A bicycle odometer (which measures distance tdb; ( q0zfru1*7 fhcjlraveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use i z0u17 l j(;crfd*hfqbt on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doeavg75oyh6*e b s a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increhbe7agyv o*65 ases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body s+h12j m++ ljxk5bgsrbe described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Expcrg;o :z/kg :dz qh+/rlain.

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 wpxc(ap,md:0*n+cc4w f ;m dxh hen your arms are stretched out in front of you? A diag f;w pxn+mcm,ap:x0c hd4(*dcram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the ppn8 b4e *+whgc5.ou k-9ykhkdedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, 4k 8 -5gowp*h9yub+h.d kkneca small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have : +(ng9ue1ol xugx. loslender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). xnu(1g+9:oxg euol l.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 beagxr/) q.gpty )m?

If the net force on a kc;3vgf5q9l uwm-vde+x7x 810it s xm,7 wfsbsystem is zero, is the net torque also zero? If the net torque o efswlg8ibcd7 , mx-5 w0sq m7vu+kv9fx3 x;1tn a system is zero, is the net force zero?

Two inclines have the same heighont133hl 2 gas+u4e we4 6gu p;5eeiujt but make different angles with the horizontal. The same steel ball is rolled down each incline. On whicew;j hn4g4aeetu o3gsi5 u3 e61 u+lp2h incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an i64th;u, x6ad pogq4t sncline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the gugx6t ; odhs6a4,4ptq reater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. w nb6w0r :m 0tio84ywtThey start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotationam:o06 w wt48rn0y tbiwl KE?

We claim that momentum and angular momentum are consg:rw//9h qztm. ogt 6us3co l-erved. Yet most moving or rota gtl s3 t/z/.gwm6hu-rco9:oqting objects eventually slow down and stop. Explain.

If there were a great migration of peops/*f.m8k: dorgrk ih;le toward the Earth’s equator, how would this affect the ; /* 8hd:ki.rmrf sgoklength of the day?

Can the diver of Fig. 8–29 do a somersault without having anyz:f35zq dua ,p initial rotation wz,5duaq z: pf3hen she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its cen.zia8zik5jtt(h.:xo3k s p d5kko9f8ter of 5o8hki 9 8f5(da.pkttx zkkj3.si oz: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 ht. koue( 8e4s y4,todoutstretched hand. If you suddenly drop the masses, wilo tetod8y 4.eus,(hk4l your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identic;wfpxfqooc/f3kcreh.1 g)+d8 8piw: al and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine wpdef hw;f)o8c3i frp qx/:8gw.ok+c 1hich is which.

The angular velocity of a wheel rotating on a horizontal axlp.o4oi5u*oj ;nio -ar e points west. In what direction is the lineai.nur-j* 5poao4 o;ior 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 o:+czgg01 w l:gyadvs(f a large freely rotating turntable. What happens if you walk towarwc 1gg:a(v+z y 0lsdg:d the center?

A shortstop may leap into the1 9xha r3vq,up air to catch a ball and throw it quickly. As he throws the ball, the upper part of hiv qx 19hpa3u,rs body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of bqho1 m9 1 p*emsrr)luewke6j4 o m+67angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propelleml+m6 1rqj1rp*eo6)mw u7sk ho4 be9er can operate to keep the helicopter stable.

  Express the following 0p r9 l aphgef)82jd6w hf4r9lpyz,o.angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coin a7zpdm7 k;wf1a0n0l9,tnxmecidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Mom01,anw0a7m9zl ;p xkfetdn7on, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, kuxtj8 +vjl6hlq+w 02-ks- kn:oabgpa 7j g 60380,000 km from Earth. The beam diverges atbh2a -0 8s:x0nawjljg7j u 6v6t-kkqlogk++p 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 rate of 6500 rp8b bk1 r 2-udmzt9xya:m. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular accelmt2 8rzdxbak:b-91y u eration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3afn7vfun.wvk4. fe)ba5 .xw2:h /n lv.5 m to another child. If the ball makes 15.0 revolutions, what is its di ke:ul74b .fv.v n. 5f )hw/awxvan2nfameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many +w9uq q.dh)bt3 ++a+lrvs9 v fjt;myc revolutions do the wha;rw+vt9lj ) mhtuq3fcqd +.bv+ys9 +eels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpooi,k 0mhvde;rm*j9 7:icvrk ;-n gd8pdn 5o9 m. Calculate its angular velocity in g5o d o9:o r;nii0;k-nm*d8rm 9p7,k ejchv vd$rad/s$ $\omega$ =    $rad/sec$
(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 revot5 :m9ad/cjua lution in 4.0 s (Fig. 8–38). (a) What is/u5taca9 j: dm the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in im g5ql/dor8 :gts orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a *enc: b,vnz+ts0*qw--xfwdrpoint
(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.;k7p ps(zlldsa8 .zn a particle 7.0 cm from the axis of rotation is to experience an acceleration kns;sz.dl lap. z( 78pof 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rgr72j99tvi2 f 7fd3xzo clo3dpm to 280 rpm in 4.0 s. Determiivo9 d d 7fcxt32jogr2l7f9z3 ne
(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 radiusc*.coug7 + qjp $R_1$ 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 Moocfw lv2 5bdu0 .3, oxu;2id*axtnu(oz n, astronauts aboard the 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 a 12-min time interval. T3z t;d,(n2wf5bvuuo.xx d0ou*ci 2 lahe 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 u:*nhfizm3*m t nzn+ 8tniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it tn* n8fi :m ztmnzht3+*urn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 i(2,z 22pebhfm, va+gcag (knv;99 ty2idz uts. 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 ford:sg9qa kl0,r i 0.4i+ xbarvb the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 comp +v.b4r,ia:rsxa kblgi09q d0lete 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 uniformly fromm)j .gcmljf6 5j -el7t6 lg-ew 240 rpm to 360 rpm in 6.5 s. How far wi5)mjlg-t 66fm7- lw ec.jgjlell a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turne (vhq.y0/co mfd off when it is running at 850rev/min It turns 1500 revoluf(0vc/ mq yoh.tions before it 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 as the car reduces its speed uniformly fr z-nj38ezbt /me- bpny2l* mq,om 95yqe m2* -nn 3-/8jbz,ezmtlpb km/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 revolu((6 w 0gjtxqwt+v 8(hj:tbwh/yjkh4utions as the car reduces its speed uniformly from 95km/h to 45km/h The tirehjj08tbv+hgwk(wxq/(:h t64 t ywj( u s 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 climbing 8rrrnp *c6qo* . ,dlz13vwhtm a hill. The pedals rotate in a rnr,3rq.tz 6 clv*hp18mwd*ocircle 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 c7.ar3 v(/ttc;gjbqf*: k) se djm s6 w.uyk:e55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is b*7ws3:jgucmjet t6 fe. .krs v);q/(yakd: cexerted
(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 th0 qgxrgcv:4ww i.x.i9e axle of the wheel shown in Fig. 8–39. Assume that a friction tov9wi. :g0g rqi.c xwx4rque 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 ma r9* 0pue4ynqrssless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculat eqn*yup04rr9 e the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tiglff**sipz7yj c32xah6:d wd +htening to a torque of 38 *d+l*fwp d:h6fsj iaz2 cy3x7$m \cdot N$ 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 ooy4vd2 g.3f.o( rcljmf radius 0.648 m when the axis of .jd4m 2ol3crfo.yv(g rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia o0zrf9ha9c p4wg6 )xn bf 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 be ignore 69p f0cbzxra)9hnw g4d (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rri6 mc5(8shn4 jcli;mod is rotated in a horizontal circle of radius 1.2 m. 84in i( j6rccms5hlm ;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 potter’s wheel rotat y xqu6 i76ajnq8zan+2ing at constant angular speed (Fig. 8–42). The friction force between her hands and in uyz6x+j7aq2 8 aqn6the 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 cp,:fq k43yuh(c8j :djwjug;b f)z z6uo-nr -array of point objects shown in Fig. 8–43 ab -j gy(w4 bd6n uj :czk-zfoup;c): rhju8q3,fout
(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 whose total masme/;rdiuo,).3 12 dce o shupms is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, ed8sa 0.lks x30eusd q.ngineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is as3x d08e0 qss. .ldukthe 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 o32;ar :oethzj f 8.50 cm and a mass of 0.580 kg. Calct3 jzarho2e:; ulate
(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 batxs58tjvof4ei n 1 j+j:, 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 han5*bmz o: xson007+j j,xuqv oid-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 xvo x:izsb 07juo5,+qmo*n0 j 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 1aetd*fjefjl4 1,. s1 zs2 y1bu0,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of sdj4ts ,111ybj2*fu.a efz el1.2 $m \cdot N$ 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 acceleratgqvo17*qb3ptc+.m(rxg i7a o es 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 action of thesu+j 8kkoooe6v-mt r*/k y07i forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is acceleratev 0k7 kijyo 8*rtke/-uoo6ms+d uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be cony )c*czkac 2c:sidered a long thin rod, as shown in Fig. 8–46.:2k ccc)*yz ca
(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 massw ofa:z mv7h30es, $m_1$ 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 jusf3y -q/wj99+ tbespn+504gs xfcs turns (revolutions) and releases it f 5-tsjsn03bg ws9fxq+ sje4p /cu+y9at 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 mvkoo6jb -j2v 0lxx g55oment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine devo9v .xk+w/oq2n cbi /gde 4vu;elops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls zusy:w 0ph wg7n9o)k8 without slipping down a lane at )po809h :ksy n7ugww z3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect tk1 jc7t;q9mtakwmq))b( r*m m o the Sun as the sum of two termt()7; wc*k rqqmk9 abmjtmm1)s,
(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.jz jo.hb7v *- xs7jjq650 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8-qz.*bxj7 hovs76 jjj .00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 k -g) ktoavz .furh (4- ab,6pxfo.p(ueg starts from rest and rolls without slipping 6uh .-4bru.afvoko- za extp, gp((f)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 vertically oblnzu bhape3.8fr, r4m15kxpkt2o q 5-vr9 ;i n its tip. It starts to fall and its lowerrz a 48it; 5 9rkovrel-qm kx5,puf21bp3nbh. end does 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 0.210-kg ball rotating on the end of a thin x*1ekiv p. go h;y*/ekstring in a circle of radius 1.10 m at an angular speed of 1ge1i e**.p/k;oyvh xk0.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8)w f3byv+wk.dr r+sc- /)bs bm-kg uniform cylindrical grinding wheel of radius 18 cm wys b ++cwsfr)b .)/br3- mwkdvhen 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 si75 w1wj h(smeli-b ry)de, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal pol7ib wwej5ys-m (rh)1sition, 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 Fig. 8–29) can reduce her moment of inertia cmo 3q6f7xlcs*+k0rp (u z1y iby a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations ix3im+7olc(cq6z r0kys* f 1pun 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 increase her spin rotation ratrb 2 :f un8rgljw) (gn9h; wai54q/xib l9kyv5e from an initial rate of 1.0 rev every 2.0 s to a final rrxl5fna gyk9hn/i;q5bj:)wi vg u89(2w4 b lrate 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 center azl 7xqs.aij 4rau802vt a frequency of 1.5rev/s The wheel can be xisz 2 ra8jula4.70vq considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular moment:dxzm6s 2k- ucum of a figure 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 7s yq5zedx7 ky 0d7q9ithe 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 ontorur.; v(he8ml an identical disk rotam(ruhr8l.; evting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsg)znl6 jek 0hizo:w:/ at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 / f6km*b;5huq m 5afbqkg stands at the center of a rotating merry-go-round platform of radiu5qkh ma6bqm ;ff5u/b*s 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating fre 7xg6c8pq-cyq ely with an angular velocity of 0.8 c6pcq7ygx- 8q $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, los(z6 m;r 0 7*b(ak xxepzmjaf,ving 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 a xvkame;j 6z*p(r (mbfzx 0,7angular 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 +j-ns / slc y v8/fpmxm hse(h45yj(1hexcess 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 1 yhxatvkx;k b19p*:+i. hqxp$ 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 resb2yfwr ,tz9,0 rnm4p kt, that can rotate freely without fri p02zym49bnk,,f rrwtction. 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 pok,0o(va, *b-pbhpotg 76 vil.wc e1the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless 6,-t ppba,ivl(og *1hecw0bo7. okpvbearings 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 with the end of theyzs)q-d:.nk .q*zgfz 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 fqgnsz:dq f.zz.-)ky* ar does the spool’s center of mass move?

  The Moon orbits the Earth such that the same sideyak4c8w,;ls28vwuj p wge7ksm,h 2)l 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 )8;lvkjk,wy7 sms2ewlaphw8,u2 gc4 case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 ml9 gqbz5.zz2 5/vygsm/$s^2$ 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 of radius 0.20 m acq/93qv stz7 sdwuires a rotational rate of from rest over as7v3s 9zwqt/ d 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 :q5ek dz e5l6: y,vjczmass 0.050 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 tovecy:,qz65 j 5zdlek: 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 spe 2ikstkrnob080oh1qqp ,h(gu b/r*9 jed 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 So0*8p4lqxz m tom zz b6 i0tloo*e. t;.gq+rr,un, 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 collab* 0q4qe tx.*rti om tzo+g6llo0zz po,rm.; 8pse 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 off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution q8jla tx(atu52hdz*+automobile is for 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 mass 240 l+dt tj*2u5h8zx( aaqkg, 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 an k,wdr 4up tr/:)l1obo$H_2O$ 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 (hoooo2jijq12 e 2mp) is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely 0qaud,rr x;f+,qasgmab (n5 8(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 acceleratio m8rx+0nq5(gu; sqbra,,afa dn 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, andjx -, rt7 ssz)6tvb0 tokr u-8ut7x2pk we want-kbr to jr-68st27ukxtx )up,vs 0z 7t to exert 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 speed v=4.2m/s on a flat roajn9.8orw sng)0ba/x6luei;qg7ej+b d is making a turn with a radius The forces acting on the cyclist and cycle are the normag 7+;xngj oqean8i0l j b.swr/ueb96) l 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 agu0beidu d:ybt/3, n: 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 , dbyntue3::iu0gdb/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 her arms i0eb8 bwm7x )ax: t0myk1( dfyas thin rods, making reasonable estimates for tdmfxab kmy)7twi x 0eb: y180(he 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 cylindrical plax-r *-d0fr(irglg u9)u tom 1utes, of xgomr gr-iud1*9ltf -r(u )0umass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 r d;n1t/ buo . fu,h-e(wxa-rpwolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop eh;f rw -no( u,bdtu/.axw1- pif 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 t/wp/ (pln-r;yqxx a4not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions a rc3 s9y*90tjnj agj7zxa08 k*cbk5zys 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 angula y9bjayjk**x0ngc7rk8tjsz3acz9 05 r 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