Sometimes people say that wa7s1nh,jdpfy5 u v-wlo 46fg0f ter is removed from clothes in a spin dryer by centrifup6u , gos-lfn4fy jd7fv05w h1gal force throwing the water outward. What is wrong with this statement?

Will the acceleration of a car be the same when the car tcap i2)feqs:l x62sq,ravels around a sharp curve at a conc :q2spsf,i e2x6l)qastant 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 the na26n v :)gpysp ,cut1car exert the gr1pus)tnc,ag y 6:vp2neatest and least 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 acting on7 z;i2gq q 16ttme/pvv;lb)np a child riding a horse on a merry-go-round. Which of these forces pvntvl/p7zmig 1p); q b;etq62rovides the centripetal acceleration of the child?

A bucket of water can be whirled in a vertical circle without the water spillyle3e-ive sck7c m,+2ing out, even at the top of the circe,y 7mc cs3 eliv2ek+-le when the bucket is upside down. Explain.

How many “accelerators” do you have in your car? 0um.dx;,.uqi- jgy dw There are at least three controls in the car which can be used to cause the car to;-g,wdu mdqxi. .y0uj accelerate. What are they? What accelerations do they produce?

A child on a sled comes flying over the crest of a small hill, as shown ino l9x+xrjoyd(9gj0b4slo6 + b Fig. 4rl69loj yobb+ d oxx+9sgj0 (5–31. His sled does not 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 r9(.z bv4+ iut2c o u+y7; mw n1ttog2,vlstmznounding a curve at high speed?

Why do airplanes bank when they turn? How would you compute the banking angjbel1cwhm; :y.7lkn ;a 5w(uw le given its speed and radius of t; wb(lm;w.nchj y1 ae57l: wukhe turn?

A girl is whirling a ball on a string around her hed wjz8yw1sswktl m(4qt; zu w.w++y3apg5.6 uad in a horizontal 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. When shoul6u+zww s;m q a5jy wt w+8u1l. (gwdz.skyp34td she let go of the string?

Does an apple exert a gravitational force on the Earth? If so, h9er1h( )+h1nhy3f ymrl lyv 4tow large a force? Consider an app1y9 h lm4n(fe1) lrv h+yr3tyhle
(a) attached to a tree, and (b) falling.

If the Earth’s mass were dou,c:kzac/ 5qdomhq ouvw:(2 n2 ble what it is, in what ways would the Moon’s orbit be diffe(, cu:m2:c/oo azhwd qqvk5 n2rent?

Which pulls harder gravitationally, the Earth on the Moon, or the Moon on tmaak)8ibu1;3piapo c w,ns 9j 2ihj15he Earth? Which accele8;a 1i s3mi,pb2k5aoa 9pnicw1ujjh) rates more?

The Sun's gravitational pull on the Earth is much larger than the Moonyl;/9ydlh9u6 rpe; ra 's. Yet the Moon's is mainly responsible for the tides. Exp 9;;hu/yl 9darl y6eprlain. [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 po,vf y1v 0,ilhu:9db,(jy vcgyles? What two effects are at work? Do they oppose , vuyg(,y,lf 10dbiyv 9j:vhceach other?

The gravitational force on the r1pay)g qtjk75d* pz *Moon due to the Earth is only about half the force on the Moon duer ypq1d5 aj )t*z*7pgk 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 larp9f6p l e j-ipggk+37zger or smaller than the centripetal acceleration of the Eaf-+k37 gzl6p g 9ippejrth?

Would it require less 5vyh shs6wexy-e hrd - :a.:,lspeed to launch a satellite (a) toward the east or (b) towar:ew e-y.l5vs:y6drx , ha-shhd the west? Consider the Earth’s rotation direction.

When will your apparent weight be the greatest, as measured by a scale i. 6ukzga0ylc6 n 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 the least? When would it be the same as when you are on the grzak6g 6.0 ylucound?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long periods in outer space could be adversely affected byun nt:(xxq;m2u/ + whkt y22kr,v- x/rzmy1s c weightlessness. One way to simulate gravity is to shape the spaceship like a cylindrical shell that /k: v mh/cnyyz;kqrsm -1+2tt2 ,xu2xurnw(x 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 “apparent weightlessi5yc;ek l ;qtgha )2e*ness” between sl2aht* qcie k)5g;e;y teps.

The Earth moves faster in its orbit around the Su2l/zq;w. glwe6 tt b5gn in January than in July. Is the Earth closer to the Sun in January, or in July? Explain. [Note: This is gl;eb6lt.5wgt / zq2wnot 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 discoversnunv.3z6 2 cti2uw 9eed to have a moon. Explain how this discuw.9s2u nz6 ct v23einovery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth is much larr*-hnpq n6dlf m--3y qger than the Moon's. Yet the Moon's is mair nlqh-3- 6-*qnfydp mnly 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 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 1980 oqumxdw:d6*) $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 Earub8 q1caud)/ zth in its orbit around the Sun, and the net force exerted on the Earth.zu8)/dubqca 1 What 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 exertptz51 /uoac -85bk rqued on a 2.0-kg discus as it rotates uniformly in a buu8 o51 prt5/zk-qca horizontal 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 th.;m:f b1gvnvm g19sl j4q-unve Earth's surface, and circles the Earth about once every 90 minutes. Find the centripetal accelergn bgm4lvj snv9 v:f-1;m u.1qation 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 a d sip0u9s5.zcl pkgjizz35 ;; l(z0gdcceleration of a speck of clay on the edge of a potter’s wheel turning at 45 rpm (revolutions per minute) if the wheel’s diametp 0d3 ck(;9 0 su;sipigzlzjzg5 5.zlder is 32 cm?   

  A ball on the end of a string is revolved u l gzs8 .a9xn1y+ 52cei/vlfbn rkc;5j0tzv5at a uniform rate in a vertical circle of radius 72.0 cm , as shn2 /vzbx9 ug flacln50yvt1 +r.5j;zs i5k8ecown 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 ac0mseo;/w fsc g9 9+hcan round a turn of radius 77 m on a flat road if the coefficigc0+wes 9o/mf;ha c9 sent 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 static frictioj oh**hvcy4 r,n be between the tires and the road if a co jvhcrh 4**y,ar is to round a level curve of radius 85 m at a speed of 95km/h ?   

  A device for training a++ kvsu 0kf;zjm:j)gj nd0-tfstronauts and jet fighter pilots is designed to rotate a trainee in a horizontal circle of radius 12.0 m . If the force felt by the trainee on her back idt);mzj j:0f knf +- s+kvg0ujs 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 rotating tuyetv7(8 2nrufzau )/6n oxlj 9 ys,du+rntable of variable speed. When the speed of the turntable is slowly increased, the coin remains f2nu)u697 no8axv(zdey f/j t s l+ur,yixed 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 when upside dow6:s 3nrmq4)vn:db ql kn at the top of a cq n6:)b lsk3vr4mdnq: ircle
(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 (iph42t ekl ra,7ih -tbf a-1ds3ncluding the driver) crosses the rounded top of a hill 3 e 1t-akfah7 lb-,is 4pd2thr(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*qbw+v;g s t ,q8oe1+ riqcvv; minute would a 15-m-diameter Ferris wheel need to make forg,qo sq8e1+*+ivv;cr vqwb t ; the passengers to feel “weightless” at the topmost point?    rpm

  A bucket of mass 2.00 kg is whirled121ui p .a8 63p 3qg tfrdp6qn+zqpfzs in a vertical circle of radius 1.10 m. At the lowest point of its motion the tensi pfziqgu66 z1q.sfn8pp1a3q r3d +t p2on 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 z8 / h(tg6sxtn9.00 cm from the axis os zgh6/ nx(tt8f rotation is to experience an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carno,s psul5nf4bch wv(mo4- x zc*x3(6bival, people are rotated in a cylindrically walled "ro5(m4s(,hxo c-sbupf 6zx bv*nwco3 l4om." (See Fig. 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 a frictionless air-hockey r f6v7cr: wuyg 3yfv-ano:m45tabletop, and is held in this orbit by a light cova n45cow7: u3 -ymr6v:grfy frd 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 cy.ejh0lgclw:c7z )w hj9 7f.t(r64mx kzi b 2weight of the ball which revolves on a string 0.600 m long. In particular, find the7 6 ihckym.74) w glrlxet2z h(:9fwbj.czjc0 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 ti,*7 gwbq5 ojhqfqf 8w 8eqwch4 6ey*.raveling 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 masses0y 0gm4wvg 4gd3l9 qdc $ 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 maneuvnvr 9p o)(dpaurp /;coe *m:d9er 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 centripetal co(bh; zlua) kwq3in d78mponents of the net force exerted on the car (by the ground) in Example 5-8 when (qahw)iub7k lnd 3;z 8its 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 f0ekb dh3 b7oo-rom the pit 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 tob hk3e-ob07 dhrough 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 with no slipping or skidding? $\approx$   

  A particle revolves in a horizontal circle of radius 2b3lf,snbt u*)rqq4mw5 ):mnh .90 m . At a particular insta*,)l3nfmn4 s:qwm)ubht5 q rbnt, its acceleration 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 gravity on ad, *r+ gtrysivyadl+ t+*. uo( spacecraft 12,800 km (2 Earth radii) above the Earth’s surface if its mass is 1350 kgg+to+(l+ r*r.s*y ,tudy di av.   

  At the surface of a certain planet, the gravitational acceleration g has a magxphd6 y8c 4l9jnitude4 8dl9phx6jyc of 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 accelera, gxiuximw;6mbw.8q5 tion due to gravity on the Moon. The Moon's radius is 1.7mmi5 xx g.w u8;bq6i,w4 $\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet hasgl)nsc4 km 0.j a radius 1.5 times that of Earth, but has the same mass)gl 0m.jnksc 4. What is the acceleration due to gravity near its surface?   

  A hypothetical planet has a mass 1.66 times that of Earth, but the same radiuydr8 mgfm +/f7:jqr-ts. What is g neafq8: 7j+r/y-mgftrmdr its surface?   

  Two objects attract each other g;1ym) da2mclk6(pp2h)f cnv ywiyt)- wg 63um ravitationally 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 va5u- 66avhlrfp lue of g , the acceleration of gravity, at (a) 3200 m 5au6frhvlp 6-   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s cenrh4ow z1:gl j4ter to a point outside the Earth where thegravitational acceleration due to the E4j 1zo 4g:hlrwarth 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 spherey9zuvgzc9k1 i9wkit7 i3xa-).1b * dxwo e0rz about 10 km i01 k * 9i3vkwb)9idw.rx 7zy9c ztgau-xo1eizn radius. Estimate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once was an a6 npd6t q)(h0ad.l ws7wh;pnt bn 3ve6verage star like our Sun but is now in the last stage of its evolution, is the size of our Moondw nlhh 6(d7.p0e )pvw6qn3;btnts6a but has the mass of our Sun. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronfmcrr9) tl0 i,auts feel weightless while orbiting in the space shuttle. Your friends rec)0 it9 rml,frspond that they thought gravity was just a lot weaker up there. Convince them and yourself 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 o x++ha aw-dw/6pfr*yacfh 5m-f a square of side 0.60 m. Calculate the magnitude and di/-x+h*cr dpa- ay+ wm 6aw5hffrection of the total gravitational 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 up on the same sidexpqk ;w :91u7i r7tbkq of the Sun. Calculate the total force on the Earth due to Venus, Jupikbrx1qt9 ikw ;p7q:7uter, 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 gravit) ss vyo6kkd74y at the surface of Mars is 0.38 of what it is on Earth, and that Mars’ radius is 3400 km, determine the 4)oykdssk6 v 7mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the kn ifnbiaz3k7qey9e5o:v87vb 1own value for the period of the Earth and its distance froq iiyo 9n ve7e 587vb3fba:1zkm 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 moving in a stable r+smeqfv78m r z) o33j5,kcvicircular orbit about the Earth at a height of 36 frj 37,ekmqizov)s8m35r+vc00 km.    $\times 10^3$

  The space shuttle releases a satellite into a circular orbit )pxzvzc9 .gn0 650 km above the Earth. How fast mustzxg) vn0c.pz 9 the shuttle be moving (relative to Earth) when the release occurs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occupants are to experccwv1b 1kkz97ience simulated gravity of 0.60 g? Assume the spaceship’s dia91z v c1bkwk7cmeter is 32 m, and give your answer as the time needed for one revolution. (See Question 21, Fig 5–32.)   

  Determine the time it takzcf 3i:keciyc4iy ,*(es for a satellite to 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 yoc3i i :ye, yzfc4*c(kiur result depend on the mass of the satellite?    s ~    min

  At what horizontal velocity would a satellite have toay-(. l ,5vak gj*6xh4/xvseq cd m17nvf:fhw be launched from the top of Mt. Eveh e k naqvx1 ,hycf lda.x*5/ g4:7sfvw-m6jv(rest to be placed in a circular orbit around the Earth?   

  During an Apollo lunar landing mission, the command module continr.: vgka nr056)(xibpfpb0i f ued to orbit the Moon at an altitude of about 100 km. How long did it take to go around th kigbvfp56r : fr i(00px)anb.e Moon once?    s

  The rings of Saturn are composed of chunks of ice tha j(;sj6u*(y qy6ow pdwt orbit the planet. The inner r*(w opd y6wqjj6us;y(adius of the rings 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 oca0cdaf pczd+ +qw/; 5nce every 15.5 s (see Fig. 5–9). What is the ratio of a person’s apparent weight to her real weight (a) at the top+f+5c dpz0cc wad/qa;, and (b) at the bottom?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut .ckkcbs/ o69y 4200 km from the center of the Earth's Moon i s6c9c.okbyk /n a space vehicle (a) moving 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 consist*i9)xlh xjw(i s of two stars of equal mass. They are observed to be separated by 360 million kmij)h xw ix(9*l 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 b (ur5sd:02kkxq s9 qlof a 55-kg woman in an elevator that9 ss52dq :qlkr 0(uxkb 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 hanrpi5l+cberj go0 *-y. gs from a cord suspended from the ceiling of an elevator. The cord can withstand a te.-p +b5ycr g* jr0oliension 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 surfac/:kac7gvlu-q ebg*,kp6w *ch e of a planet with period T , the density (mass/volume) of the planetwc,q he6lcg7p**gbv uk -k:a / 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 tcx) 7 y ogb./o9zhvfme7 6uq.the period of the Moon (27.4 d) to determine the period of an artificial satellite orbiting voh.u9q xe6ym.7v) fg ztcb7o/ery near the Earth’s surface.    s

  The asteroid Icarus, though only a few hundred meters across, orbits the Sun qckkewc8 *h-seh79.r2 :ys balike the planets. Its period iswh y:k sq9aes-e2* b.r7c8hck 410 d. What is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average distanc6*(k ky+vosbf e of 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 roughly once everwss/)z5dir; j p9rqp-e n( wq,y 76 years. It comes very close to the surface of the Sun on its closest approach (Fig. 5–39). Estimate the g i z;(q-jw erd9sps ,5npr/w)qreatest 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 uqu ,r ovudg8j-cyjf171bw 8 u;d)mh9center 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 mean distance for the four larg 3/htfr- jou *0(*kb/foosv/h8gsb ze est moons of Jupiter (those discovered by Galileo in 1609)rfb8oofvb jk/0z3hotsghe/su* *(- / .
(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 period and distanc ooaa9gylb54bk71 ll/e of the Moon. abk41log9 b loy5l7a/   $\times 10^{24}$

  Determine the mean dc+/8gabyzu g*iu9 7d6wb:ave /z.bt :hnujj .istance from Jupiter for each of Jupiter’s moons, using Kepler’s third law. Use the distance of Io and the periods given in Table 5–3. Compare to the values n:u7ab+b j/9j w 8edt / g*gi.ay:hzu.6vbucz 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 fragments (whic 8hzg3g ;vl/lbb5zaff9 2xlo .h some sz3lx g8 l; 52 ozlghvbb.f/9fapace scientists 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 artifga5jmdk ,j*9 qicial "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 distance (to make the cli5j9d a,kjm q*gmate 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 s u0uw)lk4 :bewm 96rfiwinging 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 l6bi:f rwm9u u4kew)0 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 mky.3ak/-dddg *: 5pmwg1u w ysurface will the acceleration of gravity be half what i pyg*w1 3kdu: kga5-/yw. mdmdt is on the surface?    $\times 10^6$

  On an ice rink, two skaters of equal massa.gp3)fzvw; ps50+ra(t7zw zwn,ywz grab hands and spin in a mutual circle once every 2.5 s. If we assume their arms are est+wa z3yr) ;v 70zwgw,z5f (panp z.wach 0.80 m long and their individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotate m rp*pvb k/*b oiv367h)lmt0fs once per day, the apparent acceleration of gravity at the equator is slightly less than it would be if the Earth didn’t rotate. Estimate the br3pik o*p)*6vm0/ fm7lvtb hmagnitude of this effect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from the Earth will a spacr*+hbs-c1y.l. vinz z ecraft traveling directly from the Earth to the Moon experience zero net force because the Earth a zz+li -c s..1rvby*nhnd Moon pull with equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, but when you stand on(k m*-u9chqqazmr8 8j a bathroom scale in an elevator, it says your mass is 82 kg. What is the a 8mk9(-z rqum*qjc ah8cceleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consists of a circular tubms22nzxtnqhwj wm250/+bo z .e that will rotate about its center (like a tubular bicycle tire) (Fig. 5+2 j2 .zmnmzx25nbtswwo/0h q –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 loop ( mc)tt+ u x+45jqhrda)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 accelera9lp i jf:pn *.i.d:rbmtion due to gravity at its surface and the gravitatim .*i:b9: rdp j.lfpnional constant G.

A plumb bob (a mass m hanging on a string) is deflected from the vertical bb*8 o(8gpmkjonorb2*byb 8 slhq+;e+ y an angle ;+bp8k2 8*ob*e n(s by hm8r+oobqlgj$\theta$ 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 fo*un0bnj:d bj4)uk ;ec r a design speed of If the coefficient of static friction is 0.30 (wju4 en:n bk;cuj*) bd0et pavement), at what range of speeds can a car safely handle the curve?

  How long would a day be if theh60oshs1uoaw99 h g8t Earth were rotating so fast that objects at the equator we9thos gou0 s61aw98hh re apparently weightless?    $\times 10^3$s

  Two equal-mass stars maintain a constant distand6xd e)z b6g dwi;u32nce 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 roundlb, 58ggch. u*9hp0qy xx jxv6s a curve of radius 235 m. A lamp suspended from the ceili l h9jgy*p bv68xxxcgh5, q.u0ng swings out to an angle of 17.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 timelensgl- 3xm4+ s 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 expgesm+3- nx l4lerience at the equator of such a planet. Use the 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 Telescope deduced the preseno6v 14dy 0+gnn kpqw:rce of an extremely massive core in the distant galaxy M87k 6qo n0:ryp 1gn4dwv+, 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 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 constant speed as it traversehn*uy8zmn w9i)lb ,zc1xp8 3ls the hill and valley shown in Fig. 5–45. Both th whpyc8 nu mlx),z *1nzbil983e 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 without losing contact with the road at A?

  The Navstar Global Positioning System (GPS) utilizes a group ojs2(z e.f0 vmp1 7rlagf 24 satellites 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 1l 7r.ag1jmp0f e( vsz21,000 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 Ren etfocwgt/ei(1sr; ni 801 jx6dezvous (NEAR), after traveling 2.1 billion km, is meant to orbit the asteroid Eros at a height of abrc t1; os0t8ewg/nx 1ijife6(out 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 s)1aswf ;x p jvk e1m1cry2b2d b4,7mhkpace shuttle pursuing a satellite in need of repair. You are in a circular orbit of the same radius as the satellite (400 km above the Earth), but 25 km behind i12,xykbvp21f sb ;c)kw dmjmhre41 a7 t.
(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.qr9f4ud+a :w e7.tnqd4vpua / (a) What is its mean distance from the Sun? q 44d:.d /n waqa9uuvft +r7ep  
(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 ne fpyucrs2,rgb o5a *19ed to be if you could actually "feel" yourself gravitationally attracted to someone near you. Make reasonables,c5ruroby92p f1g *a assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the Milky Way Galaxy(hb273i zvlh fq*mfs5 (Fig. 5-46) at a distance of about 30,000 light-years fv 7f* bz h(5msfh3qi2lrom 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 each sidec1tg icv; o;k ye+( uqp4ms(g). Find the magnitude q1tg ;(uev ok mgci4)scpy;(+and direction of the gravitational 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 dsplwq2n)*, s),/q pnjorgx6the Earth $ \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 hesz 0+)fvl( vhhj4a0j-/y hy.5-cm-long sweep second hand on your wrist w( /sazej f 0-vh)lhvjh0 h4+yyatch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weight around inf rb9 to0 )gin v7m8+ze;ds9acpe 2fe* a circle below you on a 0.25-m piece of fishing line. The weight makes a complete circle every 0.50 s. i2f0*ge8fvc) + bz7atd; 9po ensm er9What is the angle that the fishing line makes with the vertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in a new highway is 3p/, 0hch rtu hz0t/zkdesigned so that a car traveling kp3utrzh0chz , 0t/h/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 Acuavqqb:wz-2o*c :au l:d6q5 1z f+znpela, utilizes tilt of the cars when negotiating 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 radiq o1z-zbn:c+l :pf *d2q uaq6uvza5w: us 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.$\approx$    $\times 10^2$