Sometimes people say that water is removed from clothes in a spin dt-e;-ao (hun4wm(ovxz* go0n3zr v vrl 0:,be ryer by centrifugal fov-wmu ong- v;oa04(elzho*: (tvz3 nxb0r ,re rce throwing the water outward. What is wrong with this statement?

Will the acceleration of a car be the same when the car c:r)gey2/rwt e8y ,cvh8mfv 6travels around a sharp curve at a constant 60 km/h as when it travelvh/2e ,gmrw6r88 : yvc)ycte fs around a gentle curve at the same speed? Explain.

Suppose a car moves at constant speed along a8(fmb /7gnv;2yupexnhl;en b ;4i8sh hilly road. Where does the car exert the greatest and least forces on the road: (a) at the top of a hill, (b) at a dip between two hills, (c) onhv8y;2lhx nmbnn(f;g /bup 8si;e74e a level stretch near the bottom of a hill?

Describe all the forb85mqhip*7 zsces acting on a child riding a horse on a merry-go-round. Which of these forces pb q*him8zs75 provides the centripetal acceleration of the child?

A bucket of water can be whirled in a vertical circle/ la*ff9n dqe, without the water spilling out, even at the top of the circle when the bucket is upside down. Exp nad,9fqfe*/llain.

How many “accelerators” do you have z ouov/:)9ascin your car? There are at least three controls in the car which can be used t:zaosc9uo v )/o cause the car to accelerate. What are they? What accelerations do they produce?

A child on a sled comes flying over the crest of a 1*0mx9 gvap lhi b y2zk(n7)igsmall hill, as shown in Fig. 5–31. His sled does not leavelhk y zmv ng x*g129a0ipbi(7) 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 le. 6i6b2g9ioac h zcbjax edcme6358n2an inward when rounding a curve at high speed?

Why do airplanes bank when thbjyp8 wma; ikq3w9. uof 2)qk3ey turn? How would you compute the banking angle given its speea)3k i o p9q8fqumb.y23jkw;wd and radius of the turn?

A girl is whirling a ball on a string aroun6k5)r t b4b +jbmxzo.zd her head 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 should she let go of thej.65bxzmob)kr+4tb z string?

Does an apple exert a gravitationcit gky06d /v4al force on the Earth? If so, how large a force?ky dvtg6c 4i/0 Consider an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, in what ways would the Moo wyb/.x- zem).svz2bk w59z8- onfzhgn’s orbit be diffe) kb9m .z w2bo-wyze/ n8xg. h5zvz-fsrent?

Which pulls harder gravitationally, the Earth on thx 9u5 (a*rjz)sd .gecy8u1unae Moon, or the Moon on the Earth? Which acce)u8zruadeg1cn 5(.yj9usa*xlerates more?

The Sun's gravitational pull on the Earth is much larger than the wuzv-40eybvr5 v r(tp 2pne(2Moon's. Yet the Moon'r2n p4vy5 p2-br(tu(wev0zev s is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]

Will an object weigh more g t3uby/nz7 y*at the equator or at the poles? What two effects are at work? Do they oppose each othern /7gb3*yytzu?

The gravitational fors:-hl0 ;ry2ebuua9lcag; is; ce on the Moon due to the Earth is only about half the force on the Moon due to the Sun. Why isn’tli ; ;su2 y9ulra-ah:eg0;s cb the Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in -y0+x9duzne3 )c kg1px e sr3rits orbit around the Sun larger or smaller than the gry nxxe0z )r3e 9d3 -k+p1usccentripetal acceleration of the Earth?

Would it require less speed to launch a satm)+h ev5*0w qe. g1jzatri,e eellite (a) toward the east or (b) toward the west? Consider e*1 j.5r+m eqw etah)v,gi0e zthe Earth’s rotation direction.

When will your apparent w eev9jjq/s71 weight be the greatest, as measured by a scale in a moving ee/sj7jq 1e 9wvlevator: 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 ground?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long periods in outer space could be adversely affeklow rd0 ;bsqav 9l9 n2rx3x69cted 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 this simulates gravity. Considerd6rwa2 99xnsv30q l; 9 orbxlk (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 “apjec2f.tsffr4 n w/) f*parent weightlessness” betweenc.4*fjn 2w tf/ffre)s steps.

The Earth moves faster in its orbit around the Sun in Janu qsrj. s45dh. gvan2d3ary 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 producingadvgd3 ..h rsq jn2s45 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 o e4 l.x11+np2q2 vj.apn8mns ez1wnlnot known until it was discovered to have a moon. Explain how this discon 1q24 ze.ppn1jw1.nl oasl28mx ve+ nvery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth is much larger(25fa.ynzx yu 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 fn2y 5yua.f (xzrom 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-rm3w uy1a5ds 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 d.ve0d+mymgz c,x d(5j 6e rg5the Earth in its orbit around the Sun, and the net forccm 5 .d ve65gdx+ (rgzem,0ydje exerted on the Earth. 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 exerted on a 2.0-kg discus as it rotateshbhfc.,n3g v0i /g7fh uniformly in a horizontal circle ifhgh,/bvf3 7h g0c n.(at arm’s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is in orbit 400 km frur;zqh1 ;1vyhom the Earth's surface, and circles the Earth about once every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer iry h1 uv;h;1zqn terms of g , the gravitational acceleration at the Earth's surface. $\approx$    g

  What is the magnitude o 0ws*o.ais;hff the acceleration of a speck of clay on the edge of a potter’s wheel turning at 45 rp*i ;hf0asws.o m (revolutions per minute) if the wheel’s diameter is 32 cm?   

  A ball on the end of a string is revolved at a (u,.eea- h26 0acofhfr ba8 mpuniform rate in a vertical circle of radius 72.0 cm , as shown in Fig. 5-33. If its speedcf,eu0a b-.ohrepm ah(8fa 26 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 turn of roulo s5k o-on:ju( q2 h4bme6 lo4;-deb;t,d vadius 77 m on a flat road if the coefficient of static friction between tires and road is 0.80? Is t bn-mboo-:le u 2 seo5kvod;4qd 4t(,uhl6; johis result independent of the mass of the car?   

  How large must the coefficient of static friction be between the tires and the r2yb uu t8wbd9*oad if tudw*2y9bbu 8a car is to round a level curve of radius 85 m at a speed of 95km/h ?   

  A device for training ast i+eevbf88u 8xronauts and jet fighter pilots is designed to rotate a trainee in a horizontal circle of x8efbuive88+ radius 12.0 m . If the force 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 0vm(z bmsrwz+3v8)ipcm from the axis of a rotating turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable until a rate of 36 rpm is reached and the coi vr0+vi z)ms(m3 w8zbpn 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 downrl-44(gy eja21r 9vhp 2b yapbf h x:e(47bxxq at the top of gpr7 2yl(:4j x9 bexrp-e(4bahya q4xvh b1f2a 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 (including the driver) crosses the rou lwgg9 yg2ww,m:d:c ot9 skm7,nded top of agmk:so :gwwlt d9g2, y7w9c, m hill (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 5a2;n c ybxkcx4v7v(da 15-m-diameter Ferris wheel need to make for the passengersd( cv;v a5y2 cxk74nbx to feel “weightless” at the topmost point?    rpm

  A bucket of mass 2.00 kg is whirled in a verti*lqtdh /;l zy7cal circle of radius 1.10 m. At the lowest point of its motion thh;z/dy lt *l7qe 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.4r7me 0b5i ag)n.ia;vci+/ k e ernsb particle 9.00 cm from the axis of rotation is to experien0 .n+ii s r5c.r;4in/bame7e) kvaebgce an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotated in a cylindrically walmhhc :u1hsm1qm -w41f led "roofu 1mc1 -h1:4h mhsmwqm." (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 circycteyap9f2k 0. n)t0p le on a frictionless air-hockey tabletop, and is held in thisyet na.kp0 0)9 cf2pty 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 , precise ,(ur8wrq, oqhly this time, by not ignoring the weight of the ball which revolves on a string 0.600 m long. In particular, fin,,o8q r urwq(hd 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 0e: be/4 dz.fgk vvx +h:eoxeyl)mc;.n-whz/ banked for 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 masse(z p6c7z p,uwss $ 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 evapszxqk/* gy+ z) iud.(sive maneuver 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 components of 1ivwbmm479f jthe net force exerted on the car (by the ground) in Exa4f i9vb1mj m7wmple 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 area, i (e(xk6ac*bx5 rams *going from rest to 320km/h in a semicircular arc with a radius of 220 m. Determinaex(r*i(m5c*sk6x b ae 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 with no slipping or skidding? $\approx$   

  A particle revolves in a horizontal circle of radius 2.90p (a.; 3qmro3,waa+5lnypd 5/fvxgsx m . At a particular instant, ynp /qfm.ga5l;sv( 5o w dap3xx3ra+,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 Eartm :qcw:s ,tq4sj9. asbh’s gravity on a spacecraft 12,800 km (2 Earth radii) above the Earth’s surface if its mass is 1350 kj9a s cb:w4msq,q:t.sg.   

  At the surface of a certain planet, the gravitationalm /y1 liv 9drc)s6 w.f4ks6ktl acceleration g has a magnitude of 12 ms)tvdc4yk 9r6.slwi1 /fmk6 l/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 ogtjt6xvjxu zms 0xh/l(ye4 5u9 0+;qbn the Moon. The Moon's radius is 1.74hxvjtgb ;xl 0z4+(9q0 xse u5mutj/ 6y $\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,lnpb3j5q q ag7x(3 .g.racnl: but has the same mass. What is the acceleration due to gravity near its sg5qpanxn:r.7(lc al3g 3 b. jqurface?   

  A hypothetical planet has mwo 7)k zj k(2 6q5jibhudbx)ukd5 wv62-6hsla mass 1.66 times that of Earth, but the same radius. What ikj)dz2 wjm7d bh62kiqu5xlhou6k5v s6w)b- (s g near its surface?   

  Two objects attract each other gravitationally with a forc)k7 kj3y;5p)6juck 9g uefntbe 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 effective7jyraooo:qz0xw:*4 5p nt7qk value of g , the acceleration of gravity, at (a) 3200 m xqw*4j77q 5yzk :op o:nr t0ao   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside the Earth where trla4 gyru,.t3l5 h2fmheru42ag ., hflylm5r3tgravitational 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 sphere a k *055bbcxmgf9w ov*a,kdo , */lggplbout 10 k*g/k50lo9 ,5wvg*,dbpxf cl abomg * km in radius. Estimate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, wh4ufk t;s+vfrha+ ,n q-ich once was an average star like our Sun but is now in the last stage of its evolhs, 4kn+; f -rtf+aquvution, 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 orbiti 5j+ nyyu;v)abng in the space shuttle. Your friends respond that they thought gravity was just a lot weaker up ther)5yvu a;j nby+e. 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 of a square of sid eqi5*c jui clqcg+9zzc6.+s)e 0.60 m. Calculate uig+)c*ljc9 sqcc z 5z.+e6iqthe magnitude and direction of the total gravitational force exerted on one sphere by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred yea. zmv7q*z5bh ) x(xivmrs most of the planets line up on the same side of the Sun. Calculate the total force on th.)b5( zxq*mzxvmvh7 ie 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 t 8 tddhz5ms6,g 8c8wcyhe surface of Mars is 0.38 of what it is on Earth, and that Mars’ radius is 3400 km, determine the mass of Mas5 6cgw 8d mthd,yc8z8rs.    $\times 10^{23}$

  Determine the mass of the Sun using tuq/ebb9v *e n)he known value for the period of the Earth and its distance from the Sun. [N /q*9ueb) vebnote: 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 stablzi48i d c2d8o9-b*uz jno+reie circular orbit about the Earth at a height of 3600-bz2i9 niu dei o4o88j dc+z*r km.    $\times 10^3$

  The space shuttle releases a satellite into a circular orbit 650 km aboe6am-rhv 6 5 zq;bbml oy2e n9xbe*(;hve the Earth. How fast must the shuttle be moving (relative to Earth) when ;leab;yh96m- 2znbr(5qoehme x b6*vthe release occurs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occupants are to experience qsul6edp3 mah685 doy.f/4k usimulated gravity of 0.60 3yu eh sfdqp6/u84doa. 5klm6g? Assume 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 takes for a satell8kar4ztkt7o+ njkn6g75u9:u bw j at1ite to orbit the Earth in a circular “near-Earth” orbit. A “near-Earth” orbnot jwna1 u78:bz+ku jkka9 rt4 t7g56it 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 the top ad)e-/j r4stz2dvw nlkc (6w8 of Mt. Everest to be placed in a circular orbi( z k vnrj/wwe8ls6dd-42tca )t around the Earth?   

  During an Apollo lunar landing missineipd8e6xs1ri39 grs6 bg / p ue.;wv1on, the command module continued to orbit the Moon at an altitude of ae3ni6;pw1 s6 d ig8p1ev urexr9.sgb/ bout 100 km. How long did it take to go around the Moon once?    s

  The rings of Saturn are composed ofbn *lqu,x-3:6lxnzc e chunks of ice that orbit the planet. The inner radius of the rings is 7l- lun6cb,:xe *xnqz33,000 $\mathrm{~km}$ , 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 Figi tg2:;u73 arbjp ;2z6axe ddtph:;p r. 5–9). What is the ratio of a person’s appargap:;i 6jxpra b rt e;z;223th:pd7duent weight to her real weight (a) at the top, and (b) at the bottom?
(a)    (b)   

  What is the apparent tl f2ygp-.o p;uaz 8w: iz-3byweight of a 75-kg astronaut 4200 km from the center of the Earth's Moon -z:wiyp3 g;u8p z a oy-bfl.2tin 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 consists of two stars of equal mass. Theya rbl5f gl5 +,p6 cztkp2se1l( are observed to be separated by 360 million km and take 5.7 Earth years to orbit about a point midway between them. What slbr plf5laz (51get,ck2 6p+is the mass of each?    $\times 10^{29}$

  What will a spring scale read for the tg uw 1o96h74te+nkxfweight of a 55-kg woman in an elevator that movw41 uh79t6n eg kxot+fes
(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 hangk2 *d4pty ivx)7bs bc:l+q*lbs from a cord suspended from the ceiling of an elevator. The cord can withstand a tension of 220 N and breaks as the elevator a24d: pbsx lq)7+tl* ic b*bkyvccelerates. 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 p(yx6zy,bv 8un kr: ,nceriod T , the denvnxynb: 8,ucy, rz k(6sity (mass/volume) 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 the.6hku*b;h / xoi cku9h54 (lhj/pbr dv period of the Moon (27.4 d) to determine the period of an artificial satellite orbiting very near the Ear.9 bhuib*/; h(xprov4c/u5 h k6lhjk dth’s surface.    s

  The asteroid Icarus, though only a few hundred me e7mqh)n iel*)0qe a(kters across, orbits the Sun like the plan0mqiql*e hea))7 (nekets. Its period is 410 d. What is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average distancj3t n smor5(,t5y6-dobd 7ae fe 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 every 76 yeqd grmhr90/ lvw7e(k 4ars. It comes very close to the 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 farthe7l9erdm kr 4/(gqh 0wvst distance from the Sun.]    $\times 10^6$

  Our Sun rotates about tbfydj1 xs ,o+xv(x :(rhe 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 ma2 rnqadr7m0qe(d/j r 3ss, period, and mean distance for the four largest moons of Jupq73dq(dre ar20 /rnjm iter (those discovered by Galileo in 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 the Earth from the known period anl)s9f:fi bet9 o( n7egd distance of the Moon. : to 9)ffb7e(les 9gni   $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jupiter’s moons, using Kefw5w* d ;c-rrdpler’s third law. Use the distance of Io and the periods given icw-d;rr 5fwd*n 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 fragments (which s u 32cznves)0 8mvhoo86i ws8 oj,dl,qome space scientists think came from a planet that once orbited theeu8 lmi6oqz0 3nosvwjov8c 2)8sd, h, 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 des:t5:a5wvoac av18 x kqcribes an artificial "planet" in the form of a band completely encircling a sun (Fig. 5-40). The inhabitants lc1a: w5o8q:vax5 vkative 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 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 swingiu0jclcj 5t19 zng 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 is 80 kg, and the j zc9lct105ju vine is 5.5 m long.   

  How far above the Earth’s surface will the acceleration of gravi2t zu3kh l+ja4 a5evb:lhp/sj1i*i3wty be half what it is on the surfac3a2a4lvkstw3i*b h5:1ejjz/+p u h i le?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spino0(p tre :hnannd )l8; in a mutual circle once every 2.5 s. If we assume their arms are each 0.80 m long ar;0a8n e(ltdo)n hpn:nd their 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 w(bynjlje2pp hp o2v 09 l2ns/,v0k:y apparent acceleration of gravity at the equator is slightly less than it lyv 0p9ejyvjos2nlp/k2 ,(h:pn2w b0 would be if the Earth 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 trav 2uw92 jgs x0py3tbo -2zwmc.xeling directly from the Earth to the Moon experience zero net force because the Earth and Moon pull with e0 2jzspmy -3 22w.cox9 utxbwgqual and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, but when you stand on a bathroom scale in an ey.imij2xno 9a,d pkb 22o67 aclevator, it says your mass is 82 kg. Whatnxpi a62ja id9k2o72 oc.m, yb is the acceleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consists of a eahu .yy /r:ehig;;butu8 g1/circular tube that will rotate about its center (like 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 effeagu. heu//g1;t;by u8e:h ryi ct 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 vjq8ji)n q/e9:)z v sbzq e9krk09jx.iwx1m+ certical 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 planfnqfih n+41 j5et in terms of its radius r, the acceleration due nq4f h i5n1jf+to gravity at its surface and the gravitational constant G.

A plumb bob (a mass m hanging on a string) ibe,w6r9 c0mby oix+ ,cs deflected from the vertical by an angle xye0,crmwoc69 i+ b,b $\theta$ 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 Ify. ,2 kf3pgcibi3/cd 4 nzulu7k0 9tzq.-a hgz the coefficient of static friction is 0.30 (wet pavement), at what range of speeds can a car safely handle the curvei3,yz .czua gcghq.k4nkd90/u 3b7zp li2tf- ?

  How long would a day be if the Earth were rotating so fast thatg8)b vbj+ pl2t objects at the equator were apparently weightless+pvgb 2t8bjl)?    $\times 10^3$s

  Two equal-mass stars maintain a constant dist1.-8opo2j+wdn saznw4 cwb.dance 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 :54 xs/ xlp4y 1nidrwc0 cgp7dcurve of radius 235 m. A lamp suspended from theplc1iwy:05 / gxx4drc4dn s7p 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. Thus, it has hebp;40aiea .(ma2vq/ ) xpfz 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 experienc) p4x2.(paae0/f vmi ;zeqabhe 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 thq( u(usnpb -n8i7uc x(e 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). Tuip u7b qn-s(cuxn (8(hey 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 traverses the hill+2ym,) mu/f k ib7 tl0ierneqw. +4xol 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 .+4om,blw /ri yf0leki )+un7mx 2tqeC 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 of 24 satellites 692zzccy*8tp ued3qb orbiting the Earth. Using “triangulation” and signu c3pz b t6z9*2ecqyd8als 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 speed of each satellite.    $\times 10^3$
(b) Determine the period of each satellite. =    hours

  The Near Earth Asteroid Rendezvous (+ch/mqjfyq*( NEAR), after traveling 2.1 billion km, is meant to orbit the asteroid Eros at a height of about 15 /qf(cyqh+j *m 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 satellite in need of rkpi3hqfbhkr1o( a, 63epair. You are in a circular orbit of the bp,qf3h h3r 61oiakk(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. (a) What is its 13nosf 0dmbh4pl*-*t jf.ull mean distance from the Sun? dfuf bnm* t.sjhlpol-4*0l31  
(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 t:mm9q, f (iovio be if you could actually "feel" yourself gravitationally attracted to someone near you. Make reasonable i9,qm(vom i:f assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the Milky Way Galaxy (Fig. 5-46) ate1iixm)/ xy/ cd8 -qk9l7qmh j a distance of ab97iid1 xeylxm hc/8qkj/-m )qout 30,000 light-years from 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 siq0oymkn v6udzxf/+2 j vl/*y9de. Find the k0 v/yu92ynm d*oqvj+lf /6xzmagnitude 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 kjl z7hk92hiu++wcrh- xoq( t)g orbits the 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 1.5-cm-long swekke4ok hx-06)le* mv tep second hv*kk 6le)0-m4o kthex and on your wrist watch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a;trj s-z 0,spqbj9b6 j sinker weight around in a circle r9sj-tsbj0z p j;qb6 ,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: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in1fxoaum3 .v y6 a new highway is designed so that a car traveling at speedaom3.1y6ufx v $ 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 t +s./ao2r2pho-1dcmovi- lb qco.t 2nrain, the Acela, 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 radius oohtno2vco p-b1mq +.dsoa/i .l22- crf 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.$\approx$    $\times 10^2$