Sometimes people say that water is removed from clothes in a spilp(qz +w-td-b*9zfo p; )- bdz pv5pgzn dryer by centrifugal force throwing the water outward. What is wrong with this statem+--p(wog b;z*9vld 5z-ztb) pqp pzfdent?

Will the acceleration of a car be the same when the car t;f lvhxaeie( 1 5mvq ji+f6nvojv26; 3ravels around a sharp curve at j;ohiv16fv+ i qv a5ej6m; (lfvenx23a constant 60 km/h as when it travels around a gentle curve at the same speed? Explain.

Suppose a car moves at constant speed along a hilly road. Where does thep -*i/ yjf3oe 0;www9iq zh,mh car exert the greatest and leastqy/9z0pjwm*fe,oi3-;h hiw w forces on the road: (a) at the top of a hill, (b) at a dip between two hills, (c) on a level stretch near the bottom of a hill?

Describe all the forces act rg-y,1r 5st cnf,y;hxing on a child riding a horse on a merry-go-round. Which of these forces prgsntyxfrh1rc ,5, -;yovides the centripetal acceleration of the child?

A bucket of water can be whirled in a vxoj6jon fr(rdu 1t); .ertical circle without the water spilling out, even at the top of the circle when the bucket is dr (6u;ofj n.tx)r1jo upside down. Explain.

How many “accelerators” do you have in youzmc:ty(-8 mq +-jejb+ pui2qxr car? There are at least three controls in the car which can be used t:8qm yx tj(+-cjz2+uqmie p- bo cause the car to accelerate. What are they? What accelerations do they produce?

A child on a sled comes flpqmvl-(/o b d9bk07duk m81og ying over the crest of a small hill, as shown in Fig. 5–31. His sled does not leave the ground (he 1 b89p-(0 qk odgkldo m7mb/vudoes not achieve “air”), but he feels the normal force between his chest and the sled decrease as he goes over the hill. Explain this decrease using Newton’s second law.

Why do bicycle riders lean inward when rounding a curve at xxkx5lxw5jk; 3bxt .v43m1llhigh speed?

Why do airplanes bank when they turn? How would you compute th ; zux l)p+ud1jht+e r03h( kvv0*kpnxe banking angle given its +0*xru( th vvhej3xp kl 1);z0unk d+pspeed and radius of the turn?

A girl is whirling a ball on a string around her head in a hori8vi+;q x /*pdam-kxbe :oxn z+zontal plane. She wants to let go at precisely the right time so that the ball will hit a target on the other side of the yarx:8q;z*a b i+o/+vemx p- nkdxd. When should she let go of the string?

Does an apple exert a gravitational force on the Earth? If soa hkn i,kvm -axg)upnch+l410yu(4q 7, how large a force? Considu q7a hl( 4xpa+kg4h,nuci- v)ynm k10er an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, g*s ywwdm6e ;0in what ways would the Moon’s orbit be diffewgesy *dm60;w rent?

Which pulls harder gravitationally, the Earth on the Moon, or a*nmfet il 9ek5t8ft9o(zu7k/ 0*nrl the Moon on the Earth? Which accelerates mor9tml0*atr e/tl8uk(5 eifk9n *7zon f e?

The Sun's gravitational pull on the Earth is much la eqahdu fe70; 3tn cs;fe5am)0rger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from ;udfq e) et7;nha03 0mfea5 scone side of the Earth to the other.]

Will an object weigh more at the equator or at the poles? What two effects are /k ,xao1r6yzhu -cg5s at work? Do they oppose each other?a c,k/1z s -ohgy6xur5

The gravitational force on the Moon due twu5,ayf b/kdt z8irr q/ m3i+:o the Earth is only about half the force on the Moon due to the Sun. Why isn’t the Moon pulled away f mdw+iaiz8 kf5rqt,//:bru 3yrom the Earth?

Is the centripetal acceleration of Mars in its o4kl f2mw-cu .orbit around the Sun larger or smaller than the c-owkf m24l .ucentripetal acceleration of the Earth?

Would it require less speed to laf/pl*1oz d. 3jansat -3ei1ktrj (t d;unch a satellite (a) toward the east or (b) toward the west? Consider the Earth’s rotation direction.f3a3*; -toazerstp.j k(di 1l/1 njtd

When will your apparent weight be themhiyxg5be 9y */wrg1 * greatest, as measured by a scale in a moving elevator: when the elevator (a) accelerates downward, (b) accelerates upward, (c) is in free fall, (d) moves upward at constant speed? In which case would your weight r15g*y/9eb myhxiw *gbe the least? When would it be the same as when you are on the ground?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long periods in outer space could be adversely affteq1 a, 5wvoah9vb*5e ected by weightlessness. One way to simulate gravity is to shape the spaceship like a cylindrical shell that rotates, with the astronauts walking on the inside surface (Fig. 5–32). Explain how thisa*9hv5t w5o,aeeqbv 1 simulates gravity. Consider (a) how objects fall, (b) the force we feel on our feet, and (c) any other aspects of gravity you can think of.

Explain how a runner experiences “free fall” or oh f lzig,:3o )u1nk0x qe9u*i“apparent weightlessness” betweeikxoz3, ue19q:0 f) *u onhglin steps.

The Earth moves faster in its orbit around the Sun in bhgnnsi kf98t ))/d8z January than in July. Is the Earth closer to the Sun in January, or in July? Explain. [Note: This is not much of a factor in producing the seasons — the main factor is the tilt of the Earth’s axis r/h8 nn )9zt8gibks df)elative to the plane of its orbit.]

The mass of Pluto was not known until iu a*dk .zk+h/4 ype;ubt was discovered to have a moon. Explain how this discovery enabled an estimate*ukka ey;bdp4.uz/h + of Pluto’s mass.

  The Sun's gravitational pull on the Earth is me 90w3dvbd9k1clio-3vt u- u xuch larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in 9cou39d kt u-wvdvbx3 i0-l1egravitational pull from one side of the Earth to the other.]A child sitting 1.10 m from the center of a merry-go-round moves with a speed of 1.25 $\mathrm{~m} / \mathrm{s}$ . Calculate (a) the centripetal acceleration of the child,   
(b) the net horizontal force exerted on the child ( mass =25.0 $\mathrm{~kg}$)   

  A jet plane traveling 9la.lkbd 9vcl-t2k0p 1980 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ pulls out of a dive by moving in an arc of radius 6.00 km . What is the plane's acceleration in $g^{\prime}$ 's?    g's

  Calculate the centripetal acceleration of the Earc i:m 4drdx,oy;8sf0 m0oib9mth in its orbit around the Sun, and the net force exerted on the Earth. What exerts this force on the Earth? Assume that the Earth's orbit is a circle ;,d0:o0 md9o xif b m4c8mysriof radius 1.50$ \times 10^{11} \mathrm{~m}$ . [Hint: see the Tables inside the front cover of this book.]    $\times10^{22}$

  A horizontal force of 210 N is exerted on a 2.0-kg discus as it xd4 divzx( j h.q2(1airbe*j,rotates uniformly in a horizontal circle (at arm’s length) of radius 0.90 m. Cjd evd (rzx*b1i 4jxh.iq2,a( alculate the speed of the discus.   

  Suppose the space shuttlebc52 sv:9zxi7 edeuck idlg ); 6+)hfd is in orbit 400 km from the Earth's surface, and circles the Earth about once every 90 minutes. Fidgvchi5)7dcs2e ;9dlz: ix6+ bk ) eufnd the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of g , the gravitational acceleration at the Earth's surface. $\approx$    g

  What is the magnitude of the acceleration of a speck of clay on tyuqn y9 z8a ; 5 o2 1y:d-tnombk7bnwr5v6foo6he edge of a potter’s wheel turning at 45 rpm (revolutions per minute) if the wheel’s diametwfo6bd:n y2z roanm78ok9q51y o-u6 yn ;v5tber is 32 cm?   

  A ball on the end of a string is revolved at a uniform 20e8y qf vkgk/rate in a vertical circle of radius 72.0 cm , as shown eq0f 8k/gk2vy in Fig. 5-33. If its speed is 4.00 $\mathrm{~m} / \mathrm{s}$ and its mass is 0.300 kg , calculate the tension in the string when the ball is (a) at the top of its path   
(b) at the bottom of its path.   

  A 0.45$-\mathrm{kg}$ ball, attached to the end of a horizontal cord, is rotated in a circle of radius 1.3 m on a frictionless horizontal surface. If the cord will break when the tension in it exceeds 75 N , what is the maximum speed the ball can have?   

  What is the maximum speqs.+2rst+tll,34)9 hvz a fa qvx;ps ced with which a 1050-kg car can round a turn of radius 77 m on a flat road if the coefficient of static friction between ti z vs,t+xrv92pl;s3) sacq+afh4lq t.res and road is 0.80? Is this result independent of the mass of the car?   

  How large must the coefficient of static friction be betwe u.1i a94y8(h:qdsf5bkij jk gen the tires and the road if a car is to round a level curve of radius 85 m at 8 ujbij9s. khg5di1 fky4a(q:a speed of 95km/h ?   

  A device for training astronauts and jet fighter pilots is designed to:x seo0 c -co h5dq-o8jdxxj923p w(en rotate a trainee in a horizontal circle of radius 12.0 m . If the force felt by the trainee on her back is 7.858wn02h(xoce dcpo3j-eq - x:9jdx5so times her own weight, how fast is she rotating?    rev/s    m/s
Express your answer in both $ \mathrm{m} / \mathrm{s}$ and $\mathrm{rev} / \mathrm{s}$ .

  A coin is placed 11.0 cm from the axis of a rotatifow9eq3)dz :bar 1el1ng turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable until a rate)fo9eebad zw q11 :rl3 of 36 rpm is reached and the coin slides off. What is the coefficient of static friction between the coin and the turntable?   

  At what minimum speed must + jp*qnql8 rb0a roller coaster be traveling when upside down at the top of a np8+b0jlrqq* circle
(Fig. 5-34) so that the passengers will not fall out? Assume a radius of curvature of 7.4 m .   

  (II) A sports car of mass 950 kg (includin 75q t v,9ce/vkzqxgx*g the driver) crosses the rounded top of a hilk,7*9qczqgev5x/tv xl (radius =95$ \mathrm{~m}$ ) at 22 $\mathrm{~m} / \mathrm{s}$ . Determine
(a) the normal force exerted by the road on the car,   
(b) the normal force exerted by the car on the 72-kg driver   
(c) the car speed at which the normal force on the driver equals zero.   

  How many revolutions per minute would a 15-m-diametetfgty.o)a(x2r8)n:1ht pil w5hx b( xr Ferris wheel need to make for the passengers to feel “weightless” at the topmost p2 )8h5. ttrx g (alpniby1otwx(xf h):oint?    rpm

  A bucket of mass 2.00 kg is whirled in a vertical ce;x/t9fy gpn0fyj)b *ircle of radius 1.10 m. At the lowest point of its motion the tension in the ropf*tyx 9/;jype gn0f)be supporting the bucket is 25.0 N.
(a) Find the speed of the bucket. $\approx$   
(b) How fast must the bucket move at the top of the circle so that the rope does not go slack?   

  How fast (in rpm) must a centrifuged7mvalqt0y : qc6+;ks rotate if a particle 9.00 cm from the axis of rotation is to experience an advt:syc;0+m a 7q lq6kcceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotated in a cylindricy7m*hftt+vq o a- 00utally walled "room." (See Fig. +tmy0vu0q- atf hot* 75-35.)
The room radius is 4.6 m , and the rotation frequency is 0.50 revolutions per second when the floor drops out. What is the minimum coefficient of static friction so that the people will not slip down? People on this ride say they were "pressed against the wall." Is there really an outward force pressing them against the wall? If so, what is its source? If not, what is the proper description of their situation (besides "scary")? [Hint: First draw the free-body diagram for a person.]   

A flat puck (mass M ) is rot-iw3 8phv( gdkated in a circle on a frictionless air-hockey tabletop, and is held in this orbit by a light cord conni(v8 w3k -pgdhected to a dangling block (mass m ) through a central hole as shown in Fig. 5-
36. Show that the speed of the puck is given by

$v=\sqrt{\frac{m g R}{M}}$

  Redo Example 5-3 , precisely this time, by not ignoring the w/u f cja+jp(m8eight of the ball which revolves on a string 0.600 m long. In particular, fi8au/jj c+p(fm nd the magnitude of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ , and the angle it makes with the horizontal. [Hint: Set the horizontal component of $\overrightarrow{\mathrm{F}}_{\mathrm{T}}$ equal to m $a_{\mathrm{R}}$ ; also, since there is no vertical motion, what can you say about the vertical component of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ ?] $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ =    the angle =   

  If a curve with a radius of 88 m is perfectly banked f r8ch40e* k fs-i8na pjke1s*qpe ld6:or a car traveling 75 $\mathrm{~km} / \mathrm{h}$ , what must be the coefficient of static friction for a car not to skid when traveling at 95 $\mathrm{~km} / \mathrm{h}$ ?   

  A 1200$-\mathrm{kg}$ car rounds a curve of radius 67 m banked at an angle of 12$^{\circ}$ . If the car is traveling at 95$ \mathrm{~km} / \mathrm{h}$ , will a friction force be required? If so, how much and in what direction? $\approx$    down the plane

Two blocks, of masse8e.siak +m 5:ahqg 6 (xeo(dnns $ m_{1}$ and $m_{2}$ , are connected to each other and to a central post by cords as shown in Fig. 5-37. They rotate about the post at a frequency f (revolutions per second) on a frictionless horizontal surface at distances $r_{1}$ and $r_{2}$ from the post. Derive an algebraic expression for the tension in each segment of the cord.

  A pilot performs an evasive maneuver by diving q)o k:;a wss5l3zc1nxvertically at 310 $\mathrm{~m} / \mathrm{s}$ . If he can withstand an acceleration of 9.0 g 's without blacking out, at what altitude must he begin to pull out of the dive to avoid crashing into the sea?   

  Determine the tangential and centripetal components of the net force exifinht k19 npo2q/9h/erted on the car (by th poftnq9i9n/ 1hki h/2e ground) in Example 5-8 when its speed is 15$ \mathrm{~m} / \mathrm{s}$ . The car's mass is 1100 kg . F$_{tan}$ =    F$_R$ =   

  A car at the Indianapolis 500 accelerates uniformly from the pit a32hafk wy)en,x8 ri*wrea, going from rest to 320km/h in a semicircular arc with a radius of 220 m. Determine the tangential and radial acceleration of the car when it is halfway through the turn, assuming constant tangential acceleration. If the curve were flat, what would the coefficient of static friction have to be between the tires and the road to provide this acceleration wixi *,w32)ry awknh8efth no slipping or skidding? $\approx$   

  A particle revolves in a horizontal circle of radius 2.90 m . At a particulary*vky j3kl6 n, instant, its accelerlj*k,yyvn 63k ation is 1.05 $\mathrm{~m} / \mathrm{s}^{2}$ , in a direction that makes an angle of 32.0$^{\circ}$ to its direction of motion. Determine its speed (a) at this moment   
(b) 2.00 s later, assuming constant tangential acceleration.   

  Calculate the force of Earth’s bluxb te)qvq9;8m2e 6pxct q6v/ 51n lgravity on a spacecraft 12,800 km (2 Earth radii) above the Earth’s surface if its mass is 1350;q 28659lp qxnqvem/ et ult6c)vxb1b kg.   

  At the surface of a ce rqpe*-y2 e. pe .ykzy77g;tadrtain planet, the gravitational acceleration g has a magnitude of. z2*e r;d tp7y e7g.eay-pkqy 12 m/s$^2$ A 21.0-kg brass ball is transported to this planet. What is (a) the mass of the brass ball on the Earth and on the planet   
(b) the weight of the brass ball on the Earth and on the planet? $W_{Earth}$   $W_{planet}$  

  Calculate the acceleration due to gravity on the Moq a1ctpo2e1yp3 ttxa -7 u:nz)iv-kvqcm/n5:on. The Moon's radius is 1.74: vt3 tn)z- tqpi: n 51oxe71km/yc2vaa u-qcp $\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has a radius 1.5 times x+7teo q4zf0v that of Earth, but has the same mass. What is the acc0 +ev4 7otxfqzeleration due to gravity near its surface?   

  A hypothetical planet has a mass 1.66 times that of Ear4ifve(nr0h 6uth, but the same radius. What is g near itshu 60fer4n(iv surface?   

  Two objects attract each other gravitationalk 5e- n.op2 kkmix++qcly with a force of 2.5$ \times 10^{-10} \mathrm{~N}$ when they are 0.25 m apart.
Their total mass is 4.0 kg . Find their individual masses. m$_1$ =    m$_2$ =   

  Calculate the effective value of g , the acceleration of ks-py r4evmv ,9no 8,k:p9my /x ,hpwrgravity, at (a) 3200 mmw/ok99v,p8ev yy4m ,xh k n:-,rprsp    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point ot2u;gb r( 2inlutside the Earth where thegra; u2r ti(bn2lgvitational acceleration due to the Earth is $\frac{1}{10}$ of its value atthe Earth’s surface?   $\times10^7$

  A certain neutron star has five times thszd)6+0h wsph e mass of our Sun packed into a sphere about 10 km in radius. Estimate the surface p0 sh +z)6swhdgravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once was an d2b rotqo/4l ddu 3 w22;t0f8dlkf nc6average star like our Sun but is now in the last stage of its evolution, is the size of our Moon but has the mass of ourlw ufd; 2 tklco0r8ddob3tf/46d nq22 Sun. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronauts feel weightless while orbiting ,f2b 6o4l8gva qw t 7m*smid.fx e-0vuin the space shuttle. Your friends respond that they thought gravity was just a lot weaker up there. Convince them and yourself sl,i dv ff7xvam2b-.6q04 *weou8gtm that it isn’t so by calculating the acceleration of gravity 250 km above the Earth’s surface in terms of g.   

  Four 9.5-kg spheres are located at the corners of a square o f(r605b yz7x,gtbyqwt ztc(-f side 0.60 m. Calculate the magnitude and direction of the total graz0 tgy 57-c,tzwfbrbyx( tq6(vitational force exerted on one sphere by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred years most of the planets line upvg4l e p1i,xr1 on the same side of the Sun. Calculate the total forcegevp1x li1r, 4 on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in a line (Fig. 5-38). The masses are $M_{\mathrm{V}}=0.815 M_{\mathrm{E}}, M_{\mathrm{J}}=318 M_{\mathrm{E}}, \quad M_{\mathrm{S}}=95.1 M_{\mathrm{E}}$ , and their mean distances from the Sun are 108,150,778 , and 1430 million km , respectively. What fraction of the Sun's force on the Earth is this? $F_{Earth-planets}$ =   $\times 10^{17}$ ${F_{Earth-plantes}}$ \ ${F_{Earth-sun}}$=    $\times 10^{-5}$

  Given that the acceleration of gravity at the surface of Mars 0ofr nw4mn.zzp. tz/ qygk *9)bg2:vn is 0.38 of what it is on Earth, and that Mark*g yvgp w. r .0z)2nmnozzfq/:b9tn4s’ radius is 3400 km, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the 1)v7a:if rufe3p stz 6Sun using the known value for the period of the Earth and its distance from the Sun. [Note: Compare your:v 3r6 af)1e7ui szfpt answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite moving i1nsksui(/ c e1n a stable circular orbit about the Earteu ci(1/k1 snsh at a height of 3600 km.    $\times 10^3$

  The space shuttle releases a satellite intoczh(6az+k 5z f a circular orbit 650 km above the Earth. How fast must the shuttle be moving (relative to Earth) whenz5h+kcza6 (zf the release occurs?    $\times 10^3$

  At what rate must a cylindrical spacyb7uj-e ;7qjyeship rotate if occupants are to experience simulated gravity of 0.60 g? Assume the spac-ey ju; qyb77jeship’s diameter is 32 m, and give your answer as the time needed for one revolution. (See Question 21, Fig 5–32.)   

  Determine the time it takes for a satellite tkf8-wlv l-kp5z1.fyx 6gj0j eo orbit the Earth in a circular “near-Earth” orbit. A “near-Earth” orbit is one at a height above the surface of the Earth which is very small compared to the radius of the Earth. Does your result depend on the mass of the sa 5.j-lz8xkyfge-w 6kvpljf01 tellite?    s ~    min

  At what horizontal velocity would a satellite have to be launched from the t +7jlbg: cw5aq0b5p2 l6psm vqop of Mt. Everest to be placed in a circular orbit arounplqsa+ jm2lwb 05cgv qb :p756d the Earth?   

  During an Apollo lunar landing missyx8i,crq )hzp ;k( m;rion, the command module continued to orbit the Moon at an altitudekmh( c ,zpr;yx); qr8i of about 100 km. How long did it take to go around the Moon once?    s

  The rings of Saturn are composed of chunks of ice that orbitx;..pcd abrgcy(hg(7 the planet. The inner radius of the ringh yb c(.c g(r;apd.xg7s is 73,000 $\mathrm{~km}$ , while the outer radius is 170,000$ \mathrm{~km}$ . Find the period of an orbiting chunk of ice at the inner radius and the period of a chunk at the outer radius. Compare your numbers with Saturn's mean rotation period of 10 hours and 39 minutes. The mass of Saturn is 5.7 $\times 10^{26} \mathrm{~kg}$ .


$T_{\text {irner }} $ =   
$T_{\text {outer }} $ =   

  A Ferris wheel 24.0 m in diameter rotates once every 15bd6)hvp w 9tq/.5 s (see Fig. 5–9). What is the ratio of a person’s apparent weight to her real weight (a) at the top, and (b) at th /qvbpwd)96hte bottom?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut 4200 km fmo a02p; n/ciprom the center of the Earth's Moon in a space vehicle (a) movm;0co/p pan2iing at constant velocity,     ----待选择----towards the Moonaway from the Moon 
(b) accelerating toward the Moon at 2.9 $\mathrm{~m} / \mathrm{s}^{2}$ ?     ----待选择----towards the Moonaway from the Moon 

State the "direction" in each case.

  Suppose that a binary-star system consists t6j0d9y mf7a hw .e04ihkj)porv/ f)xof two stars of equal mass. They are observed to be separated by 3600ohi./kj6hexfj am9pw 7 ryt4 )dvf 0) million km and take 5.7 Earth years to orbit about a point midway between them. What is the mass of each?    $\times 10^{29}$

  What will a spring scale read for the weight of a 55-kg woman in ae.kb.v5 xqdd 2n elevator that52. ebxkv qdd. moves
(a) upward with constant speed of 6.0m./s   
(b) downward with constant speed of 6.0m./s   
(c) upward with acceleration of 0.33 g,   
(d) downward with acceleration 0.33 g   
(e) in free fall?   

  A 17.0-kg monkey hangs from a cord dm;lbny +e w*1suspended from the ceiling of an elevator. The cord can withstand a td*w +e1byn; mlension of 220 N and breaks as the elevator accelerates. What was the elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orbits very near the surface of a planet with p2qyci j e;477cvkrxqfh h p75)w5 ozh8eriod T , the density (mass/volum vj8q7cf557x4 z )k7iye w;oqhhrhpc2 e) of the planet is $\rho=m / V=3 \pi / G T^{2}$ .
(b) Estimate the density of the Earth, given that a satellite near the surface orbits with a period of about 85 min .

  Use Kepler’s laws and gw4 w m.;*7y0eb jydygfl18em the period of the Moon (27.4 d) to determine the period of an artificlwe*.e m4yb7;0gmwy y d18 fgjial satellite orbiting very near the Earth’s surface.    s

  The asteroid Icarus, though only a few hundred meters alqx+in y8j:a 5,/dptqcross, orbits the Sun like the planets. Its period is 410 d. What is its mean distance from the t/,p5+ nidy8qjalqx: Sun?    $\times 10^{11}$

  Neptune is an average distance of sz hj0u6bsz719bj, in 4.5 $\times 10^{9} \mathrm{~km}$ from the Sun. Estimate the length of the Neptunian
year given that the Earth is 1.50 $\times 10^{8} \mathrm{~km}$ from the Sun on the average.    $\times 10^2$

  Halley’s comet orbits the Sun roughl:ng7u y qjz t(+((teliu/ c)boy once every 76 years. It comes very close to the surface of thez t7c(itoe:+g/yq)bulu (j (n Sun on its closest approach (Fig. 5–39). Estimate the greatest distance of the comet from the Sun. Is it still “in” the Solar System? What planet’s orbit is nearest when it is out there? [Hint: The mean distance s in Kepler’s third law is half the sum of the nearest and farthest distance from the Sun.]    $\times 10^6$

  Our Sun rotates about thei+z: l 859q x*cgp4 q(j2i1h jsmeexo ww)vm-s center of the Galaxy $\left(M_{G} \approx 4 \times 10^{41} \mathrm{~kg}\right)$ at a distance of about 3 $\times 10^{4}$ light-years $\left(1 \mathrm{ly}=3 \times 10^{8} \mathrm{~m} / \mathrm{s} \times 3.16 \times 10^{7} \mathrm{~s} / \mathrm{y} \times 1 y\right)$ . What is the period of our orbital motion about the center of the Galaxy? $\approx$   $\times 10^8$

  Table 5–3 gives the mass, period, and m0bb)xy 8bhc/ mean distance for the four largest moons of Jupiter (those discovered by Galileo in 1 ybxh/b)0 cb8m609).
(a) Determine the mass of Jupiter using the data for Io.$M_{\substack{\text { Jupiter- } \\ \text { Io }}}$ =    $\times 10^{27}$
(b) Determine the mass of Jupiter using data for each of the other three moons. Are the results consistent?
$M_{\substack{\text { Jupiter- } \\ \text { Europa }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Ganymede }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Callisto }}}$ =    $\times 10^{27}$

  Determine the mass of the Earth from the known perina ( ec*f2-fjaod and distance of the Moon. eca-naf(f *j2   $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jupiter’s moons, using x6y4t f eicv-b3:vb t;Kepler’s third law. Use the distance of Io and3y:x - cveft4b 6v;itb the periods given in Table 5–3. Compare to the values in the Table.
$r_{\text {Farip }}^{r}$ =    $\times 10^3$
$r_{\text {Guarymale }}$ =    $\times 10^3$
$r_{\text {Culimio }}$ =    $\times 10^3$

  The asteroid belt between Mars and Jupiter consists of many qt4f9 p-, 9s.zaxophcy+urgk -lyb13 fragments (which some space scientists think came 4-yt+,fhk apob99cpys- ul z1.3qx gr from a planet that once orbited the Sun but was destroyed).
(a) If the center of mass of the asteroid belt (where the planet would have been) is about three times farther from the Sun than the Earth is, how long would it have taken this hypothetical planet to orbit the Sun?$\approx$    y
(b) Can we use these data to deduce the mass of this planet?

  A science-fiction tale4hzotaq)0iz)g5/ b k qm65z bk describes an artificial "planet" in the form of a band completely encircling a sun (Fig. 5-40). The inhabitants live on the inside surface (where it is always noon). Imagine that this sun is exactly like our own, that the distance to the band is the same as the Earth-Sun distanceb5hqz6m )k4)qoi bz/0za5 gtk (to make the climate temperate), and that the ring rotates quickly enough to produce an apparent gravity of g as on Earth. What will be the period of revolution, this planet's year, in Earth days?   

  Tarzan plans to cross a gorge by gg/o: fc+gf2di9q ww9swinging in an arc from a hanging vine (Fig. 5–41). If his arms are capable of exerting a force of 1400 N on the vine, what is the maximum speed he can tolerate at the lowest point of his swing? His mass io c gwdff:w2g+9/ 9qigs 80 kg, and the vine is 5.5 m long.   

  How far above the Earth’s +elkn g;d-bal28 ) wlysurface will the acceleration of gravity be half what it is on the surfacen)weyll 2k; abg- 8l+d?    $\times 10^6$

  On an ice rink, two skateroco31 ttk0ja 4/biw 4 vao*kn*s of equal mass grab hands and spin in a mutual circle once every 2.5 s. If we assume their arms are each 0.80 m an3cka ito vt/0bo 1j*4w*k4 olong and their individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates once perjj+ dsf9. ial i/,:zyh day, the apparent acceleration of gravity at the equator is slightly less than it would be if the Eartsi,.y ljf9ai/z:+hjd h didn’t rotate. Estimate the magnitude of this effect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from the Earth will a spacecraft travqcc +m ;mv a-le,e3-nyeling directly from the Earth to the Moon v+ea- cl qec mn-y3;,mexperience zero net force because the Earth and Moon pull with equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, but when.-gz 1bc 1nlcd you stand on a bathroom scale in an elevator, cc1bznl1g.d- it says your mass is 82 kg. What is the acceleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consicb(*t4g )w lu1gbp3olsts of a circular tube that will rotate about its center (like* 1o l bg3ublg)4(ptcw a tubular bicycle tire) (Fig. 5–42). The circle formed by the tube has a diameter of about 1.1 km. What must be the rotation speed (revolutions per day) if an effect equal to gravity at the surface of the Earth (1.0 g) is to be felt?$\approx$    $\times 10^3$

  A jet pilot takes his aircrafttxgpd+b.,b4w y pj3xc*s2l 5i in a vertical loop (Fig. 5–43).
(a) If the jet is moving at a speed of at the lowest point of the loop, determine the minimum radius of the circle so that the centripetal acceleration at the lowest point does not exceed 6.0 g’s.    $\times 10^3$
(b) Calculate the 78-kg pilot’s effective weight (the force with which the seat pushes up on him) at the bottom of the circle    $\times 10^3$
(c) at the top of the circle (assume the same speed).
   $\times 10^3$

Derive a formula for the mass of a planet in terms of its radius r, the acceleraeco jn3el8cri)-v vey4 v43e6l)lfj,tion due to gravity at its surface and the gravlfc l cev4)vn-8 jvj3o3 e6,le)4 riyeitational constant G.

A plumb bob (a mass m hanging on a string) xyaak d. 3n7a+(7a .m flpwp 2of33lhdis deflected from the vertical by an.(pflnx2m. kopa7+3 a3afdwhday3 7l angle $\theta$ due to a massive mountain nearby (Fig. 5-44). (a) Find an approximate formula for $\theta$ in terms of the mass of the mountain, $m_{\mathrm{M}}$ , the distance to its center, $D_{\mathrm{M}}$ , and the radius and mass of the Earth. (b) Make a rough estimate of the mass of Mt. Everest, assuming it has the shape of a cone 4000 m high and base of diameter 4000 m . Assume its mass per unit volume is 3000 kg per $ \mathrm{m}^{3}$ . (c) Estimate the angle $\theta$ of the plumb bob if it is 5 km from the center of Mt . Everest.

A curve of radius 67 m is banked for a design speed of If the coeffipeny2p u;m .c) kw n29)acai3ocient of static friction is 0.30 (wet pavement), at what range of spee3puk922y;wieanmanc) pco .)ds can a car safely handle the curve?

  How long would a day be if the Enb w/l/bxfuyx 3-f0ow(/c(zt arth were rotating so fast that objects at the equator were apparently wet0lfnwy/o(( uwbz/x 3c- b/ xfightless?    $\times 10^3$s

  Two equal-mass stars maintain a constant mtl7eb82ln) u6w2 mqg distance apart of 8.0 $\times 10^{10} \mathrm{~m} $ and rotate about a point midway between them at a rate of one revolution every 12.6 yr . (a) Why don't the two stars crash into one another due to the gravitational force between them?
(b) What must be the mass of each star?    $\times 10^{26}$

  A train traveling at a co)m*gn-- hpj 4m/ u-dk ghaxpx.nstant speed rounds a curve of radius 235 m. A lamp sus*pngjk-)g 4p -.x-/da uhh mxmpended from the ceiling swings out to an angle of 17.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times as massive as the Earth. Th+/ei9ff 7 bq sz)9 5*:ae pguasu7odcrus, it has been claimed that a person would be crushed by the force of gravity on a planet the size of Jupiter since people can't survive more than a few g 's. Calculate the number of g 's a person would experience at the equator of such a planet. Use the followinsffucg:z5s9dieb erq9 u7 *o a7+ /pa)g data for Jupiter: mass =1.9 $\times 10^{27} \mathrm{~kg}$ , equatorial radius =7.1$ \times 10^{4} \mathrm{~km}$ , rotation period =9 $\mathrm{hr} 55 \mathrm{~min}$ . Take the centripetal acceleration into account.    g's

  Astronomers using the Hubble Space Tv 0:g 9et otn *3re9ycf:j9aor (5lwipelescope deduced the presence of an extremely massive core in the distant galaxy M87, so dense r:p3:0fjyo ae9e*o r t5g 9wnc(9itlvthat it could be a black hole (from which no light escapes). They did this by measuring the speed of gas clouds orbiting the core to be 780 $\mathrm{~km} / \mathrm{s}$ at a distance of 60 lightyears $ \left(5.7 \times 10^{17} \mathrm{~m}\right)$ from the core. Deduce the mass of the core, and compare it to the mass of our Sun. v =    $\times 10^{39}$solar masses =    $\times 10^9$

A car maintains a coni-j3: hi/xxjcstant speed as it traverses the hill and valley shown in Fig. 5–45. Both the hill and valley have a radius of curvature R. (a) How do the normal forces acting on the car at A, B, and C compare? (Which is largest? Smallest?) Explain. (b) Where wou -h/xiji:3 xcjld the driver feel heaviest? Lightest? Explain. (c) How fast can the car go without losing contact with the road at A?

  The Navstar Global Positioning System (GPS) utilizes a grouc+sa 18fsln.rnwzi0)u e c9a(md1;6 ppp-gz rp of 24 satellites orbiting the Earth. Using “triangulation” and signals transmitted by these satellites, the position of a receiver on the Earth can b9c1 n zig )upaaw1m;(r6l nps-df8.e+rcs0zpe determined to within an accuracy of a few centimeters. The satellite orbits are distributed evenly around the Earth, with four satellites in each of six orbits, allowing continuous navigational “fixes.” The satellites orbit at an altitude of approximately 11,000 nautical miles [1 nautical ]. (a) Determine the speed of each satellite.    $\times 10^3$
(b) Determine the period of each satellite. =    hours

  The Near Earth Asteroid Rendelrxmk/ tc6f, *xw u* 9dcil(0gzvous (NEAR), after traveling 2.1 billion km, is meant to orbit the asteroid Eros at a*0*w /ckiul ,(txgcfm rdx9l6 height of about 15 km . Eros is roughly 40 $\mathrm{~km} \times 6 \mathrm{~km} \times 6 \mathrm{~km}$ . Assume Eros has a density (mass/volume) of about 2.3 $\times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ . (a) What will be the period of NEAR as it orbits Eros? $\approx$    $\times 10^4$sec
(b) If Eros were a sphere with the same mass and density, what would its radius be? $\approx$    $\times 10^3$
(c) What would g be at the surface of this spherical Eros?$\approx$    $\times 10^{-3}$

  You are an astronaut in the space shuttle pursuing a satellf mpj6y)+ +evgite in need of repair. You are in a circular orbit of the same radius as the satellite (400gm+j) vfy6e+ p km above the Earth), but 25 km behind it.
(a) How long will it take to overtake the satellite if you reduce your orbital radius by 1.0 km?$\approx$    h
(b) By how much must you reduce your orbital radius to catch up in 7.0 hours?$\approx$    $\times 10^3$

  The comet Hale-Bopp has a period of 3000 years. (a) What is its meanx e*5z n0wik(m: -lwy9ecy0lz distance fromzl eyec9n0w* mixlw:50-kzy( the Sun?   
(b) At its closest approach, the comet is about 1 A.U. from the Sun ( from Earth to the Sun). What is the farthest distance?   
(c) What is the ratio of the speed at the closest point to the speed at the farthest point? [Hint: Use Kepler’s second law and estimate areas by a triangle (as in Fig. 5–29, but smaller distance travelled; see also Hint for Problem 59).]   

  Estimate what the value of G would need to be if you could pzrk87r2:ct y l 4um r 0y8qcbo;1a0lpactually "feel" yourself gravitationally attra;pormcl2a:4 zpr0k1br8cqly y708 tu cted to someone near you. Make reasonable assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the Milky Way Galaxy (Figoi2kl h *--j hjk6u5dl. 5-46) at a distance of about 30,000 light-years odlhu2h 5 -k*jli-k6 jfrom the center $ \left(1 \mathrm{ly}=9.5 \times 10^{15} \mathrm{~m}\right)$ . If it takes about 200 million years to make one rotation, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun $\left(2 \times 10^{30} \mathrm{~kg}\right)$ , how many stars would there be in our Galaxy?$\approx$    $\times 10^{11}$

  Four 1.0-kg masses are located at the corners of a square 0.50 m on exd lygxo4 uts2o 6oo4)r6f)j8ach side. Find the magnitude and direction of the gravitational force on a fifth 1.0-kg mass placed at ouol62s jfd6yoo4 x)xr4g)8tthe midpoint of the bottom side of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 kg orbits the Ea645jum kcnh 7:x9fhfi 5 2mtpvrth $ \left(\operatorname{mass}=6.0 \times 10^{24} \mathrm{~kg}\right)$ and has a period of 6200 s . Find (a) the magnitude of the Earth's gravitational force on the satellite, $\approx$    $\times 10^4$
(b) the altitude of the satellite.    $\times 10^5$

  What is the acceleratlezepb.:iqs,7pj5-xi p9 m a7ion experienced by the tip of the 1.5-cm-long sweep second han app7.ej7,:lz 5 meib9sxq-p id on your wrist watch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weight around in a ci4(oh mu6 u2lckrcle below you on a 0.25-m piece of fishing line. The weight makes a complete circle every 0.50 s. What is the angle that the fishing line makes with the vertical? [Hint:uk4hm6(o2 clu See Fig. 5–10.]    $^{\circ}$

A circular curve of radb*c;(3z143 jd j ivlpnhkycf)ius R in a new highway is designed so that a car travelc c31(khn*3 ;v4)lpf jidyj zbing at speed $ v_{0}$ can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from $v_{\min }$ to $v_{\max }$ . Derive formulas for $ v_{\min }$ and $ v_{\max } $ as functions of $\mu_{s}, v_{0}$ , and R .

  Amtrak’s high speed train, the Acela, utilizes tilt of the cars when ne + dai67osamolrt fe+z:*,gnevet*: / gotiating curves. The angle of tilt is adjusted so that the main force exerted on the passengers, to provide the centripetal acceleration, is the normal force. The passengers experience less friction force against the seat, thus feeling more comfortable. Consider an Acela train that rounds a curve with a radius of 620 m at a speedmof: l+s7ae*e/z,arvg i tno*t:ed6+ of (approximately ). (a) Calculate the friction force needed on a train passenger of mass 75 kg if the track is not banked and the train does not tilt. $\approx$    $\times 10^2$
(b) Calculate the friction force on the passenger if the train tilts to its maximum tilt of 8.0º toward the center of the curve.$\approx$    $\times 10^2$