Sometimes people say that water is removed from c5*ke:vq1,c) gu so dydlothes in a spin dryer by centrif) kv:*degd51u cqos,yugal force throwing the water outward. What is wrong with this statement?

Will the acceleration of a car be the same when the wzy,tpn8o vwo 2o/8( 7wjrno9car travels around a sharp curve at a constant 60 km/h as when it travels around a gentle curve at the same8jp zo,onvryww tw2 oo9n87(/ speed? Explain.

Suppose a car moves at cz(z4nyt j ns/1wfy s/ r6 p(pu:ql-1wconstant speed along a hilly road. Where does the car exert the greatest and least forces on the road: (a) atwp yn:l(s s1/ u-jzzfyn1tp c(/wq6 4r the top of a hill, (b) at a dip between two hills, (c) on a level stretch near the bottom of a hill?

Describe all the forces acting on a child riding a horse on a merry-go-6rnx3rf42aa qround. Which of these forces provides the cen qn6rax3f42ra tripetal acceleration of the child?

A bucket of water can be whirled in a vertical circle without the water qg :vpd-h6p,o spilling out, even at the top of the circle when the bucket is upside down. - g:p,hdpv q6oExplain.

How many “accelerators” do yowp,pr 6fwrj..kh rs2;u have in your car? There are at least three controls in the car which can be used to cause the car to accelerate. What are they? What accelerations do they pro ph;fs,r6.rpwr. wjk2duce?

A child on a sled comes flying over the crest ol6 d1vvtw+t7k f a small hill, as shown in Fig. 5–31. His sled does not leave the 6l7vwv1tk dt+ ground (he does not achieve “air”), but he feels the normal force between his chest and the sled decrease as he goes over the hill. Explain this decrease using Newton’s second law.

Why do bicycle riders lean inward when rounding a curve at high speed1f 6: wmlw3mi-y;e nad?

Why do airplanes bank when;4yls: m.gmgk they turn? How would you compute the banking angle given its speed and radik;.y gg: l4smmus of the turn?

A girl is whirling a ball on a string aroun*wwx*t8 f1hr,:bypgj a5h 0eqd 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 the st0qa brj5,yh** tw xgp f18:ehwring?

Does an apple exert a gravitational force on;by;61if i k wz9qfli6 the Earth? If so, how large a force? Consider a ;6b ikf 6l;iz1ifwy9qn apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, in what ways wky9fg u6. nhf7ould the Moon’s orbit be differen7fg .6hfun 9kyt?

Which pulls harder gravitationally, the Earth on the Mo v75og6wd pou)ba26gn on, or the Moon on the Earth? Whichp5g 6u2d 7gb)oaw 6nov accelerates more?

The Sun's gravitational pull on the Earth is much larger than the Moon's. Ye ,vs*j3on8 icit the Moon's is mainly responsible for the tides. Exjsin 8v*3oic ,plain. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]

Will an object weigh 9 r cpik+yym0wyd/(x4q9b kw8more at the equator or at the poles? What two effects are at work? Do thb49r(iqkc 0yywy /89dxpw+mk ey oppose each other?

The gravitational force on td3ar86gfz*mx p 0lo4hhe Moon due to the Earth is only about half the force on the Moon due to 03 86rghxplmo * fzad4the Sun. Why isn’t the Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in its orbit around the Sun larger orksw4e.el, 9,lv im)po smaller than the centripetal acceleratioe4, vliek,w s9mo). pln of the Earth?

Would it require less s6ta.oq uh15v -*ejwpz peed to launch a satellite (a) toward the east or (b) toward the west? Consider the Earth’s rota*vjoq et w5-z1 u6h.pation direction.

When will your apparent weight be thec*x3w -,hvx sv greatest, as measured by a scale in a moving elevator: when the elevator (a) accelerates down* 3s,chxv-wxvward, (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 affectedmj,)eu, xcn u- by weightlessness. One way to simu-u,cmej,ux)n late gravity is to shape the spaceship like a cylindrical shell that rotates, with the astronauts walking on the inside surface (Fig. 5–32). Explain how this simulates gravity. Consider (a) how objects fall, (b) the force we feel on our feet, and (c) any other aspects of gravity you can think of.

Explain how a runner experiences “mw8ehml-bn r64jci enbi4 h( 14z7xw-free fall” or “apparent weightlessness” between step jb eew 6i-4whl4cmhbnnmx81z 4i-7r(s.

The Earth moves faster(.5g c xd)rbap in its orbit around the Sun in January 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 producing the seasons — the main factor is the tilt of the Earth’s axis relativegr( b) .acd5xp to the plane of its orbit.]

The mass of Pluto was not known until it was discovered to have a mobl 4jkxp0wn l6+7 6aupon. Explain how this discovery enabled an estimate oflpnaw+x b7u 60plk4j6 Pluto’s mass.

  The Sun's gravitational pull on the Earth is much largerz 7fgdthjxd,umo:6wq f6/6e3 than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from ojf6we/d 6fd,u73 h6t qzm:gxo ne 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 19g 0*,oc,ozg mq xtn39d80 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 orbitv41d0d2t n bg) cpdtb4 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 radius 1.td g c)1d4bpv0d4bt2n50$ \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 vw4p+s9 18pwe )ferl/hzb yo8 a 2.0-kg discus as it rotates uniformly in a h ew8wzpl9sb)4+/h e 8fpr1yov orizontal circle (at arm’s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is in orb( goh r6-f:vjoit 400 km from the Earth's surface, and circles the Earth o6 jfo: g(-vrhabout 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 Earth's surface. $\approx$    g

  What is the magnitude of tht3h7l )+w-b uorr. 6k ccjj(5vp8ykine acceleration of a speck of clay on the edge of a potter’s wheel lkj.i7yuc3- v) 6(pnt 5b8kcrwjo+r h turning at 45 rpm (revolutions per minute) if the wheel’s diameter is 32 cm?   

  A ball on the end of a string is revolved at a uniform rate in a vertical cipinlp )5j sc1d.7:oldrcle of radius 72.0 cm , as show:l id)c1lp7p sj. do5nn in Fig. 5-33. If its speed is 4.00 $\mathrm{~m} / \mathrm{s}$ and its mass is 0.300 kg , calculate the tension in the string when the ball is (a) at the top of its path   
(b) at the bottom of its path.   

  A 0.45$-\mathrm{kg}$ ball, attached to the end of a horizontal cord, is rotated in a circle of radius 1.3 m on a frictionless horizontal surface. If the cord will break when the tension in it exceeds 75 N , what is the maximum speed the ball can have?   

  What is the maximum speed with which as3sq; bpe9 -fifye, 0,2 pwffk 1050-kg car can round a turn of radius 77 m on eb 0 s wkes;ffp-p9ff,, yi2q3a flat road if the coefficient of static friction between tires and road is 0.80? Is this result independent of the mass of the car?   

  How large must the coefficient of static friction be )8e2r) hmyduo between the tires and the road if a car is to round a level curve of radius 85h8)u)e2r odmy m at a speed of 95km/h ?   

  A device for training as-baue/z 3j dq14ivgr2v* zi o,tronauts and jet fighter pilots is designed to rotate a trainee in a horizontal circle of radius 12.0 m . If tizbvv * ja21,rz q3-g4/oiud ehe 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 cm from the axis of a rotating turntable of s7f0 thud5f s-.ve -qcvariable 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 ctfss5u.q-7cv0d fhe- oin slides off. What is the coefficient of static friction between the coin and the turntable?   

  At what minimum speed cm2ckj*pu9n0 z, o zt5must a roller coaster be traveling when upside down at the tm 92n kucj *z0tp,zoc5op 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 (including the driver)5c6zz,j)j euf crosses the rounded top of a hill (radijj,6z5u cz f)eus =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 9jo+ovcbvdzv )8 n-fmh5( x1xFerris wheel need to make for the passengers to feel “weightless” at the topmost poi 8ozm1xd5fbv)v nx-v+j c 9(hont?    rpm

  A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1.10 m. A0 s: eyt 3qil5xv -xb s.,b(8v:yxkggft the lowest py kgbb :lxs5vve0x.y-q i8:t,s3f(gx oint of its motion the tension in the rope supporting the bucket is 25.0 N.
(a) Find the speed of the bucket. $\approx$   
(b) How fast must the bucket move at the top of the circle so that the rope does not go slack?   

  How fast (in rpm) must a centrifuge rotate ifcj ;2 8uj t(uf0g2yldp a particle 9.00 cm from the axis of rotation i2; ju8pdfc2(tu jyl g0s to experience an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotatedyxbe+ce2ui2kx8rwdjl6 2 0t 0 in a cylindrically walled "room." (See Fig.w2y8d6jr02k +c ut2x leixeb0 5-35.)
The room radius is 4.6 m , and the rotation frequency is 0.50 revolutions per second when the floor drops out. What is the minimum coefficient of static friction so that the people will not slip down? People on this ride say they were "pressed against the wall." Is there really an outward force pressing them against the wall? If so, what is its source? If not, what is the proper description of their situation (besides "scary")? [Hint: First draw the free-body diagram for a person.]   

A flat puck (mass M ) is rotated in a circle on a frictionless air-hockey tlxg1ec xc3m:p9j) k,babletop, and is held in this orbit by a light cord connected to a dangling block (mass m ) through a central hole as show1)b kcm9xxc 3jp:l, egn 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 ig8 1kpee wlcm nd(u,-7lgkht.4noring the weight of the ball which revolves on a string 0.600 m long. Inp (,.m4gncl7t- k1ew8hkdule particular, find the magnitude of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ , and the angle it makes with the horizontal. [Hint: Set the horizontal component of $\overrightarrow{\mathrm{F}}_{\mathrm{T}}$ equal to m $a_{\mathrm{R}}$ ; also, since there is no vertical motion, what can you say about the vertical component of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ ?] $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ =    the angle =   

  If a curve with a radius of 88 m is perfectly banked for a car traveling 7wt9 vicx/4e 5y5 $\mathrm{~km} / \mathrm{h}$ , 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 massesl 4qga:kl dw8:y6)pf t $ 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 b.qsp*qpya y 6.y 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 the net force exerted on dsp;fewi.i(: p hz2z7 the car (by the ground) in i hs(.ipfz:e7;d pz2wExample 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 acceleratess/c ka5t:w f(g 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 the curve were flat, what would the coefficient of static friction h5t cf/:kg sa(wave 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 . At a particul8y xf;)4e3up km6y ksdar instanu3mpk)fy4 ksy;d 8ex6t, 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 Ear(zq/x:u6u ) mlqb.xx wth’s gravity on a spacecraft 12,800 km (2 Earth radii) above the E.zu(: /wqx)lmb6 qxuxarth’s surface if its mass is 1350 kg.   

  At the surface of a certain pq*ylz:9v tg*4e14lsv bi q* ualanet, the gravitational acceleration g has a magnitude of 12 m/v e*q*gtva*1 y94zql s4ibl:us$^2$ 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's ralu;e-q+ho*spd zl);i dius is 1.74 e hozpl+;- *)uisqd;l$\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 mes zvxjswo )b e+cb;,1-:4xn-svkgiyhw . /+that of Earth, but has the same mass. What is the acceleration due to gravity near its surface?y,o+h evnx4 -v/ 1 w:gwbesc.z+ki )-;xmssbj   

  A hypothetical planet has a mass 1.66 times that of Earth, but the same ra kyx.nxp(,2j vdius. What is g near itsxp2ky(njxv,. surface?   

  Two objects attract each other gravitationally with aylt )c2 hd8t)a force of 2.5$ \times 10^{-10} \mathrm{~N}$ when they are 0.25 m apart.
Their total mass is 4.0 kg . Find their individual masses. m$_1$ =    m$_2$ =   

  Calculate the effective value of g , the acceleration of gravit6x :bj6hl vc v)m-guv:y, at (a) 3200 m xv jb cvug6vl:):6mh -   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside pbka6n 1exag/i (27-vl ;iieq the Earth where thegravie6a;17nq vig2e-kbli/xap ( itational 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 4d11k 6penmjd kv mb-,krys24sphere ab2 ,s61-ne1jrvkmb4 kydd4k m pout 10 km in radius. Estimate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once was an average star 0y eb r(+y1/gn/u,onchsnt 9tlike our Sun but is now in the last stage of its evolution, is the size of our Moon b n01b tos/(/ ,ynehtcu+y9nrgut has the mass of our Sun. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronauts fyo;; b2,0on70cztrcv x uqs02 zb.bkaeel weightless while orbiting in the space shuttle. Your friends respond 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 tc72roxbcb0 zk.qs; ua, z02ob;v0ny Earth’s surface in terms of g.   

  Four 9.5-kg spheres are located atf9i5fw5ieh7h h( g;j3 wt8s sk the corners of a square of side 0.60 m. Calculate thekigh8(9w7 sf5eh ;f5h 3swtji 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 years most of the planets line upqpkra i*4 ((xg 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 4(arx(g*pqki 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 accele/h 2d)xheijf *-zy iclb0k)j 3ration of gravity at the surface of Mars is 0.38 of what it is on Earth, and that Mars’ radius is 3400 km,i ce)h ji)b k* dhf3x2-z/0yjl determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun ub4 / xr) 4etela-5rtk:s+lykgsing the known value for the period of the Earth and its distance from the Su/ -er:r yab+t)xl45 l kt4ksegn. [Note: Compare your answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite moving w6l swfu7os *+in a stable circular orbit about the Earth at a heighwfo7ws*s + u6lt of 3600 km.    $\times 10^3$

  The space shuttle releases a satellite into a circular orbit 650 km above thtthx /2-5lsiqe Earth. How fast must the shuttle be moving (relatili h/qt 5t-2sxve to Earth) when the release occurs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occupantrewyxr)y :fo43(ekkv7b b+ucz w 0o)2 s are to experience simulated gravity of 0.60 g? Assume the spaceship’s diameter is 32 y+rb eou x(2vceb 7wofz y)r4:k3 )0kwm, 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 circulrx5oefp5c o9;n /w yw z715ccgar “near-Earth” orbit. A “near-Earth” orbit is one at a height above the surface of the Earth which is very small compared to the radius of the Earth. Does your result depend on the mass of the satellite? og7wo;5zwncc 1ercxfy5/59p    s ~    min

  At what horizontal velocity would a satellioy;hpj j-(i.g te have to be launched from the top of Mt. Everest to be placed in a circular orbit around the Ea.iy( ;jo- hjgprth?   

  During an Apollo lunar landing mission, the command module conts;l.ct gn/j eb;a-- byinued to orbit thajel/-bb s-. gyn;t;c e Moon at an altitude of about 100 km. How long did it take to go around the Moon once?    s

  The rings of Saturn are composed of chunk lhi6;-u,for pqji(om) -4rhes of ice that orbit the planet. The inner radius of the rings ismo,)q- re4f pjo;-6hl u(hiir 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 /3f79m:t 4+oktcy3eaiv)zsrvh i2s ym in diameter rotates once every 15.5 s (see Fig. 5–9). What is the ratio of a person’s apparent weight to her real weight (a) at the top, and (b) at the 4if 3tv:)2ih 3kyacs+ ev9tzrs o m/7ybottom?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut 4200 km from the center of the r4egs+w2w rl-Earth's Moon in a space vehicle (a) moving at constant velocityw +lg42er wr-s,     ----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. Thexy5h+p+xdza (y are observed to be separated by 360 midxxh (p 5az+y+llion 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 wei.0 9+wzqth y.ltrfjr*ght of a 55-kg woman in an elevator that moves+rfj .9hqw*zrtt.0 ly
(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 u3 ebs+m9a7iu a cord suspended from the ceiling of an elevator. The cord can withstand a tension of 220 N and breaks as the elevator acc7+3u9m subeaielerates. What was the elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orbits very near the surface ote-hk ds(2y-of a planet with period T , the density (mass/volume) of the s2(-yo h ed-tkplanet 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 twbh6+1 gja)e4m frzj (he period of the Moon (27.4 d) to determine the period of an af14ej+(zbmjgr 6a) w hrtificial satellite orbiting very near the Earth’s surface.    s

  The asteroid Icarus, though onlyn2q(8nj k6d k ,/ynvap a few hundred meters across, orbits the Sun like the plane6y(pd qnv/2, akjnn8k ts. Its period is 410 d. What is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average v: zoub(w(dw j*w f3;rc/bq 3ndistance 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 evd +2.ie; yy (*vtrm;wvj(m zfxery 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 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 am(; rd ij. ty;*ef(y2w xvvz+mnd farthest distance from the Sun.]    $\times 10^6$

  Our Sun rotates about the centc dgb*q ly9i5.er 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, pe df+*4m4vsxmtwa 3o6v 8pey 4aa( 7lvtriod, and mean distance for the four largest moons of Jupiter (those discovered by Galileo in 1pmv6a+8w3v(t4 t*7ax y 4l4mfeadvos 609).
(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 peri1 (f6do tsgc0k *2ezmvod and distance of the Moon. 1dmevs2 c* 0fg6 t(zko   $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jupiter’s moons, ud; ddb u y76v md,p(61rjgpr/tsing Kepler’s third law. Ubp / udr71 ddtgv(6,jyd 6rp;mse 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 between Mars and Jupiter consists;/ aa4gy/xxttu.jy f+ of many fragments (which some space sci+4fjtu ga; x/ /yaty.xentists think came from a planet that once orbited the Sun but was destroyed).
(a) If the center of mass of the asteroid belt (where the planet would have been) is about three times farther from the Sun than the Earth is, how long would it have taken this hypothetical planet to orbit the Sun?$\approx$    y
(b) Can we use these data to deduce the mass of this planet?

  A science-fiction tale describes an artificial "planet" in the formqr)ban44b,( v g .bvmm of a band completely encircling a sun (Fig. 5-40). The inhabitants live on the inside surface (where it is always noon). Imagine that this sun is exactly like our own, that the distance to the band is the same as the Earth-Sun distance (to make the climate r4)vvmab. g,(nqbm 4b 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 swfb)3 zr. pjpk5zh(m a7inging in an arc from a hanging vine (Fig. 5–41). If his arms are capable of exerting a force of 1400 pbr(.3z)75 m kjfahpzN on the vine, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 80 kg, and the vine is 5.5 m long.   

  How far above the Earth’s surface wilbq +cgf-l u7 hhq,5di9l the acceleration of gravity be half what it is on the-bqq7hcgfih9 udl +,5 surface?    $\times 10^6$

  On an ice rink, two skatera*for nk6, ,mc1rkiv9 s of equal mass grab hands and spin in a mutual circle once every 2.5 s. If we assume thec i9n,vk rmr61 oaf,*kir arms are each 0.80 m long and their individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates once pal dvd8 p. kapm69-5pher day, the apparent acceleration of gravity at the equator is slightly less than it would be if the Earth didp v 9kd6h.8d5laap-pm n’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 directly y4uhu6k-n0z8s5yoc p from the Earth to the Moon experience zero net force because the Earth and Moon pull wno5y-yscu h06zk8pu4ith equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, but wa x +s4.xvrf3khen you stand on a bathroom scale in an elevator, it says your mass is 82 kg. Whatxafk4v3.+xr s is the acceleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consists o tnw kmy.g.:gf.ruj*5h z9o0zf a 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 effect equal to gravity at the surface of the E95y.t .:grjznwuf* k. gm 0hzoarth (1.0 g) is to be felt?$\approx$    $\times 10^3$

  A jet pilot takes hihn ;q blme7n,n*n p1a9. h0kjws aircraft in 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 tqi3u y1a4sgluoc zl+q,6;aa ( he mass of a planet in terms of its radius r, the acceleration due to gravity at its surface and the gravitation g6c ;lqi1z4+q aus(a,y 3uaolal constant G.

A plumb bob (a mass m hanging on a string) is deflected frzpff ni31 w6wf()fk*xom the vertical by an*w1 npff63 k(if)zw fx angle $\theta$ 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 If the caill. hs 56du4x k5i)coefficient of static f lxikls4i5)d 5a.c6uhriction is 0.30 (wet pavement), at what range of speeds can a car safely handle the curve?

  How long would a day be if the Earth were rotating so fast/ gbo8jq.w u+ cys*i3( pjfl9s that objects at the equato*oj .y(c+fqsj pwls39b g/ iu8r were apparently weightless?    $\times 10^3$s

  Two equal-mass stars maintain a constant distance aparns5p/ iej).f0k w jit7t 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 cons;7 b3ebqcfp;v um pw;hd.qt vwo76u+ 3tant speed rounds a curve of radius 235 m. A lamp suspended from the ceiling swin7;uo.mcb 3bpv;e;63 f+d7pqhut wwq vgs 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 m -ys9y8b5 xnh8,taxbEarth. Thus, it has been claimed that a person would be crushed by the force of gravity on a planet the size of,hby-yan5mt x8b 98sx Jupiter since people can't survive more than a few g 's. Calculate the number of g 's a person would experience at the equator of such a planet. Use 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 Spacnz lch/2a1o;q e Telescope deduced the presence of an extremely massive core in the distant galaxy M87, so dense that it could be a black hole (from which no light escapes). They did this by measuring the speed of gas cloal z;n2oc1 h/quds 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 sp9v pvw+zvmd.c.ce) s 8yf3+dm eed as it traverses the hill and valley shown in Fig. 5–45. Both the hill and valley have a dcmsv8)9 p wc. f3z.+yevm+vdradius 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 sa w)ukzmg(vc6;p 99dpitellites 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 eachupcw9(;kdg6 ivz 9 p)m 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 billion km, ism 9,k ;w*qo+ru dpx)ik meant to orbit the asteroid Eros at a height of about 15 km . Erosk , md9;w) ip+koqxr*u 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 dw0ak9r- -e:d 16yhepdoy 7buneed of repair. You are in a circular orbit of the same radius as the satellite (400 km above thodehb0kdedr 61 pyy7aw :9-u-e 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 ufc4qmh2r9mqzy,l;brbcue1 9 /f/ *fperiod of 3000 years. (a) What is its mean distance from the Sun9,//q; uqm2lrc*fzby1frm 4 heuc bf9?   
(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 coul9xdans 44cx 8zm42c wali:fw.d actually "feel" yourself gravitationally attracted to .:n9 fwa4aw 4ls2z 8xdcimxc 4someone near you. Make reasonable assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the Milky Way s irja90 7qj5tGalaxy (Fig. 5-46) at a distance of about 30,000 light-years j5q0tiasr 9j7 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 side.vjhlb(sf60/b,k(ans j. 4 v4z5as hgq Find the magnitude and direction of the gravitational force on a fifth 1.0-kg6v n0jbjhsk5hz, 4sqbls.v 4( fgaa(/ mass placed at the midpoint of the bottom side of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 kg orbits 04qk.o6m f9,w8 )cybdf ikbgtthe 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 sweep s* zmthjqbys 2 vaocq,u-6t809econd hand on your wr yq-qc m6usb*o h j0ta28zvt9,ist watch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weight arm*t/wfb tlo64;byzf (+l 8ggxound in a circle below you on a 0.25-m piece of fishing line. The weight makes a complete circle every 0.50 s. What is the angle that the fi6bwgglyf*/;xfl8 (4+bmto ztshing line makes with the vertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R iu1*4lp vh pv;+jx bey+n a new highway is designed so that a car traveling ajvx y*b4 lv1 +up;hep+t speed $ v_{0}$ can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from $v_{\min }$ to $v_{\max }$ . Derive formulas for $ v_{\min }$ and $ v_{\max } $ as functions of $\mu_{s}, v_{0}$ , and R .

  Amtrak’s high speed train, the Acela, utilizes tilt of tuz9 mzs s.as690 dq6jkhe 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 of 620 m at a speed of (approximately ). (a) Calculate the frictq69d.6z0u ka s9jzsmsion 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.$\approx$    $\times 10^2$