Sometimes people say that water is removec 78 i8qk qbiq4b26zdjd from clothes in a spin dryer by centrifugal force throwing the water ouzbq dij28qbi48 c67kqtward. What is wrong with this statement?

Will the acceleration of a car be the same when the car travelsmxtqo(l0dn*s-:g h t- around a sharp curve at a constant 60 km/h as when it travels around a gentle do qhg-mx(*s- t0nl:t curve at the same speed? Explain.

Suppose a car moves at coufjw e.1foip b nqiej+938-9*qb9vsbnstant speed along a hilly road. Where does the car exert the greatest and least forces on fwjsfb9 o9uq jebni* i+9e3.1p-v qb8the 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 acting on a child riding a horsxbqpwk u*6 (h*e on a merry-go-round. Which of these forces provides the centripetal acceler*p xk(q*hubw6ation of the child?

A bucket of water can be whirled in a vertical circle without the water spilling0oirxhd3fmv luea936r 0/o+x 7 5duu h out, even at the top of the circle when thxoi a/m 3u0l+ 67derhh5r9 udvoxfu03 e bucket is upside down. Explain.

How many “accelerators” do you have in your car? There are at least tzr hsey 7p3s6t:5eh l3hree controls in tpz: 3 sh675telh eyrs3he car which can be used to cause the car to accelerate. What are they? What accelerations do they produce?

A child on a sled comes fly2tp 1skm.9j bting over the crest of a small hill, as shown in Fig. 5–31. His sled does notpm2 19tjs bk.t leave the ground (he does 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 high sp9 ),tsmzdg,4b*pescm-p o2ho eed?

Why do airplanes bank when they turn? How would you compute the bdso 0cn4ym iun/6v .gr(kqa02anking angle giv24r(u06g0ionavsndm / k.cyq en its speed and radius of the turn?

A girl is whirling a ball on a string around her head in a horiyh (d-; + w9vc k 95+r;xfex:wy5wr5cwcl zaezzontal 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 yard.; w59e vrhxxwacey +5 rdc 5+zw; k9-:ywcl(zf When should she let go of the string?

Does an apple exert a gravitgq+ .b il;x4dpational force on the Earth? If so, how large a force? Consider alxp+g ;q4d.ib n apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, in whaa)8aw.q.:3e c 2tnf axkmj lqi6e s5-rol6g)yt ways would the Moon’s orbit be da l-lcrm e 6)2q)gat:sk wa.ei5f3jyq8.o 6xn ifferent?

Which pulls harder gravxytoi2 3l)m,m y 34xhfitationally, the Earth on the Moon, or the Moon on the Earth?,m xx thof23 m4yi)y3l Which accelerates more?

The Sun's gravitational pull on the Earth is much larger than the Moon'nhmm51 sy707l skuqp8s. Yet the Moon's is mainly responsible for t5pul17mqs s0h m 78kynhe tides. Explain. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]

Will an object weigh more at the equator or at the poles? What f7 y3( gaqzl0/wxt6vwtwo effects are at work? Do tww73yzvl/tfag (x0 6 qhey oppose each other?

The gravitational force on the Moon due to the (mec xz,0;jh 2i.uyy wEarth is only about half the force o( u0jz.e hxi;,ymyc2w n the Moon due to the Sun. Why isn’t the Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in its orbit around the Sun largwxwfri 2sl .*w48zim +;(vxz ger or smaller than the centripe8+w wf42rg ;v *l.w xszzx(imital acceleration of the Earth?

Would it require less speed to launch a satellite (a) toward ther)dh4+dsls q8vh3x t2 east or (b) toward the west? Consider the s24) 8d+rxsh3 hldq tvEarth’s rotation direction.

When will your apparent weight be the q;c cnmezplu; ,5-bx +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 be tu - +xe b;n,lcmz;q5pche 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 affected behdwl8.u (vg/a.k :pvy weightlessness. One way to si.wv8.vhl(ue kgp/d : amulate 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 this 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 “aw62( a6r *tqp)pdwqkfpparent weightlessness” between st6r wq(ppdq6k*w2t)fa eps.

The Earth moves faster in its orbit around the Sun i8cgx 6dp c;lrzpc (;+(qg,(/vb* dteb a:gxomn January than in July. Is the Earth closer to the Sun in January, or in July? Explain. [Noc xce*d o at p+xg6qg,( zvp;lbdr(:;b8mg(c/te: This is not much of a factor in producing the seasons — the main factor is the tilt of the Earth’s axis relative to the plane of its orbit.]

The mass of Pluto was not known until it was discovered to have a mo4cy.ja,4 e+zoe;3 o 5tkfmvi von. Explain how this discovery enabled an esty.v 4mv;4eec5a 3 tfj,+o koizimate of Pluto’s mass.

  The Sun's gravitation5u 5 v :x29dqtea(v-w pljj3myal pull on the Earth is much larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]A child sitting 1.10 m from the cen5j v2 lyt qxvpua(53j9-:wmdeter 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 b6 b hz 2rovn+lixkm.552hzv3 w4z0ui1980 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 x.z5l2/2w0jskddvvdaglq 35 acceleration of the Earth in its orbit around the Sun, and the net force exerted on the Earth. Whatz5.5v sqad d lwd 32g2jkvl/0x exerts this force on the Earth? Assume that the Earth's orbit is a circle of 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 exyajfd*p2(y,3 xrob vyx (a ;2 jsxo64herted on a 2.0-kg discus as it rotates uniformly in a horiza(a ,obxpf6dyyjyxj h;2 v32s (x 4o*rontal circle (at arm’s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is in orbit 400 km from the Earth's surface, ;c tkyxw 2v4fp6bx7q)and circles the Earth about once every 90 minutes. Find the centripetal acce7v;wxtb 62 4pcyqfk) xleration 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 the edge o zqjv.-ijg:62 *e bgrmf a potter’s whejge i.mb-:q2*v6grj z el turning at 45 rpm (revolutions per minute) if the wheel’s diameter is 32 cm?   

  A ball on the end of a st8gh ,4l)klmwiscb8 a5y2 ;uwyring is revolved at a uniform rate in a vertical circle of ra;sl,bgli k 52h c8 m)8uwywya4dius 72.0 cm , as shown 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 speed with which a 1050-kg car can round a tufouoe;gx9a a54 )h+..y2 qs (omdsox lrn of radius 77 m on a flat roaeoo). sx2 fh4sg;d9a. (muoqo5a xl y+d if the coefficient of static friction between tires and road is 0.80? Is this result independent of the mass of the car?   

  How large must the coefficient of stw /b2k0xyrf lgy0uj7 2atic friction be between the tires and the road if a car is to round a lev 2grjw70y2 x0u lbfky/el curve of radius 85 m at a speed of 95km/h ?   

  A device for training astronauts and jet fighter pilots is designed to rotate a p 5e4r-f/2 hur-d +v f7ioeikjtrainee in a horizontal circle of radius 12.0 m . If the force7-p/i-o5+4ihdfef er rv2 kj u felt by the trainee on her back is 7.85 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 rotatqhq zc89i8* 4uz bv8rving turntable of variable speed. When the speed of the turntable is slowly increased, the coin remainqzr*uh i8vb 88cq49z vs fixed on the turntable until a rate 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 a roller coaster be traveling whenlflk8 u g)7*by1zb0ku upside down at the top of a circlylf8*)k0uz g 7u1 kblbe
(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 (including the driver) crossh: u++5-rowwakpz3ukbg 54 ux es the rounded top of a hill (ra5 axgur zpwo-u:+hb5uwk k43+dius =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-diameter Ferris whezm 9u9)n0: x1yvlpxr oel need to make for the passengers to feel “weightless” at the topmost 919xrxluozy0 :mp nv)point?    rpm

  A bucket of mass 2.00 kg is whirled in a vertical circle of kd6oi8l)kqqm; :y2djradius 1.10 m. At thk:y;)olmqdkj86d2 iqe lowest point of its motion the tension in the rope 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 centrifuge rotate if a particle 9.00 cm9c4uqpcf*b 8t from the axis of rotation is to experc9cpf u*q 4t8bience an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotate-czrli9zf+ 6wd in a cylindrically walled "room." (See Fig+ l-wizc6zrf9 . 5-35.)
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 rotated in a circle on af8v wloel6-n,aonv9 3 g( i02xo: xspu frictionless air-hockey tabletop, and is,oo g6(89i el0afx3-:osv vnn2pxu lw held in this orbit by a light cord connected 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 sm o h */l*ar1jf sak2,ck5nf6weight of the ball which revolves on a string 0.600 m jm 1asnsa*o2, 6rk5 fk cl*hf/long. In particular, find 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 for a car traveling 75 l n*ei263lgvz6m* i9au5bev l$\mathrm{~km} / \mathrm{h}$ , 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 masseo,bbw n* j:s1jb-;k vfs $ 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 /;lrkde.yd lkidw))y0 ic 2 b(by diving vertically 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 centg jsk+v6l9hg +sy5:xqripetal components of the net force exerted on the car (b5sh syq+6kgx+v9 l:jgy the 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 pi1d9bnf.n c tmjky6v80 t area, 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 woultn1df8 9n .v cj0ybmk6d the coefficient of static friction have to be between the tires and the road to provide this acceleration with no slipping or skidding? $\approx$   

  A particle revolves ia59n 6 ywhmzm :7kcip2n a horizontal circle of radius 2.90 m . At a particular instant, its acceleration is 1.knawz h ic952p6ym:m705 $\mathrm{~m} / \mathrm{s}^{2}$ , 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 gravity onz(anl szr5h68 bh8o,m a spacecraft 12,800 km (2 Earth radii) abmlr 8(sz 5b8h6,aohznove the Earth’s surface if its mass is 1350 kg.   

  At the surface of a certain planet,dw07,9r ;d5wny8tko 3aqnq1 kdbi.xu the gravitational acceleration g has a magnitude of 0nw,; 3.qdk18 atq7b95urokxinw ydd12 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 Mo-jwwvij8u equ. z)hcvu 165/don. The Moon's radius is 1.7.duv 5-wze 8i1) wch qju6j/uv4 $\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 that of Earth, 2uqhz (ggyn3m.( 1uk9fu()jdsif(x x but has the same mass.(nzj (9m2 xfk(h (g)uis.fxg1u udqy3 What is the acceleration due to gravity near its surface?   

  A hypothetical planet has a mass 1.66 times that of Evk (vn8r6xx td. + 1uvrvc)w2oarth, but the same radius. What is g near itsdunw x ko2t1xv)rr8+vv6(. cv surface?   

  Two objects attract each other gravitationally with a force of f2 c l9yicq/uby+z ligaw 6,)-g,) xrv2.5$ \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 qplfo 5 wuae(5z 1:.kqacceleration of gravity, at (a) 3200 mo5z:wq5l1a(eq uf p k.    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s cj 3k1z )kpl)lrenter to a point outside the Earth where thegra)zplk lrj1k)3 vitational 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 the mass of our Sun packed into a sphere3qbld8lc 4v0.nbefu4 l3b7 lc about 10 km in radius. Estimate the surface gravity on this monb0.bd3cl7ql lb4 l8u3venf c4ster.    $\times10^12$

  A typical white-dwarf star, which once was an average star like ourlrdj2 u:oy) okfa*a74 Sun but is now in thefra7o yo*kd :) 4u2jal last stage of its evolution, is the size of our Moon but has the mass of our Sun. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronauts feel weightless while orbiting in t1or)x)mx m(qqhe space shuttle. Your friends respond that they thought gravity was just a lot weaker up there. Convince them and yourself that it isn’t sq(mq) r mxox1)o 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 of si iropr0f4 2d,/yjye) sde 0.60 m. Calculate the magnitude and direction of the total gravitational force exerted on one sphere 0/pj4r , )dfy2ioryesby the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred years most of the planets line up ontn 0rto(:h;l*un3btrau2ewq o* .dl1 the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in adtn r0t.ob 2*u l :eutqrw (n3*l;oh1a 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 gravv 3sh -bl8zhm r7fwt06ity at the surface of Mars is 0.38 of what it is on Earth, and that Mvm hf7l83trswhz-0 b6 ars’ radius is 3400 km, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Su(aau2axp-+ k3s xes cea45w5kn using the known value for the period of the Earth and its dista 5u 23(kxsa4-a ksxwaecp+ea5nce from the Sun. [Note: Compare your answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite mocw hd9zobx(m+5x-3hqm xb5 9 : z6cpaivnf-g-ving in a stable circular orbit about the Earth n-ad: b5b6vp(5fgm i-xxxc+-omc 99 hhq3 wzzat a height of 3600 km.    $\times 10^3$

  The space shuttle releases wne0h3. zby h+a satellite into a circular orbit 650 km above the Earth. How fast must the shuttle be moving (relative to E+h e3bhzw.n y0arth) when the release occurs?    $\times 10^3$

  At what rate must a cylindrx0 99tnuvfw- lical spaceship rotate if occupants are to experience simulated gravity of 0.60 g? Assx 9fu09-wtvnl ume the spaceship’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 take*yskj)96f6zn knr hj-s for a satellite to orbit the Earth in a circular “near-Earth” -n6 z6jr*j9sfkhy) nkorbit. 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 satellite?    s ~    min

  At what horizontal velocity would a satellite have to be launched from th d qzc-f0wji2,e top of Mt. Everest to be pl0 ,ijwfcz2 -dqaced in a circular orbit around the Earth?   

  During an Apollo lunar landing missio6nsd7rkr+m +n*zaa * on, the command module continued to orbit the Moon at an altitude of about 100 km. How lonmn *dozs*nr+6ar7a k+g did it take to go around the Moon once?    s

  The rings of Saturn are compofd5qqy/r jy422s vlm0sed of chunks of ice that orbit the planet. The inner radius of the rings is 74l2 2sy/myjr 5fdv0q q3,000 $\mathrm{~km}$ , 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 15.5 s (see Fig. 5–c.+f(mc :4 y t6nt-iraatz7or9). What is the ratio of a person’s apparent weight to her real weight (a) at the top, and (b)ar+n7(ati tf4 mz rccy ot.6-: at the bottom?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut 4200 km frole:j j :zr1pntoz z4*+m the center of the Earth's Moon in a space vehicle (a) movioejt+zzjrlz :*p 1n:4ng 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 of two stars of equal mass. They arzy a9vb2y9)fj e observed to be separated by 360 million km and take 5.7 Earth years to orbit about a point midway between them. What is the ma)j b92vazfy9 yss of each?    $\times 10^{29}$

  What will a spring scale read for the weight of1.u7m9wsjen x; ru yl2p2x.rop04ev x a 55-kg woman in an elevator that moves um e9 ;124 vp.pxyr o2ul.rwx7 0ejnxs
(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 suspended from the ceiling of an elevator. Tbhu eb.5sv/-x cxr;-.zk(eujhe cord can withstand a tension of 220 N and breaks as the elevator accelerates. What was the elevator’s minimum acceleration (magnitude and direction)? ar-x/b;-.zxcv bjuu 5k e(ehs. =   

Show that if a satellite orbits verqm +cd 84gk88bni -zgry near the surface of a planet with period T , the density (mass/volume) of tq+ -inbc8d mzr88k 4gghe 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 the period of the Moon (27.4 d) to determine the pergqk4ba 2auz *ago1 fn s+3;p9bcdhu. -iod of an artificial satellite orbiting ve huod9 szb*ngb aqa u;p3ac.f21k-+g4ry near the Earth’s surface.    s

  The asteroid Icarus, though o- srx 5)zukog;d t9o t899qekqnly a few hundred meters across, orbits the Sun lik-tk)9kuz go9q xe 5roq;98tsde the planets. Its period is 410 d. What is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average distance of 4.5 z,ztom-hbd5bp - ,r;2 tn p.tv$\times 10^{9} \mathrm{~km}$ 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 orbit7hnf+w /i 45qn/jeuak9k.ivu s the Sun roughly once every 76 years. It comes very close to + ij nun4 a/u9k/.f evhwk75iqthe surface of the 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 the center of th0rhgjiqp +( f)e 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 mean distac8rtz8,oe a e8d94c2*zgrq-ucfo j 9tnce for the four largest moons of Jupiter (those discovered by Galileo cduc8aj 4o et8-fz tzr,g9qr9*co82 ein 1609).
(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 t:iv kgsv/ ys+y:up.,che Earth from the known period and distance of the Moon. u ys:p.vg:+ ,/ scivyk   $\times 10^{24}$

  Determine the mean disxx rx07hr924rllad x .tance from Jupiter for each of Jupiter’s moons, using Kepler’s third law. Use the distance of drx7 hr40lxrx l92a.xIo and 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:1m9b uk*fvq r-7i,fec9k qj qd1pla+ and Jupiter consists of many fragments (which some space sciq9 -1,1+df:kjlefcq umka7*i pvrb q 9entists think came 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 tale describes an artixja //dugm1y-,ht5jj ficial "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) jagytjuxd ,/j-/1m5h. Imagine that this sun is exactly like our own, that the distance to the band is the same as the Earth-Sun distance (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 swinging in an axhup x(tl10/by 1f hvwc yx9:x/z(vb,rc from a hanging vine (Fig. 5–41). If his a1w(pcbx u(,fxlhhv/9y0b:1/xz v tx y rms 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 is 80 kg, and the vine is 5.5 m long.   

  How far above the Earth’s surface will the acceleration of pzrf g4,7g.i;b1b2 ,mw celjrgravity be half what it ir,ge,cmfj 4p1 gi.z b7wb;l2 rs on the surface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and9 g3ut3we.hlc c9r; lv spin in a mutual circle once every 2.5 s. If we assume their arms are each 0.80 m long and theirl;39 w3r9tcvheglc. u individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates once per day, the apparentinurs 17zmsfd,3 3; ri acceleration of gravity at the equator is slightly less than it would be if the Earth didn’t rotate. Estimate the magnitude of zr3ss r,1idu mn 3;if7this effect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from the E9kndo (ezuwl,(( ;yprarth will a spacecraft traveling directly from the Earth to the Moon experiek9,uo;n pw(ld zye(r(nce 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 you sta:+ ajm z8 a4xtwl6uap4nd on a bathroom scale in an elevator, it says your mass is 82 kg. What is the l a+pzauwxat 8 64jm:4acceleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space stac3ox.hwsy7 z0emy q7ab9 +d l3tion consists of a circular tube that will rotate about its center (like a tubular bicycle tire) (Fig. 5–zay9o+d sql7e073mhy x.3cbw 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 aircraft in a vertical loop1(h9u5xdbpvci 6zp l. (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 ac yzhxh w(sqfj,znt, 6,ir, *4vceleration due to gravity at its surfachfiz n*y,wrhvqst,( 6,,j4z x e and the gravitational constant G.

A plumb bob (a mass m hanging on a string) is deflectea31bb 5bj pz73bead. xd from the vertical by an 7zeb33 1pjd5bba xa. bangle $\theta$ 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.iav1x9v1k;/lzg txbj(mpnh3 w v-- f a design speed of If the coefficient of static friction is 0.30 (wet pavement), at wha3ka iw vv g;jx 1-p9xnmt1z(-lv./fhbt range of speeds can a car safely handle the curve?

  How long would a day be if the Earth were rot7,-pv1qy fstb, 11tg w8kfule ating so fast that objects at the equator were apparently weigh, wy1 1bsqepg8f ttuvl,f17k-tless?    $\times 10^3$s

  Two equal-mass stars maintain a constant distanceceqdf65 rz7q ts08 bn, 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 constant speed rounds a curve of radiedj+vr)f8h 0s us 235 m. A lamp suspended from the ceiling swings out to an angle of 17. r0vj+ df)se8h5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320eoib 8g/d*l 1r times as massive as the Earth. Thus, 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 t g1 rli/edb*8ohe following 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 Telescopn+lk 1zp 2qo:ne deduced the presence of an extremely massive core in the distant galaxy M87, so dense that 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 7 nql2ko+zn:p 180 $\mathrm{~km} / \mathrm{s}$ 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 constant speed as it traverses the hill andmuk w:6)x jnka l2t(mnja( 78x 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 would the driver feel heaviest? Lightest? Explain. (c) How fast can the car go withou(k(na7k au:8x mx6jm2n tj) lwt losing contact with the road at A?

  The Navstar Global Positioning System (GPS) utilizes a group of 24 satellites //2v1gbhn hrr orbiting the Earth. Using “triangulation” and signals transmitted by these satellites, the position of a receiver on the Earth can be 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 sh1v/nr/ b2hrg peed of each satellite.    $\times 10^3$
(b) Determine the period of each satellite. =    hours

  The Near Earth Asteroid Rendezvous (NEdi(ri(af8*j vAR), after traveling 2.1 billion km, is meant to orbit t8( vjirdif*( ahe asteroid Eros at a 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 vwrk t1hz;n*u.7e9l f space shuttle pursuing a satellite in need of repair. Youwk*rztl9f 7eu.; vhn1 are in a circular orbit of the same radius as the satellite (400 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. 4fvits:lg j-.2;k,j-4rl0g f okrld c(a) What is its mean distance from the Sun?i; 24l:-o- 0vjlklrrcg, gtd.sjk 4f f   
(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 vavubj 2v/vv 2epmxks62nnj9-fd20z + jlue of G would need to be if you could actually "feel" yourself gravitationally attracted b 90s2/pdvvfv-2 j+jkv6n2jeuxn z 2mto someone near you. Make reasonable assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around thevp6tzmop0 ubc 3 .z80a center of the Milky Way Galaxy (Fig. 5-46) at a distance of about 30,000 light-years from thecu m8ozzv. 3tpbap060 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 *j;mn55sl: st.ka:wr3zg drs 0.50 m on each side. Find the magnitude and direction of the gravitationa5janwzrs:lkd 3;.5g :*smtsr l force on a fifth 1.0-kg mass placed at the midpoint of the bottom side of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 kg orbits the E scckk:y( 7o5ti)lpx bl e1gv,p5fw7*arth $ \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 acceleration experienced by the tip of the 1.5-cm-long sweep sjlb9/7tqiwbf/t tus9hc - x*9econd hand on your w7-q bhtbct tw/iu9/ l9f*js9xrist watch?    $\times 10^{-4}$

  While fishing, you get bored and--5un*sp jlvt -tnah+ start to swing a sinker weight around in a circle 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 vl5vuhjpnt-sa +* -t-nertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in a ;0z3 sgze-6+ydqnvjf new highway is designed so that a car traveling ae3- f+gvqj0 z;nsy 6dzt 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 vp b do9 x6f4x,zyrs 51h0)0eoenfl2fgotiating curves. The angle of tilt is adjusted so that the main force exerted on the passengers, to provide the centripetal acceleration, is the nor pd z1enx20efxo06 5 hfs4r,oyf9l bv)mal 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 speed 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.$\approx$    $\times 10^2$