A bicycle odometer (which measures distance traveled) is attached near t2lu 9i.r64gwylitjoiw cucg+.a.2 j2he wheel hub and is designed for 27-i wt .2uy2 o.j+l6clij gruia wg49c2i.nch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angula(xnurji, wjh6 *7.dj yr 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 component of linear accxnhdi j,jyw*67j(r.ueleration change?

Could a nonrigid body be desccwywg2 262zybr r)7f kribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque thafwf-z3c; n(p 7ejrfyy 87k4qpn 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 sitxx*; .vk4/huz v u u/xb 3)trwqw(0djg-up with your hands behind your head than when your arms are stretched out in front of you? A diagra jkuv wxhuz)x 4g 0x/;q uw(3/b.rtv*dm may help you to answer this.

A 21-speed bicycle has seven sproco2r9r- oo: hjifi-n al3:qw jrw:6pd:kets 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 which gear is it harder to pedal, a small front sprocket or anl 9ph-dqior:: 6 a-:r2rjwo3:wijof large front sprocket? Why?

Mammals that depend on being able to run fast have slender lowdz d jsnr.f02x:q:l/ yer legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the : fq.djlz2/ny:r d0 sxbasis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkyy:;e9an:gwj n4w-en :rh+vg ers (Fig. 8–35) carry a long, narrow beam?

If the net force on a0rlddj* p q)j.5are 7r system is zero, is the net torque also zero? If the net torque on a system is 7pqjl.rr*0d j)da5e rzero, is the net force zero?

Two inclines have the sam cnv4m6e0w*n5b gwa s7) dwlo1e height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be great7 l4amcwdw vgnn)be o*1w0s65 er? Explain.

Two solid spheres simultaneously start r+c,t(o6 (sefmi j k6wbolling (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 haswoi+be( k6s tcj6m,( f the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and 2bhlj,f13.y u67sd8sc ( mw g0.9tb rcdmju pwthe same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Whisgjfdb c.plj s9 (th2.,y318wcmb6 u 7wudrm0ch has the greater rotational KE?

We claim that momentum and angular moma8vpmk 8fs23 bjjoc( )entum are conserved. Yet most moving or rotati8)2m pjjf8s(v kbac3ong objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, hnp eckv1 06dw(ow would this affect the length of the dw1pc0v6 (ken day?

Can the diver of Fig. 8–29 do a somersault wit3*40wfw 4v7+8zo 0xos*u l tkttkyxughout having any initial rotation when shet3l*0v k u0xxwu 84toszwoyf74+ tkg * leaves the board?

The moment of inertiah0dsy)4n1sa/j cfri28c0(fc vmknqf* vx;a 4 of a rotating solid disk about an axis through its center of mcsrak /niq ma(;c 1 2fv8jnfdx0440f yhs) c*vass 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 g-)cx eyg30o lfuhw(ro( p* ,za 2-kg mass in each outstretched hand. If you suddenly drop the masses, o* uo)p,x(f3h0ze gcy r( lwg-will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howmre3et8o69i m1jcv 8n ever, one is hollow and the other inmrmi3c88 o vtje961es solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal ax8h 95 r+*dk b3dgzinii .yse8xle 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 a s+iig3b rni.d xh 9e8*5dyz8kngular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotatin)h6b elxa56tha(9ncug turntable. What happens if you walk tol h5)ea(hx9c6a tnu b6ward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As1s 0rli d6culm,9)pl i-kn0ng he throws the ball, the upper part of his body rotates. If you look quickly you will notice th0d r 9-s,0ulilmnkn1 gpcl) 6iat his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the 6;x-1 kvmqgniot1ac boza 4 0*law of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propo 10 m64;taxv a1kq ni*gc-bozeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angl/akjgl8,n, coes 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 b8oxc*bru /.;:c0zwjpf,;zqvl/ 6 doyip kf 7mecause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Suj;xl8yirmcc/ d*z ,.p0ufwzb p;/q6o:fokv 7n and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,0v. fr05-*se5qj efbfgkdz lyn/ ym 0:pma4r,+00 km from Earth. The beam diverges at an anglea/5.m 0fv ml*,rjfkeg4y0 dzs5ye:q +-bfnr p $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotg*.0ipxikd z;ate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is thei.p*0x igd; zk angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to a h dpotdj9*-w b*07oelo*awb5: 3aja.g bzau4nother child. If the ball makes 15.0 revolutions, what is its diameter? w * bho:dtdawlb4oj *7zaga.u-eaj 5ob 90p*3    m

  A bicycle with tires 68 cm in diameter travels 8ml1,ehpsg8xz91 +vr0a7x p 8cidnny, .0 km. How many revolutions do the wheels make,mdpe8,1 x rin 8gl9+1s y0czvahp7xn?    $rev$

  (a) A grinding wheel 0.35 m in diameter r0zm oy0 efj-j .4a,jicw-z2 (ynux+wpotates at 2500 rpm. Calculate its angular velociec( j-iyjf0w 2jnpz,0+zoym.4a-u x wty 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 4.0pa e+7 -,2xkn3cjjddy*a-fbx-p7n hh s (Fig. 8–38). (a) What x*n y2kxd-ea73p-jbhjp nfdac+h- , 7is 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 thezg*fz i s4ghpv ;j740msv5m x1 Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe qtt(zqlm.)q9b qr clenq;4( -ed 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 texg1 i+m n1a)8 dww./yf;hzxparticle 7.0 cm from the axis of rotation is to experience an acceleration of /1 +e;z)fxw8 tg.wmxahd1y in 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abouttv)lu gpal90 d1m rp8: its center from 130 rpm to 280 rpm in 4.0 s. Deuvl8 1):0plrp at9dmgtermine
(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-ddj k v:0q)tk $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themsel+mx 5tfv.2v4lckrfzp 6lzl30ves 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 mincl.rzzlf 6+0vt4x3f5 pl mv2 kute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 scos*jgclh - f;s5m5 6s. Through how *of ls5s hc- 5sjc;g6mmany revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 450003v/;.ftkp s wl +mgdr 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 a1hd xxew -5w/vi)q(w o whirling “human centrifuge,” which takes 1.0 min to turww5)1 d- /ohxxi(vew qn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter acceleratemw+r.4f-ca p p7qn9iqs uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have travr7nmi qfaq 49-+wc pp.eled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It 2s( jbi*1:2t puwte(9 mmoyt gturns 1500 revolutions before it comes (o :us*t mjweiyb29gtm ( 1pt2to 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 reduces0.;bu lp6j+*h n ld.mzgq,tzd its speed uniformly from 95km/h to 45km/h The tires have nu+.m6dj, z* qgb. lzhdt0lp;a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car rfri u+9r 7pgsl:bg- m6educes its speed uniformly from 95km/h to 45km/h The tires havegrrf b +6l 7mgs-ip:9u 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 weightvcd.ti4;an+uv 84:a 3 dyoznu on each pedal when climbing a hill.y 4:ond 4u8v3 nui;avatz +d.c The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm witbovrn+upx6( o/;f y2de. What is the magnitude of the torque if the for/t(yb;2v6urxno +op f ce 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 the wheel shown in Fig. 8–39. Assavm 9q5a*5tuwmy/m.s ume that a friction torque of 0.4 ms5 qmwyuta.va/9*5 m$m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, arf,qv( e e an9kc3hd((ue attached 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 the net ,va heqf9(u(kedc n( 3torque on this system.

  The bolts on the cylinder head of an engine require tight; azaz 1+4pn6f9lqpl qening to a torque of 38 lfl+n z64p9qazpa ;q1$m \cdot N$ 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 ou xs s (2gb,0kqun;oy z102cfbf radius 0.648 m when the axis of rotation is through its cente1,sxzonck g s2ub;uq00 (yf2br.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel as6) eb/xrx 0z66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The r6sb0xe)x z a/mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizontayt e2 1y)f chevgg,/3bw:zn-fl circle owfgy)2 v/ng ec,:e y-bt13zhff 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 potter’s wheel rotating at constant angular s,1k vv(d ( 9rye /u)jphnrc4ggpeed (Fig. 8–42). The friction force between her hands and the cla4hk( n/ cjyv (grpr,d)v9ug1ey 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 ofbgdnbxg 7xy bt4/i;m4x.( ni * the array of point objects shown in Fig. 8–43 abot7.ixmd ;*xgn (y/bbn gib44xut
(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 cs *fs0fh2z-n donsists 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 cylindrical satellite spinning at th9zp3e-kgr cv4e correct rate, engineers 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 the requkvp e g4z-c93rired 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 anxmxkz5:uyw : wtp(a:((p4d9eol y; nx d a mass of 0.580 kg. Calculaupk (wx 9o p:mz5x lnx4;t:y(e(aw :dyte
(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 rebd52:jz9 3t nmtbwdi( 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 hand-driven lrpfow -re95;rp00 fkmerry-go-round and is able to accelerate it from rest to erlp0 rr5 fo;-k f9wp0a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 1 ik7lwrrjab6nt;g; -8bnv 0v 30,300 rpm is shut off and is eventually brought uniformly to rest by a frictional tor0r vt;ai-7;nv rj36gknbl8b w que of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball at 745k/h mmk bdzqpt z1c(,yd3m 4 $m/s^2$ 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 the forear -y 0;sacsul46cd6bri m, which rotates about the elbow joint under the action ofy6udlac; s 6rbc04 s-i the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade canbh i(7 thp1uma82ht/a be considered a long thin rod, as shown in Fig. 8–46.hiu1h2th8tma/ p 7 b(a
(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 masses,r7nqn / dvqi.v52 yk8yz8.wvk $m_1$ 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 tw:ldt) p: hrqy2u)ny3urns (revolutions) and d:y h2u3qrnwyl: )t)p 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 vm/h95 1qrucna moment 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 develops a torquesbzn k 3-jbe;- 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 without slipping doww (n*t:7bf *:1 bmtdxwcm8afdp es9;wcv5s* wn a la1 x9 (wcwsm* *d w8:dn*fs7a tbp;w:ebm5 vtcfne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of thdsg-c7c4vt ;ejg ;1fzj ka5y- e Earth with respect to the Sun as the sum of tw f4agkjz;gs715;v-dccye- jt o terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg a1t/taj koy l- eiq2d06nd a radius of 7.50 m. How much net work is required to ac /etk0dy2-jal oi6q1 tcelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cml4 lgl1-1p+9e o/tzqj vn thv7 and mass 1.80 kg starts from rest and rolls without slipping down g1 9t+nv t7v l4-/ jh1eloplqza 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 t ws+vs j+xgljtxz3 89 7im6ru187rmjb ip. It starts to fall and its lower end does not slipm3sl 9si868uwx mg xr jj71btv j++zr7. 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 o +s+f. vzy*qjgf a thin strig.* +f sj+zqvyng in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg u:y:3 (zroeo hgniform cylindrical grinding wheel of rado yrh:e 3g:(ozius 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 hi,+jz2xwq7 rx fckd:s wph 7;3j3 -xyjm xcx91ks side, on a platform that is rotating at a rate of 1.3re9wxj7xdrjk z3m,q:whs y+jc1fxpcxkx23 ;-7v/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shh 5a v.cdg r8bw*+drp7own in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck positihw7 8dprg.a drcbv+5 *on. 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 increase her sp tsovl)9 4z/piz56 jfein rotation rate from an initial rate of 1.0 rev every 2.0t)es if965z4ozl/ jvp s to a 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 aroun4uto :jr al7oua) .c2rd a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws.uua j):7 aolrtr24o c 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 figure skater spinning at tsmb/ l(l(t9t 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 ,arg (m -op3dh;; rcgwftz::gd:ui (uEarth
(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 momvtb4 vuaoj 1w63)fifsq8v, j3ent of inertia I is dropped onto an identical disk rotating at,a wt4 jubf)s631fviv3joq8v angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ata6nglv4fmdln 0 c8(z 4 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 rotati)k:ys5n w97d91deu o t*rszo/ktzr 3z ng merry-go-round platform of radius 3.0 m and moment of inerti d z/tkzu:zker91w9tno)*3 5yr7 ssdoa 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 with1j/yhwfg l9 3(qxc/yz an angular velocity of 0.8 y/ z(gjhl9/c3xy1wqf $rad/s$ 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 dwarz)7lj90 ohznaa .ei k3f, 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 new rotation rate be?(round to the nearest ino.a7lh9ezn) aji 0 kz3teger)$\approx$    $rad/s$ (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 excessx0l1 fby;z+p:rj 9xjt 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 (eml j3 jpkk c.3l*gwj-l6lp: wdb -d7$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, thaejex-v kc2/s+l,q9 rz t can rotate freely without friction. The moment of ic z+-e jsq9xvl,/rek2nertia 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 syzjju 9.:m8 kltands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertiaj:lk9 zmuyj 8. 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 gro- jo5hp 0x+ .xzaiq4ntund with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. Whap-jati5x0z4 .+nh q xot 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 sid4e h j,k4fm/nvw5;dg,ofg :jhhom - l2e always faces the Earth. Determine the ratio of the Mmjl4hn me ;wj2ho- gvk f,dg5 /,:o4fhoon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rateunr+q.o, 8ysdg6 g0j l of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acqui0ow qomkz*ks)j 9b7 a.res a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. k0qm)sozk ja*bw.7 9o Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid )sfh t5/oa;d a61hvgh cylindrical disks, each of mass 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 to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it f h ;a1/hgtav5dhs)o6 is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how99z; o.dh 9ysek9 jzan is 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 S xsh ic71(6d xh7l*h4,j 1pvxz3jkgusp 53s ciun, 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 v65sp7(c3ujh xs1,ip1hs7zlhd4 cixxkg *j 3 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-pollutiox1lxit5)k7.t ks3jjy n automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car h7k )yt5sk3 ixlxtj.1 jas a total mass of 1400 kg, uses 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 illustrates .jt2c1siib 7c 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 hori(b j3 ( zl.o+* kgnivgpw9sf)ozontal 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 yg s1e1 8qh6phfreely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.g 61hhqsep 8y1]

A wheel of mass M has radius R. It is standing vertica vl *)7kyqj1exlly on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which jevy7l xk*q1) it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed9 ,ay.sfms 3z) ic-xcp v=4.2m/s on a flat road is making a turn with a radius .-szmxcc)sa3fi9,p y 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 0.50-kg rock /i.v5zetzjs-+yyx*fitj( +zinto 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 sv++ ( ezt*-f.yj z/i5zxitj ytorque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a sol r c//(z*heotlid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly againsrhco/tz/ (l e*t her body.   

  You are designing a clutch arsc)wxk-umg )5: d+ddg ksv22ssembly which consists of two cylindrical plates, ofs)g xr2)ddgwk:sck+5v 2 udm- mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped ro24b n(c, 3a zuzbpch6gugh 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 point of the loop nb 3czu zpa4b2(hg c6,without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but 2vgqhbwl hs mef0/;x,f*q98 f;zq 6yido not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as thezcoagjtj- ;92)ia)g m3:fem v car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wa): e;9ga3fat-2i gjc zmjo vm)s 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