Sometimes people say that water is removed f /aghd ,r/djxbimp b)f;v12:q) gff7vrom clothes in a spin dryer by centrifugal force)/ jd :7b phb2iv x gmqgrv/;f),fadf1 throwing the water outward. What is wrong with this statement?

Will the acceleration of a x8vpk4f;; wi3 i9o4tyr. uj btcar be the same when the car travels around a sharp curve at a constant 60 km/h as when it travels around a gentle curve at the same speed? Explain4tt ;u 4.i;xkj3pfwv8r oiy9b.

Suppose a car moves at ou) tvwgu(z8:constant speed along a hilly road. Where does the car exert the greatest and least forces on the road: (a) at the top of a u:8wozv gut ()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 meajz/y/l (hv r .gt2p-q(ne v3qrry-go-round. Which of these forcv-rqjyqg 3p(/.2z/l(n eav thes provides the centripetal acceleration of the child?

A bucket of water can be whirled in a vertical circle wi sqb(yk.8esvwxixf*pt;0 b )hjlj. svn+ /,(athout the water spilling out, even at the top of the circle when the bucket is upside down. E+)bwev0lx/tbhyq(8s.j.as* v p; nji,s (fkxxplain.

How many “accelerator innbaus n2er6g*29w 6s” do you have in your car? There are at least three controls in the car whichbwragnn 66n 92i* seu2 can be used to cause the car to accelerate. What are they? What accelerations do they produce?

A child on a sled comes .j21at r /guo 4jyt ba:5ox2reflying 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 between his chest and the sled decrease as he goes over the hill. Explain this decrease using Newton’r5xou1aa brt2 ./tg yj24o:ejs second law.

Why do bicycle riders lean inward when rounding a curve at huqhly 5:0hcy2igh speed?

Why do airplanes bank when they turn? How rq28(vssk jt /would you compute the banking angle giv /s(tqks82v jren its speed and radius of the turn?

A girl is whirling a ball on a string aro rux2)7/ 0l, pq h /qwhnxzzc)lx;he)tund her head in a horizontal plane. She wants to let go at precisely the right timq0ztelhx,rql2 hp /)xx/ 7wu;)nh)c ze so that the ball will hit a target on the other side of the yard. When should she let go of the string?

Does an apple exert a gravitational force on the Earth? If so, how la)adsbk 1g* f 9)blkw;mrge a force? Conside)l f 9kbdkgwm)s;*a1 br an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were doub g0lc*hn;lp 5:y0eb4bgwumn )le what it is, in what ways would the Moon’s orbit be diff0ehnyb g :uw n0mgpcl)4l b5;*erent?

Which pulls harder gravitationally, the Earth on the Moon, or the Moon on the E3 .ii2oywdo2t arth? Which accelerates moi 2woy .d3io2tre?

The Sun's gravitational pull on the Earqo25vel2 cdc0th is much larger than the Moon's. Yet the Moon's is mac2qc0 l5ve2odinly 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 moe+lnkft,mq n ,j4.wsi+ z43 atre at the equator or at the poles? What two effects are at work? Do they nzl af,q4mi4tsj .3t+w n+k,e oppose each other?

The gravitational force on the Moon due to the Earth is only a8hdstxi7, y:uvc7u 7rbout half the force on the Moon due to the Sun. Why isn’t the Moon pulled awayrx8th,:7iy d v7su7c u from the Earth?

Is the centripetal acceleration of Mars in its orb.ketw 9 ri(:ou-gfu f70*gth iit around the Sun larger or smaller than the centripetal acceleration wuitif hfgo7k:.9 (u-0teg*r of the Earth?

Would it require less e.2 nvfflgjff ,wn,6)k .3l mnspeed to launch a satellite (a) toward the east or (b) toward the west? Consj.len2.,3lkm )f w,vf6fgf nnider the Earth’s rotation direction.

When will your apparent weight be the greatest, as measured by a scale( 6qcfuxn.ok m)5vzk78;7n3in grkqd in a moving elevator: when the elevator (a) accelerates downward, (b) accelerates upward, (c) is in free fall, (d) moves upward at c kknnk7 diu; gxz m73rq)86fvc5o(qn.onstant 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 coqir( *k9wu*dlmr 83h auld be adversely affected by weightlessness. One way to simulate gravity is to shape the spaceship like a cylindrical shell that rotates, with the astronautsku wd8**ai3r mh lr9q( 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 weightlessness” beqvye/ gx0y5rcp(,2c uxtg:8e* rcu 4ptwee*: pggx, te p(rcue cy2r8qv54y/xuc0 n steps.

The Earth moves faster in its orbit around the Sun in January than in July. su (:ids 3)qrr hsgn6sr5 :1yuIs the Earth closer to the Sun in January, or in July? Explain. [Note: This is not much of a factor in psuy3s1 5) rqsiu nrsh(:6:d grroducing the seasons — the main factor is the tilt of the Earth’s axis relative to the plane of its orbit.]

The mass of Pluto was not known until it was discovered to hb:.rm4hcilxnp yh5-2p.y pa2v+x oq )ave a moon. Explain how this discovery enabled an e+onq2p . a-rvbx4h)hlpcm 2ypy5:ix. stimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth is much adcm044ve nyk+(uwakh. -lj1 larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the diffel uwc.a4kkd+m -v jah(y1e0n4 rence 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 d;*vo0 bg( s i3voq1 eiv1d:pi$\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 orbitgoy6tp h g 8ei9i.*z-x around the Sun, and the net force exerted on the Earth. What exerts this forc x*.zi89yie hg-pt g6oe 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 rotateso1b* e7bml d7)mr+aad-fbzz, uniformly in a horizontao) m1,bb l7zd*er+mzd - 7fabal circle (at arm’s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is km+/ lmwu+ nz-pmq;z; 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++;;-zp/zqmm wlum k n 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 acceleoe8mjpf 2+ uhj93*wgi,ivj- r ration of a speck of clay on the edge of a potter’s wheel turning at 45 rpm (revolutionjvir3wpejf8+ ug -ih ,mo2j*9s per minute) if the wheel’s diameter is 32 cm?   

  A ball on the end of a string is rwogc8 gm9r2 (sevolved at a uniform rate in a vertical circle of radius 72.0 cm , as shown in Fig. 5-33. If it rg8cwm2o g9s(s 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 9s,2j /quwlsk9 uw3dv with which a 1050-kg car can round a turn of radius 77 m on a flat road if the coefficient of static friction between tire swk uuqj9lw39,/2vsds and road is 0.80? Is this result independent of the mass of the car?   

  How large must the coefficient of static friction be between the tires and tuc8mvox u g82p xn;3s3he road if a car is to rocgos x x23vp m;83nuu8und a level curve of radius 85 m at a speed of 95km/h ?   

  A device for training astronauts and jet fighter pilots is designed t0x +5oisn vnfd( oct:d+bo;bl ,khk8 4o rotate a trainee in a horizontal circle of radius 12.0 m . If the force felt by the trainee on h+(ilvnk5d xbo4f+ c:td;s8ho, o0 nkb er back is 7.85 times her own weight, how fast is she rotating?    rev/s    m/s
Express your answer in both $ \mathrm{m} / \mathrm{s}$ and $\mathrm{rev} / \mathrm{s}$ .

  A coin is placed 11.0 cm from the axis ojcv*qeuy /m3.8 /cec lf a rotating turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntcqv3 elcuj/c y. 8*/emable 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 travelingf2u+uz gg.2n6dszuh 1 when upside down at the top ofg2.26uzg sufdz+ n1uh 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 (including the driver) crosses the roundedv8l6gkq 0c qlgc408em top of a qq8 lvk4cc60m gl8ge0hill (radius =95$ \mathrm{~m}$ ) at 22 $\mathrm{~m} / \mathrm{s}$ . Determine
(a) the normal force exerted by the road on the car,   
(b) the normal force exerted by the car on the 72-kg driver   
(c) the car speed at which the normal force on the driver equals zero.   

  How many revolutions per minute would a 15-m-diametebqw,a3eza*o a ro8x*-k7;o j ir Ferris wheel need to make for the passengers to feel “weightless” at the topm*j7 rxbwkoq8za;oa -o3,i a*eost point?    rpm

  A bucket of mass 2.00 kg is whii:m/i)mij ea 63;vl, :qlosh uhpg,1grled in a vertical circle of radius 1.10 m. At the lowest point of its motion the tension in the rohq),sl ge6l1/v m hmgp ; uii,3oa:ij:pe 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 centrifuld k5k; s;ous.l)rgtz0 y16fege rotate if a particle 9.00 cm from the axis oo)01g;5dktzlyr ;k u s6fel.sf rotation is to experience an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, peoplehg k t a.9y/ .hxldx/ohm/x5;q(n w3hl are rotated in a cylindrically walled "room." (See Fi tlyh/dhnq5h;h(.ok gx.xl3x/ m 9aw/g. 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 rotate* jx7su l j4+j.xh6appd in a circle on a frictionless air-hockey tabletop, and is held in this orbit by +. jpj64ah*s x7xupjla light cord connected to a dangling block (mass m ) through a central hole as shown in Fig. 5-
36. Show that the speed of the puck is given by

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

  Redo Example 5-3 , precisely this time, by not ignoring the weigra; oblpr9x8;fjm8ie wnd9.;ht of the ball which revolves on a string 0.600 m long. In particular, find 9x p 9m;8e rdli.;jrafnb8wo;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 p;7a i u rpv5hvf7ja8o6(tz4p uerfectly 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 masseug (01gjj ff(ts $ 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 b/i.5vl1 x owk0( ahtxdiving 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 the net /z8zevpr l59 l/7xxm vforce exerted on the car (by the groun vez8x7p5 /mx9v/zllr d) 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 Indianapoli.bs xkj5- wvm+)y2lyzs 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 through the turn, assuming constant tangential acceleration. If thy )mk. w-j+zv2yslxb 5e 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 horizontal9-upjdxcl-30 p: nujc circle of radius 2.90 m . At a particular inn0 -x:ppuc3j9 lcj-u dstant, 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 q13m0y* nfqs sp;knu. gravity on a spacecraft 12,800 km (2 Earth radii) above the Eartf nypqmss0 *; u1.kn3qh’s surface if its mass is 1350 kg.   

  At the surface of a certain planet, the gravitationalydp3l.24bq n m acceleration g has a magnitud4l ny2bpm q.3de 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 acceleration due tri5no )x0ip3k)s mc-eo gravity on the Moon. The Moon's radius is 1.74 )s-kniemc o p5r0xi )3$\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet hagb w kg, 8n, +2alg rpu-dcqle*.xkzzv/c54a;s a radius 1.5 times that of Earth, but has the same mass. What cnrkz ;ag 4.w,+-2ca z*5 gqk dvxulgp8/bel,is the acceleration due to gravity near its surface?   

  A hypothetical planet has a mass 1.66 times that of Earth, but the sam0*t a*vdu 25xxqplr(tv l0 li+e radius. What is g p* ui (lv5ql0rx2 d t0*+ltvaxnear its surface?   

  Two objects attract each other gravitationally with aipa rb0qq06;. b w *xiuo50juj force of 2.5$ \times 10^{-10} \mathrm{~N}$ when they are 0.25 m apart.
Their total mass is 4.0 kg . Find their individual masses. m$_1$ =    m$_2$ =   

  Calculate the effective value of g , the ap;+cmvn au83dd nf(z76cm2 a2nz 2amxcceleration of gravity, at (a) 3200 mauzcfmv+a;x 7p8m zm2dnnan2(c 3d 62    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside the Earth wher-ie k.(00v:ckrh anjn8g z/ dde thegravitational acceleration due to the Ea nra dh0. kn:kz /(8vijgdec0-rth 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 uwhxi suh 22braqg 4 lz 3pz9*wl)7/l2into a sphere about 10 km in radius. Estimate the surface gravity on this mo 9 /hi l3pra4q lshu7*2w uw)z22zlxgbnster.    $\times10^12$

  A typical white-dwarf star, which.xwe+m+d tid+ t9w qxq 5d7gr: once was an average star like our Sun but is now in the last stage of its evolution, is tt qdx: e gwi 7+d+md+.5rwx9qthe 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 astronrr,cy /4f(5nc jvf o,xauts feel weightless while orbiting in the space shuttle. Your friends r r(,frv 5/c 4yjfnxo,cespond 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 of a square of side 0.60 m. Calcuay, dvgagbs17 q /3f4plate the magnitude ab 1pd 3sva fq/7,gy4gand 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 years most of the planszpv9 pz210 5f0d+ gp6 vjmtgfets line up on the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in a line (Fig. 5-38). The masse69pjm+21 0tsp 0zgfv d5gvzpfs are $M_{\mathrm{V}}=0.815 M_{\mathrm{E}}, M_{\mathrm{J}}=318 M_{\mathrm{E}}, \quad M_{\mathrm{S}}=95.1 M_{\mathrm{E}}$ , and their mean distances from the Sun are 108,150,778 , and 1430 million km , respectively. What fraction of the Sun's force on the Earth is this? $F_{Earth-planets}$ =   $\times 10^{17}$ ${F_{Earth-plantes}}$ \ ${F_{Earth-sun}}$=    $\times 10^{-5}$

  Given that the acceleration of gravity at the surface of Mars kl xuy7yt2bjmhw5(d j 32,x1iis 0.38 of what it is on Earth, and that Mars’ radius is 3400 km, detd7bi32,umt kxxj 2w(15jyhylermine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value for the perioez1en;pf9jiv a7 fs+z++thhz 4*k0nmd of the Earth and its distance from the Sun. [Note: Compare your answer to that obtained using Kepler’s laws, Example zif*pnj+hez+; 7 0h 4sevtzf +a1kn9m 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite moving in a skz0y xv 7rm8p0b +x.wktable circular orbit about the Earth at a height of 3600 kp8 mx xkkbz.700y vwr+m.    $\times 10^3$

  The space shuttle releases a satellite into a circular orbit 650 km abov smxh8 7ut()koe the Earth. How fast must the shuttle be moving (relative to Earth) when themht(7s8 )x oku release occurs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occupqq79 2lq5vfazants are to experience simulated gravity of7qv qfl5a2z9 q 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 Earth in a circula.gfixw 2v knivih /8.8r “near-Earth” orbit. A “near-Eah .82.ifn/ivg x8vkiwrth” orbit is one at a height above the surface of the Earth which is very small compared to the radius of the Earth. Does your result depend on the mass of the satellite?    s ~    min

  At what horizontal velocity would a satellite have to be launched from the +p/vbq7fmueo *ci/)fmj l :z;top of Mt. Everest to be placed in/ucbivf*:mqlf o/+zem j)7; p a circular orbit around the Earth?   

  During an Apollo lunar landing missiry3ri26bd;tx m,sj a8qe6rt 2on, the command module continued to orbit the Moon at an altitude of about 100 kmrrt2e qyb,x8 sd a26;6irt3mj. How long did it take to go around the Moon once?    s

  The rings of Saturn are composed of chunks ofqurs 6q:kt.nps+ at* / ice that orbit the planet. The inner radius of the rings is ua k/rp6:n+*qt.s stq73,000 $\mathrm{~km}$ , 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 (sna2eh173kijlz1pxi-m: hz:cc 6k v kg-v(2 efee Fig. 5–9). What is the ratio of a person’s apparent weight to her hvikeg6jzp1 -:1c2 7e (2-fvk3 m cnkialhxz: 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 from the centerzwvpl ) uz:qpn:+k0/o of the Earth's Moon in +:ln vpp zwzkuo:/)0q 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. They arc/f4y okav4j+js 6n)p-lvjl. e observed to be separated by 360 million km and take 5.7 Earth years to orbit about a point midway between them. What is the mal.s6 c/ofj4-ny)p4v avj lk+jss of each?    $\times 10^{29}$

  What will a spring scale read for the weight of a 55-kg woman in an elevator t)znrvnbb:,1v hat 1brvn nv,b)z:moves
(a) upward with constant speed of 6.0m./s   
(b) downward with constant speed of 6.0m./s   
(c) upward with acceleration of 0.33 g,   
(d) downward with acceleration 0.33 g   
(e) in free fall?   

  A 17.0-kg monkey hangs from a cord suspen (ljj8u mo+*uk,maf8pro +.kgded from the ceiling of an elevator. The cord can withstand a tension of 220 N and breakjmf8(ug,mj k*.+ukro8p l + oas as the elevator accelerates. What was the elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orbits very near the surface of a planet with cp2;e ar+/ zqnuxv 7t-period T , the density (ma -tcuzex;2/vnq7rp+a ss/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 theapoen 2l f:/w+se*9tt Moon (27.4 d) to determine the period of an artificial p:w/ale29*one ttfs +satellite orbiting very near the Earth’s surface.    s

  The asteroid Icarus, though3q tbc 33,ytsk only a few hundred meters across, orbits the Sun like the planets. It3t3qbck 3 ,syts period is 410 d. What is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average distanceud6 ,nxg)i cs1 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 ever1fxp.sc zl+6gn0 mp h9y 76 years. It comes very close to the surface of the Sun on its closest approach (Fig. 5–39). Estimate the greatest distance of the com+f9mxlpz hs1.gp6 c n0et 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 t)3hsp6ug n v8khe 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 distance for the four largest m,wvj34k x)+l dnisa4j(x6jbjoons of Jupiter (those discovered )bsnljjjx4i3+w (dkja4xv 6 ,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 and distance ou)ip0t b ,lmi;mllu +3s0-rqtf the Moon. rl)mi-s ib qu;3+t0,ltlp 0um   $\times 10^{24}$

  Determine the mean distancew,xzy v0-f 2tx 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 in the Table.xx,y -v20ftw z
$r_{\text {Farip }}^{r}$ =    $\times 10^3$
$r_{\text {Guarymale }}$ =    $\times 10^3$
$r_{\text {Culimio }}$ =    $\times 10^3$

  The asteroid belt between Mars and Jp ,qi9 k:k( 2h*slrzdhupiter consists of many fragments (which some space scientists think came from a planet that once orbited the r:9(lid2z kk shq, ph*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 artificial "planet" k rmx5a0o)8qvu;0uylin 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 o50q8) rmuxl u;ky0oavwn, 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 sgg(ro6a7 84vn hf3hez4 *flyby swinging 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 is4z*7rh sng6g yfae4h( f8vl3o the maximum speed he can tolerate at the lowest point of his swing? His mass is 80 kg, and the vine is 5.5 m long.   

  How far above the Earth’s surface will the acceleration of gravity b-q,gb/ l.z+o sv svz)k+ mlbg)e half what it is on tz-gv)l. /gosb+ +lmbqzv,) sk he surface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spin in a mutual circlerqn:guu7b)r fr( v e81 once every 2.5 s. If we assume their arms are each 0.80 m long and their inr (r1 r8ug 7:)bnufveqdividual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates once per day, the apparent a9gy adx-40whv cceleration of gravity at the equator is slightly less than it would be if the Earth didn’t rotate. Ey4w- 9dhav0xgstimate the magnitude of this effect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from tb6wxo.)g olc,y8g 8kvt2 q5i)lu o,cxhe Earth will a spacecraft traveling directly from the Earth to the Moon experience zero net force because the Earth and Moon pull with equal and oppobo.8) lw k6 x)gxlcvg,u5cyt oio, 28qsite forces?    $\times 10^8$

  You know your mass is 65 kg, but when you stand on a u wa(c3l9u15rj uj4uvipr/b(:xu4)hi3 b lu hbathroom scale in an elevator, it says your mass is 82 kg. What is the acce/ 9u3c3l u)uu ih:a4 hil14 (xr5 pujbjwvurb(leration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consists of a circular 8f0 q3uw7 +7arb gfkkhtube that will rotate about its center (like a tubular bicyclegk7bkua qff3hw0r8 7 + tire) (Fig. 5–42). The circle formed by the tube has a diameter of about 1.1 km. What must be the rotation speed (revolutions per day) if an effect equal to gravity at the surface of the Earth (1.0 g) is to be felt?$\approx$    $\times 10^3$

  A jet pilot takes his aircraft vsuqn)okn46;1at (j6vkfr f1+ irz 8rin a vertical loop (Fig. 5–43).
(a) If the jet is moving at a speed of at the lowest point of the loop, determine the minimum radius of the circle so that the centripetal acceleration at the lowest point does not exceed 6.0 g’s.    $\times 10^3$
(b) Calculate the 78-kg pilot’s effective weight (the force with which the seat pushes up on him) at the bottom of the circle    $\times 10^3$
(c) at the top of the circle (assume the same speed).
   $\times 10^3$

Derive a formula for the mass of a planeikmf*j c + tj0f82d-lpt in terms of its radius r, the acceleration due to gravity at its surface and the gravitatmp fit*j jf2c8 lk0+-dional constant G.

A plumb bob (a mass m hanging on a string) is .l/9z03qiearvdv33xnm9g k t deflected from the vertical by an angle la03knm9dxet9rv / 3 .zq3gvi$\theta$ 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 specf,pe t bu.e(fe-:1x nd+- otsed of If the coefficient of static friction is 0.30 (wet pavement), at what rps+ed(etfu.-1- f,b tne : oxcange of speeds can a car safely handle the curve?

  How long would a day be if the Earth were rotating so fast that objects at t;ix y4) tpmel,) 0+vr ,fl5o8bakqrnahe eqte yn+)ap ,abmr5lv);fi8 xoqr ,k40l uator were apparently weightless?    $\times 10^3$s

  Two equal-mass stars maintain 46lpy/w5,f6rod yhdncd 8e,s 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 sp0g8)bg+s xn afh5ta 4peed rounds a curve of radius 235 m. A lamp suspended from the ceiling swings out to an )8gg+s04 axatfpbn5hangle 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 been claimed l:+5f u0r pqocnv:;4bn+o hii 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 goci+ovr q4;5l: funn:h+0pbi 's. Calculate the number of g 's a person would experience 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 deducryd ho/k 4/3b3 )n.y kmhha/2esdu0jged the presence of an extremely massive core in the distant galaxy M87, so dense thau3j3n)2 /0dyh /rhsdgkh ob/e4kyam.t 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 traverserc3 2; cptvmt;s the hill and valley shown in Fig. 5–45. Both the hill and valley have a radius of curvature R. (a) How do the normal forces acting on the car at A, B, and C compare? (Which is largest? Smallest?) Explain. (b) Where would the driver feel heaviest? Lightest? Explain. (c) How fast can the car go without losing contact wit tv rc;p;c2mt3h the road at A?

  The Navstar Global Positioning System (GPS) utilizes a group of 24 sa q rjzfm (ho3yc pf);;q);o6qkm9/hul tellites 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, allowim;) /3j l;ozk f;rh6qyqhcpum9)oq(fng 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 Rendezhunpt( /9:s(tvn;idwokp 9:;qjvt8 a vous (NEAR), after traveling 2.1 billion km, is meant to orbit the asteroid Eros at 8tnqk/i pv;;t (s(tap: 9: njwuvdoh 9a height of about 15 km . Eros is roughly 40 $\mathrm{~km} \times 6 \mathrm{~km} \times 6 \mathrm{~km}$ . Assume Eros has a density (mass/volume) of about 2.3 $\times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ . (a) What will be the period of NEAR as it orbits Eros? $\approx$    $\times 10^4$sec
(b) If Eros were a sphere with the same mass and density, what would its radius be? $\approx$    $\times 10^3$
(c) What would g be at the surface of this spherical Eros?$\approx$    $\times 10^{-3}$

  You are an astronaut in the space shuttlrki8+nh5 mkij04 y zc3e 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), bk 4c0m+ j8ik rhin3yz5ut 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 30/5wus4v y8+6njl) lz+ec iss o00 years. (a) What is its mean distance from the s 8s4s lze+ivywnu/ljo6)c5+Sun?   
(b) At its closest approach, the comet is about 1 A.U. from the Sun ( from Earth to the Sun). What is the farthest distance?   
(c) What is the ratio of the speed at the closest point to the speed at the farthest point? [Hint: Use Kepler’s second law and estimate areas by a triangle (as in Fig. 5–29, but smaller distance travelled; see also Hint for Problem 59).]   

  Estimate what the value of G would need to be if you g tg))utr w3p7vz;l9scould actually "feel" yourself grav7lr)t9s) zgtpu3w ; gvitationally attracted to someone near you. Make reasonable assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the Milky Way Glyi; syx4m *g 7 276o.l7j sqkygkl,cyalaxy (Fig. 5-46) at a distance of about 30qy*gjk 2kg7ll7;y,4 y7liyo.6mcs sx,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 o ;jw,lgpfhg- yt g-:e3n each side. Find the magnitude and direction of the gravitational foryggl-;tj 3-he wg:p f,ce 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 5507zk3i0bn/asuzass940c+ s ql0 kg 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 experi9t;2utj.exaf ib/s 4a enced by the tip of the 1.5-cm-long sweep second hand on your wrsa94bf.eixj t/;a 2utist watch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sifo;5- )hho bwxnker weight around in a circle below you on a 0.2oxbho) f-; hw55-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 rahtr6nv)rt039 s- q iuvdius R in a new highway is designed so that a car traveling at spvq utn6 i srt0)rv-3h9eed $ v_{0}$ can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from $v_{\min }$ to $v_{\max }$ . Derive formulas for $ v_{\min }$ and $ v_{\max } $ as functions of $\mu_{s}, v_{0}$ , and R .

  Amtrak’s high speed train, the Acela, utilizepwdyw-,y m q 4+9/ocans 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 nd/4mpww +q-o cya ,y9friction force against the seat, thus feeling more comfortable. Consider an Acela train that rounds a curve with a radius of 620 m at a speed of (approximately ). (a) Calculate the friction force needed on a train passenger of mass 75 kg if the track is not banked and the train does not tilt. $\approx$    $\times 10^2$
(b) Calculate the friction force on the passenger if the train tilts to its maximum tilt of 8.0º toward the center of the curve.$\approx$    $\times 10^2$