Sometimes people say that water is removed frodvv /rgk4eso14:4c k5jqw 4i g* ;jwdom clothes in a spin dryer by centrifugal force throwing the water outward. Wvcesd d:g4 4kov4* k;5jriwg1jow4q /hat is wrong with this statement?

Will the acceleration of a car be the same when the car travelbvf x(m:b f.b xaf10(ws around a sharp curve at a constan0afb(1mw(: bxfv.b xft 60 km/h as when it travels around a gentle curve at the same speed? Explain.

Suppose a car moves at constant speed along auyfwu +3hz8x6 i-tv h3h 7sj-b hilly road. Where does the car exert the greatest and least forces on the road: (a) at the top of iuw-8juh h73tsxzb+f-h y36v 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 horse on a merry-go-round. Wh+izw*nmdn0s og6*kr8k-h l .dich of these forces provides the centripetal acceleration of the child? r8o6*ks*+ kl.0dg dzwnn- ihm

A bucket of water can be whirled i ntw2klz+ 7-: o0h)(ie zkbftin a vertical circle without the water spilling out, ev hni(e+o2 zk:)wtlit7-zk 0bf en at the top of the circle when the bucket is upside down. Explain.

How many “accelerators” do you have in your car? There are at least th +6nx+h(k lzye k ajt)xa uan(3:e7ue*ree controls in the car which can be used to cause the car to accelerate.7a6xh (3z nj):yu *ukeexket+(l+a an What are they? What accelerations do they produce?

A child on a sled comes flying202h,zxl2aqd6n ht fd over the crest of a small hill, as shown in Fig. 5–31. His sled does not leave the ground (he does not achieve “air”), but he feels the normal force baf 6 xdq,0h2zl2htdn2 etween 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 jbxd;o 72mil1k4 -hrxqlqh: ,a curve at high speed?

Why do airplanes bank when they turn? How would you compute the banking angle gioa; gg+ ti f9yl)nq8e6ven its spe+yi6 f8n l)geo9;qa gted and radius of the turn?

A girl is whirling a ball on a string around her head in a horizontal plane. x(1ueba)c uq* t0v m ltge67 *t*v,zbvShe wants to let go at precisely the right time so that the ball will hit a tartvbv*61q7uumz) t, **(bagtlvcex 0e get on the other side of the yard. When should she let go of the string?

Does an apple exert a gravitational force on the Eay uit mj7 kz1k68s0.jsrth? If so, how large a force? Con.8u tm6s1 ykks7j0ji zsider 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 Moon’s pdm2tuv25 .7 ss fvp-l1/giylorbit be ddu ipm .1st s/52 lp2fv-v7ylgifferent?

Which pulls harder gravitationally, the Earth on the Moons;2q *y ux*zad vrp5p(, or the Moon on the Earth? Which accelerates morazuq5s2dr ;y(*vxpp *e?

The Sun's gravitational pull on the Earth is much larger than the Mooq1cikf n c8;,* d jiozg3;8laxn's. Yet the Moon's is mainly responsible for the tides. Explain8 n o*1 kli,3dfza;8c jxqcgi;. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]

Will an object weigh more at y),wk4 ubmc o(the equator or at the poles? What two effects are at ,m(4 cw)ubkoy work? Do they oppose each other?

The gravitational force on the Mo:o2bm/fiu *j.3c a4o lu9a wwpon due to the Earth is only about half the force on the Moon due to the Sun. Why isn’t the Moon pulled away from the Ea 39f:2*.iumbw wu/ ocpa la4jorth?

Is the centripetal acceleration of Mars in its orbi*qv)5jkh jo aa4m1(lxt around the Sun larger or smaller than the centripetal accjahk( lx)j5o* maq 1v4eleration of the Earth?

Would it require lessw4 x7z 18wnyvv speed to launch a satellite (a) toward the east or (b) toward the west? vv7 xy48zww1nConsider the Earth’s rotation direction.

When will your apparent weight be the gr2 wtw97b /thryeatest, as measured by a scale in a moving elevator: when the elevator (a) trtyh /w27w9baccelerates 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,a(c:*e o ltva in outer space could be adversely affected 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. Consider (a) ac(v t*:l,eaohow 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 expra 7nf6+9mlv5prp xv) ad0 c9ueriences “free fall” or “apparent weightlessness” between steparc)dp5v0f7vau x +m 6ln9r9ps.

The Earth moves fasterj vvp -ngtr/*h7md:h1 in its orbit around the Sun in January than in July. Is the Earth closer to the Sun g-rh1djh:tn7*vmp /vin January, or in July? Explain. [Note: This is not much of a factor in producing the seasons — the main factor is the tilt of the Earth’s axis relative to the plane of its orbit.]

The mass of Pluto was not known until it was discoverec u,f:,hwa;za +b fr7hd to have a moon. Explain hr+h cw7u az,b:f;,ahfow this discovery enabled an estimate of Pluto’s mass.

  The Sun's gravitationayvs :j h f f3)5y3yt(o6ii7qz):tnreyl 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 tt )rz76eihn:oy3qt( : 3svj fyify 5y)he 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 194*okgf.j woj080 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 Earth in its orbi68k7x3zvg7+llmih rl t around the Sun, and the net force exerted on the Earth. What exerts this force on the Earth? Assume that the Earth's orbit is a circle of radi7riz xm67glkh8 v3l +lus 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 2gb4r2m dd; ) g(5h*li*kkn3zcc qy;y0c2bzy a.0-kg discus as it rotates uniformly in a horizontal circle (at arm’s length) of radius 0.90 m. Calculate the speed of thy5 (2y0zqra kbh;l2m )nk d*c byg;d 4i3cg*cze discus.   

  Suppose the space shfvmyr27fynwm3) ,o;auttle is in orbit 400 km from 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 in terms of g , the gravitational acceleration at the Eaywnf ;or ym)a2mf73 ,vrth's surface. $\approx$    g

  What is the magnitude of the acceleration of a speck of clay on the edge of a poe 37aom7*ec eqb+nwdn-wgp -g0 *gi+jtter’s wheel turning at 45 rpm (revolutions per minute) jq-n cdpnmwo e 7+0g*i*7ae3b-e ggw+if the wheel’s diameter is 32 cm?   

  A ball on the end of a string is revolved at a uniform rat*ig. 05hot8e i,z pxtae in a vertical circle of radius 72.0 cm , as shown in Fig. 5-33. If its speed i,eht xg*.a p5oz08 tiis 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 l0tc;z. vwp(me5dxfeof65x 6 turn of radius 77 m on a flat road if the coefficient of static frwfm (p6zx5ov;0 l5dect. x6feiction between tires and road is 0.80? Is this result independent of the mass of the car?   

  How large must the coefficient of static friction h u:ixtqdk kw;z (t8s gj+-5 788jgbvabe between the tires and the road if a car is to + bag8 z5:t;gjqv7-kuhjsx 8 (iwk dt8round a level curve of radius 85 m at a speed of 95km/h ?   

  A device for training ar/:wo*s+4mnd ;zuox fvgicg,r,8qm1stronauts and jet fighter pilots is designed to rotate a trainee in a horizontal circcmr 4 g;m,zqw,* sfuo dg+8vion/1xr :le of 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 k)tswn4rtg 0)cm from the axis of a rotating turntable of variable speed. When the speed of the turntable is slows)n)tr 04tgkwly increased, the coin remains 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 wm/9o9s tvp xfu zd40c9hen upside down at the ttu9 4/xosmz9 vf c90dpop of a 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 (includingq)qc*5,2qcb:p bexnv the driver) crosses the rounded top of a hill (radiuspqc,nvq:)beb25 *c qx =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 wheel nee a*yi7rj 2c(i bmir.x5s:vb: ud to make for the passengers to feel “weightless” at the bau(2rs :*y mbvi7j.r5ic:ix topmost point?    rpm

  A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1.10 m. At t bs/zrgn6f1*:dhut d(;hj im7 he lowest point of its motion the tension in then*dums6i1 gd (7z/:rhf;bht j 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 i)68s yabvlhkf/f )5q;hl g gn/f a particle 9.00 cm from the axis of rotation is to experience an acceleration of 115,g)gyn/6;sla q 5bl hff8/vh k)000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carniv; j(mxxu4;vs g9utd 4,zljq2d al, people are rotated in a cylindrically walled "room." (See Fig. 5-3l j9x;mq4j s tdvzgx;4, ud(2u5.)
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 )b-s 69q +*hkko-: yzpmvd-i ao is rotated in a circle on a frictionless air-hockey tabletop, -pm6+zba*k9qh: -ooikys - dvand is 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 timejx 8)c4oubkl(l o g5q1, by not ignoring the weight of the ball which revo l 5ugjc1qbx)8(4lokolves on a string 0.600 m 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 trt nq,l-, *y igul/kmc1aveling 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 massesui .i643ctbto $ m_{1}$ and $m_{2}$ , are connected to each other and to a central post by cords as shown in Fig. 5-37. They rotate about the post at a frequency f (revolutions per second) on a frictionless horizontal surface at distances $r_{1}$ and $r_{2}$ from the post. Derive an algebraic expression for the tension in each segment of the cord.

  A pilot performs an evasive maneuver by divij83tnp9wu sg)w lc6- hng 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 8 rkhz ixl.n*rfyzr9n)*78p u the net force exerted on the car (r u7l ri.k*n 9nrx*z8hypfz)8by 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 Indianapoegf40dtc 4lcsdm/kp i/ 74f k7lis 500 accelerates uniformly from 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 throug7dp4dfis4ek/t 7c kg 0mclf/ 4h 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 2.90 m . (vlgj*rqcg,xy 9x0; fAt a particular instant, its acceleration is 1.q(vj,cf9r gygx0 xl; *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 a spauqu r2a 1s/4 nvz3*hxicecraft 12,800 km (2 Earth radii) abo* r4zsaqn/h2xu1 viu3 ve the Earth’s surface if its mass is 1350 kg.   

  At the surface of a certain planet, the gravitational3x eevf wdq+1d1zu:he ad* p(+ acceleration g has a magnitude of 1fz*(u3 ve+d:eehwxpda+ q1d 12 m/s$^2$ A 21.0-kg brass ball is transported to this planet. What is (a) the mass of the brass ball on the Earth and on the planet   
(b) the weight of the brass ball on the Earth and on the planet? $W_{Earth}$   $W_{planet}$  

  Calculate the acceleration due to gravity on the Moon. The Moon'speyx *xw zg8dl(9avlzn/udnk+: 3 3*d radius is 1.74yp39uen/x 3xd(*l vwld+z8ga k:* ndz $\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has a radius 1.5 timhoq;4:-ie i7h4nfvaj es that of Earth, but has the same mass. What is the acceleration due to gravity near it4vieaq-h4 fijh; :on 7s surface?   

  A hypothetical planet has a mass 1.66 times that of Earth, k-l. 6lttv08ru9t ji sbut the same radius. What t. u9rl vti-jl06k8stis g near its surface?   

  Two objects attract each other gravitationally with a force of tbltavev/tf5g)9e 3l48g /z m 2.5$ \times 10^{-10} \mathrm{~N}$ when they are 0.25 m apart.
Their total mass is 4.0 kg . Find their individual masses. m$_1$ =    m$_2$ =   

  Calculate the effective value of g , the accelex7 1krmhte og ngo 82:a4j5a (qdbs8z,ration of gravity, at (a) 3200 msa k ,4jg:h71mdbo5o8r2qz ne t8ax(g    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance frxq4f r;g8n*w t)efcb0) kjgz/ om the Earth’s center to a point outside the Earth where thegjtw brx;0 zf kq/)g*8c) 4nefgravitational acceleration due to the Earth is $\frac{1}{10}$ of its value atthe Earth’s surface?   $\times10^7$

  A certain neutron star has f-,fnwsr : jolba/pkrdn, 254iive times the mass of our Sun packed into a sphere about 15asn lk4:p/ 2f,b-odrn rw,ij0 km in radius. Estimate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once was-d i*al i.8aci an average star like our Sun but is now in the last stage of its evolution, is the size of our Moon but has the mass of our Sun. What is the sur- i*a .ii8dcalface gravity on this star?    $\times 10^7$

  You are explaining wsn06s evxi 9yr.x1m e(hy astronauts feel weightless while orbiting in the space shuttle. Your friends respond that they thought gravity was just a lot weaker up there. Convince them and your xv6 9x1er0(in msey.sself 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 u,jegp+w2e l oih( phq8w6)e-of a square of side 0.60 m. Calculate the magnitude and direction of t(g-owp 2e ph6hqliue8)wj+e, he 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 theqo/lifq0 t0r.2m2wm fsgme 3; same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in a limfm2m 0 le23oqw0q;irts .g/ fne (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 accel(.wfd x l7n0x8tocw3 teration of gravity at the surface of Mars is 0.38 of what it is on Earth, and that Mars’ radi t7x3(nw8xl.w 0ftcod us is 3400 km, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known val;;y k1+ pw mha85tnvisue for the period of the Earth and its distance from the Sun. [Note: Compare your answer to that obtaiky 5ipt 1+;awm vh8;nsned using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite moving in a stable circular orbit,bq0f73vkcai about the Earthkvbia,cq 7 3f0 at a height of 3600 km.    $\times 10^3$

  The space shuttle releases vg)ns iaz v /*fk;9x,qa satellite into a circular orbit 650 km above the Earth. How fast must the shuttle be moai xnsk )f/;vg*zvq9,ving (relative to Earth) when the release occurs?    $\times 10^3$

  At what rate must a cylindrical spaceship roat+ e)l 8 qvfiox4cl+1tate if occupants are to experience simulaeoq xlilav)++4cf8 1tted gravity of 0.60 g? 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 satellite to orbit the Ea.n 0o hk)-)0od yx(v,yjuhb yurth 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 tokjhv ().bx 0 nu0d,)oh y-oyyu 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 toppl0icl c+g; h0 of M;g0i+chc p0llt. Everest to be placed in a circular orbit around the Earth?   

  During an Apollo lunar landing mission,r *r(3(9exc3alw 5l4t wuifqz the command module continued to orbit the Moon at anx aw tf3iw4uz *3e9r(rcl(lq5 altitude of about 100 km. How long did it take to go around the Moon once?    s

  The rings of Saturn are composed of chunks of ice that orbit the planet. T35iw;c5j: jf w8klxoe he inner radol5;jw jw:c8e fi3x 5kius 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 once every 15.5 s (see Fig. 5–n33u a*8buj8r/jzt o k9). What is the ratio of a persobt/* uzja8nk3uj 8o3 rn’s apparent weight to her real weight (a) at the top, and (b) at the bottom?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut 4200 km froma ;z-wixb0s,dl eq waz9g-1mz-.f :an the center of the Earth's l-wz:aq f1za, ngmb.wdx-z 90;eisa -Moon in 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. T: q wf:d)v5yi dj:bkv,hey are vw qb 5f::kidjd)v,:yobserved to be separated by 360 million km and take 5.7 Earth years to orbit about a point midway between them. What is the mass of each?    $\times 10^{29}$

  What will a spring scale read for the weigh8x,as -2fiyuvm+m9 s ot of a 55-kg woman in an elevator that ox8myv +s9uaim2,-s fmoves
(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 suspc7+- uo, kfflwibsyd 8 cx2 nq))tucblp,080m ended from the ceiling of an elevator. The cord can withstand a tension of 22yn0l,lkdb 8oxtu7mp qb8w )c-+suc i)fc,0f2 0 N and breaks as the elevator accelerates. What was the elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orbits very near the surface of a planeg.prrb h o d8g9p;qq)/,eulr7t with period T , the denp7qg9 r8/rl.bpo,;erg )u hqd sity (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 period of the Moon (27.4 n2pj3 6w*ejz td) to determine the period of an artificial satellite orbiting very near the Earth’s su2jzn36 t we*pjrface.    s

  The asteroid Icarus, though onlmik;ud/j. x 5)qyfa) rlh;j2fy a few hundred meters across, orbits the Sun like the planets. Its period is 410 d. What is its mean distance from the q/y )m ;ahjk5x;.jflfu ir2 d)Sun?    $\times 10^{11}$

  Neptune is an average distance of+o kqa2)t 1ma4wg w;uy7bxw n8 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 years. It comes vd:b/u ynr mp:mdc59p)1xk he 6ery close to the surface of the Sun on its closest approach (Fig. 5–39). Estimate the grea r )p5ynd1/mmx :6:uckbphd9etest 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 ta jg:4gztdrp5)p h7gq/au9z iv/t8i-ckt: w*he center of the Galaxy $\left(M_{G} \approx 4 \times 10^{41} \mathrm{~kg}\right)$ at a distance of about 3 $\times 10^{4}$ light-years $\left(1 \mathrm{ly}=3 \times 10^{8} \mathrm{~m} / \mathrm{s} \times 3.16 \times 10^{7} \mathrm{~s} / \mathrm{y} \times 1 y\right)$ . What is the period of our orbital motion about the center of the Galaxy? $\approx$   $\times 10^8$

  Table 5–3 gives the mass, period, and mean di*;a 2n18 e1hpjdwkf wpstance for the four largest moons of Jupiter (those discovered by Galileo1; pk2h fep* a8wj1dnw 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 and distance of 2v s25u ;gcq; dcpmjsw78ab+pthe Moon. 2qd;j bs v7gs;u a85 cpwm+cp2   $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jupiter’s moons, usingd+; zi vc9dqt0vd2ecq:ba, g1 Kepler’s third law. Us2tq,a d: ; id bcvzcev0dqg91+e the distance of Io 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 betwe3j9 0).mwuj yyk +fmlten Mars and Jupiter consists of many fragments (which some space scientists think came from a planet9jj+)3 y mmy.u 0ftlkw 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 a2gv/ h bf)ro2ln artificial "planet" in the form of a band completely encircling a sun (Fig. 5-40). The inhabitants live on the inside surface (where it is always noon). Imagine that this sun is exactly like our own, that the distance to the band is the same as the Earth-Sun distance (to make)rog/ 22hfbvl 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 arc 28x ut+n 8dbk/ h:vsdtfrom 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 cv2:8 u ndh/xts8dbk+tan 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 Earw0 ahkc6vy2 /nth’s surface will the acceleration of gravity be half what it is on the6akh0 cvn/ wy2 surface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spin ih9ozfyvef9 sa (z661c n a mutual circle once every 2.5 s. If we assume their arms are each 0.80 efa 1c9yh6vzs6 9of( zm long and their individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates+tn5,hn*j6vy -h fipj once per day, the apparent acceleration of gravity at the equato+t6jfn ,5h p*vhy jni-r is slightly less than it 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 traveling dire::y xo4qyl0c/ pom/b q2 eu9eh0ta pu.ctly from the Earth to the Moon experience zero net force because the Earth and Moon px4ybu0:/o tepq cpl.m hq/02euo a:y9ull with equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, but whqtpb 3-32 9jwp 4zzwqqen you stand on a bathroom scale in an elevator, it says your mz 2b-pt3zq w93pq jq4wass is 82 kg. What is the acceleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station co5 tj mb.z.xwc,nsists of a circular tube that will rotate about its center (like a tubular bicycle tire) (Fig. 5–4w,5cjx. tb m.z2). 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 vertir6 r8g/.+fmo/jb4loi 1il *nd ol 5srccal 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 )mq,5 z5/aha mj2pgf zthe mass of a planet in terms of its radius r, the acceleratioq m/ hjz,a5fm a5g)zp2n due to gravity at its surface and the gravitational constant G.

A plumb bob (a mass m hanging on a string) is deflected from the verticascl.bx 6var7d; fgj4l 46i,hdl by an anh .v il4c g6dlxd 7jr;4,sf6bagle $\theta$ 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 spees;di: r**zk n1.srx:*vtrxx f d of If the coefficient of static friction is 0.30 (wet pavement), at what range of speeds can a car safzrri*:*n1x.r* s ;:s dkf vxtxely handle the curve?

  How long would a day be if the Earth webvb qah;;5zu/h4png+: gun t 0re rotating so fast that objects at the equator 5+hnatq 0; nvu/bp:g;uzhb 4g were apparently weightless?    $\times 10^3$s

  Two equal-mass stars mai 3va.bgd 3 g wj*sn4ysk/.gy o8xi2w(bntain a constant distance apart of 8.0 $\times 10^{10} \mathrm{~m} $ and rotate about a point midway between them at a rate of one revolution every 12.6 yr . (a) Why don't the two stars crash into one another due to the gravitational force between them?
(b) What must be the mass of each star?    $\times 10^{26}$

  A train traveling at a constant speed rounds a curve of radius 235 m. A zw7vjz8/49 lgszt(:0ee di cxlamp suspended from the ceiling swings ou:ie/x9 ved84zcl zgst w07(zjt 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 taws4r* dlg(-ux p8)aghe 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 experien4 gsg8(up- a*wd)xa rlce 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 k,*iul2 snoupy0:q k 0bm s76pdeduced the presence of an extremely massive core in the distant galaxs7qsy mu*bn ik0,puo6 k2p l:0y 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 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 and valley showmez+ 7cqo/4sz n in Fig. 5–45. Both the hi7z/m4sco +ezqll 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 of 24 sau bi119sf(xm9+eue 9n t(ea eatellites orbiting the Earth. Using “triangulation” and signal(aei1 xm9 9 9eunetf+1e ub(sas 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 (NEAR), after traveling 2.1 bkejc+sn*s fm,,kv n5sfq dn 5b 555ck3illion km, is meant to orbit the asteroid Eros at nqf, fsdk5k*cjb 3sn 55+5enmkcs,5va 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 b:8 j2 adlcr52v-nb-jd aeyl0the space 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 2alabjbj5d2l--:c vnrd e08y 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 qqt6sfl.ctaw 7i 25 4 n(whz3mits mean distance from 72qitsan .53(qhftcl6w4 zwm the Sun?   
(b) At its closest approach, the comet is about 1 A.U. from the Sun ( from Earth to the Sun). What is the farthest distance?   
(c) What is the ratio of the speed at the closest point to the speed at the farthest point? [Hint: Use Kepler’s second law and estimate areas by a triangle (as in Fig. 5–29, but smaller distance travelled; see also Hint for Problem 59).]   

  Estimate what the val r38,app)aaz 2vcfpdl)y 7z/ nue of G would need to be if you could actually "feel" yourself gravitationally attracted to someone near yoa a8r2dz/7np)lc) pa fzp,3vy u. Make reasonable assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the Milktm,ew/g vw.jo bpwhx+ y5+i;)y Way Galaxy (Fig. 5-46) at a distance of about 30,/o gbh yv,+)mt p .iw+jw;xew5000 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 ec-:-s- a4ijkyzqy (l.50 m on each side. Find the magnitude and direction of the gravitational force on a fifth 1.0-kg mass plac:l4qc-jayisz y(-ek -ed 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 Earthbsfq(2 p vx;d7 $ \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 sw)kyi f3te5y;/7k;y ties fmy)eep second hand on yourk yy) fikm5e/y 73;festiy;t ) wrist watch?    $\times 10^{-4}$

  While fishing, you gebr ks.i*a)9xunf39u ,d 4dyibt bored and start to swing a sinker weight around in a circle below you on a 0ix 3kunfi. )bus9 ,dd ay4br9*.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 in a new highway is dyu,+ jbb.9lu pesigned so that a car travel p+by.ul9j,ub ing 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 trainpav6 4i)rrl)oysg 9)2zf re*s , 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. Considr evis29apysg ))rl *ro f6)4zer 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.$\approx$    $\times 10^2$