:m7damr,, pr *e 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 ac f3 ;zxblq27p/ nl7ebxtion of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accx;l2 xz be7q/p7ln bf3elerated 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 bkszfur.qd pyqr. )w(9g7 9 u.tlade can be considered a long thin rod, as shown in Fig. 8–46 9u.dzwyq.ksr)9(grftp 7qu..
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two ma, fec9sduh*sjqbn/ 65 sses, $m_1$ 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 full turns (r6q fitqbv 16jrn /n*2jevolutions) a2n61q*b6tjfnj q riv/nd releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia o9dn1s/z i;fejy1uq+hldk8( a f $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a toro3l d; 7f1iu*+osiwsvfpemmk 18.sc 0 que 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 radiu+yb xzpc23d8 ss 9.0 cm rolls without slipping down a lane at 3p c28+3s ybdxz.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the:-n v z g+:wf-rk*6dy rvfv7dy sum of two terms, -ddy vfz6k*gr:vr:7wv f+y n-
(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 tz;q oxkn7(f86p u(3kj 0h vikmass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it k0fn(k;3h6zp(vqi7 j 8 oukxtis a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg j m ,gb0u 3fha,du()qztx 3ys4starts from rest and rolls without slippifjdt),uyg sab,xh03 u 3z(m 4qng down 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 verticall+war0pm97 snlc(ob8ky on its tip. It starts to fall and its lower end dosnlb pa7 (+98k o0wmrces not 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 ke/o g/ :5pds l6i(roa qhv8w(0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an aneqvior/kd gw:(56sah(/ o pl8gular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8- qqp1lscia.*r22k x9tbdz h *-kg uniform cylindrical grinding wheel of radius 18 cm wh psa * d2k12*hcqiqx9zt b-l.ren rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate*i8ds0m2 1oikuf r ;he of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotatio 1muife *h0ri; d28skon 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. 8–29) can reduce her moment of inzie8 yc,u ya7x5bp4f:ertia by a factor of about 3.5 when changing from the strp8ayizc, :bx5 yf7u e4aight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can 4p sz gg*v he9,p,j)x2q kk0dvincrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate o,vk)ve g20 ,pq xdj9phsg4k*zf 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 a tllvtgdkkh h/e2677* xis 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, approximately shaped as a flat disk of radi t hht27vllked67* g/kus 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 spinning at 3.5 o+s .m dyl -o1.aagz-v$rev/s$ 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 jutd/*vl: c/ o x(dwwc f1/hm7Earth
(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 o-dr1oqezhzf55gv kh+ c++i8mf moment of inertia I is dropped onto an identical disizez+g+ dokmh5 vh q-+5cr1f8k rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at )wo 3n+5kmtpz kim;p1 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 mob eb vmxbo/)2e))8 mdyk jx1*erry-go-round platform of ra )) kmj/bodbe* v)xmey1o8b x2dius 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-w 9iw)np5q z2y7 6mnbfround is rotating freely with an angular velocity of 0z2 wnp5ib7y n )mw6f9q.8 $rad/s$ 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 gxy82t-qg1r6eqrk ylt v / fczd a3 90nx51r2linto a white dwarf, losing about half its mass in the process, and winding up with a radius 1yg2 qa f1k1-c 0/g9t 6x lrt8zrdxy25nerlvq3.0% 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 excess otb lu.0jufrah44k j6(f 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 otebfm o0a, 5)(y3jnj$ 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 platfr i3j 43p78cswayn e4xorm, initially at rest, that can rotate freely without friction. The moment of inertia onej384rp a3xy 7 i4wcsf 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 merryzi f1w 6s0bbq7w evu,md 9qe2.-go-round turntauwie9vz6 bd.2q,e qfm s1b0w7ble 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 rolls on the ground w8 xk +x*0unx0o-uurolith the end of the rope lying on the top edge of the spool. A person grab0x+ rx-k*ul u0 ouxn8os 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 does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Determl)ht bpw;9. bdine 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 p h.;t)db9wlb Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at asx o bjp. ec2ugd/n86* 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 cylindertrv-3.; j z4abu3g atx of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torx 33b;atru g4v-zja.tque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 *p+vxvu- f9v hkg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg f h-+v puv9x*vand 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 i n;;nou7c9bvweod ar 4 51z1uis the 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 of our Sun, but with mass 8.0 times as great, were rotat,mkswiad rz)7v0y; v0bm87 vming 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 carries of)d80 m v77ky0zvsi,r wmmvab;f no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to wjqyo /vpo 5v6bv981wbvt8 t 2use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, 8ov8v y/wpvjb5t69v w1t q2b ouses a 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 illustrarn. tm4qn ,u-m.ejj 82 tf+lejtes 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 horizontal s zheo,fkc0a 57u c6yt+urface 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 i wuv7g)maxy riwt/ -3k5 1)/wjnyoya .gnore friction) about wyo/ yv.ymauwr5 1i w)j/nx-atkg3)7 a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verthbvikbde6p+;0o i8sv*/9 ;o9 ilrkbjically on the floor, and we want to exert6 0beikod9/rpi*vi k b vb;j89lhs+o ; a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with spbwka:r/o qw- 7h ct(m5eed v=4.2m/s on a flat road is making a turn with a rbk:-r/a (7hwco5qw mtadius The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.506ccsgpor ifuo*v6fp2;.0b f gz* 9b4z-kg rock into a sling 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 feabc66i;of p**.02sb zrg4o upfgv 9cfzt, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, . :vp 3ptr a-ymtybq1nwv-7*lmaking reasonable estimates for th.1y vt*pq-trl ya37p -mv nb:we dimensions. 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 which consists of two cylindri*r;2 j r w:w.oax0 garow7z3y4wx of6ccal plates, of mfwrx2. wa * ro6xra0zcoj3w:y g; 4w7oass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track -: u9ifxow86ixpz m0rfeq te+bc+)b:.a l5so of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to rulf.ri 5+:z staqebx:c bo-i x0mf) opw9e86+each the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dyfqt cz:tgzck)7/:2hkc.mh u) y;nt3o not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformlh)dpb16r,qtt d, 1xlx y from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) Whadqr )dtp6 1,xxb,l1h tt 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