Sometimes people say that water is removed from clothes inqcq ;6 vudf+w- a spin dryer by centrifugal force throwin; qfc wu6-dvq+g the water outward. What is wrong with this statement?

Will the acceleration of a car be the same when the car travels around 6 bb(hwz2o6o3 m; qhuyfh x1:,lf3tqba sharp curve at a constant 60 km/h as when it travels arounfxfblwq (umh2hbo63to;q1, y h3:zb6d a gentle curve at the same speed? Explain.

Suppose a car moves at constant spsx0sxzn1j.u 4beqat 6:)n6tl eed along a hilly road. Where does the car exert the greatest and least forces on the road: (a) at the top of a hill, (b) at a dip between two hills, (c) on a level stretch near the bottom ofl qsntxa1 4x)6tj0u .szn: 6be a hill?

Describe all the forces acting on a child ridinga vmt cx+uf.-ht0)3dt a horse on a merry-go-round. Which of these forces provide.hcxm da-u ttvf)30t +s the centripetal acceleration of the child?

A bucket of water can be whirled in a vertical circle without 2m ijnaxay z9n0s1 0n;the water spilling out, even at the top of the ciria;nn x m2ys j91zn00acle when the bucket is upside down. Explain.

How many “accelerators” do you have in your car? There are at least three contro:bw+*g9hkf7jd rwd nppk25 .vls in the car which can be used to cause the car to accelerate. What are they? What accelerations.g :rwvj7fnh*d2w9p+5 kdkp b do they produce?

A child on a sled comes flying over tjp awnk 05k5tce1,9ggu*l1gzhe crest of a small hill, as shown in Fig. 5–31. His sled does not lgtlp 1 ezk 10*9uj,a 5kwgcg5neave the ground (he does not achieve “air”), but he feels the normal force between his chest and the sled decrease as he goes over the hill. Explain this decrease using Newton’s second law.

Why do bicycle riders lean inward when rounding a curv)i /fs0cpid 6ze at high speed?

Why do airplanes bank when they turn? How would you compute the v2 l:4qhkgd( kbanking angle giv4:hlkd (kg2v qen its speed and radius of the turn?

A girl is whirling a ball on a string around her head in a horizontak l j57+u:qc4jzb;)ojut, ri gl plane. She wants to let go at precisely the right ric4 tz;gj+,7)u kjq :lu5bjotime 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 7 s6dex,o.b 0,e7ok in4syr-ru kf)x mEarth? If so, how large a force? Consider an app6,br) kn dxx uofosry4kmes770,ie .-le
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, in whagi af7d;1jj v-: t52:2zcr(zcf emrha t ways would the Moon’s orbit be div h czzj:jmc27g(2afdf1t ; rea-5i:r fferent?

Which pulls harder gravitationally, the Earth on the Mo ylig(e p7(2nc cjr d-mv.:l;son, or the Moon on the Earth? Whligr7j.cc -2n((se:ypv ;dmlich accelerates more?

The Sun's gravitational pull on the Earth is much larg.hetd :wz qkqqpd4 3vw g;(8-ver than the Moon's. Yet the Moon's is mainly responsible forqz :vw(q-w 4d;d 8ep3tqhkg v. the tides. Explain. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]

Will an object weigh more at the equator or at the poles? What tqorso u 3jcdx:;ce9*02jui 2rwo effects are at work? Do thq:9 r03c2rc2 djosiueux*j ;oey oppose each other?

The gravitational force on the Moon due to the Earth is only about half the foy(k/vjr u:ugu4w iw r(/8zx2 crce on the Moon due to theg(2yjrx84u( cz: wr/u /uik vw Sun. Why isn’t the Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in its orbit around the Sun larger or s5trklg9); oic maller than the centripetal ac;)ior95k lctgceleration of the Earth?

Would it require less speed to launch a satellite1dca3 duclx( m4yz e.- (a) toward the east or (b) toward the west? Consider the Earth’s rotation dird.uced-x1 acl 4 m(zy3ection.

When will your apparent weight benkv2m n1oic4sk 19jw4 the greatest, as measured by a scale in a moving elevator: when the elevator (a) accelerates downward, (b) accelerates upward, (c) is in free falli2k4ns km1 nj491voc w, (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 affected by p3bpd:e7nk 4 kweightlessness. 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) how objects fall, (b) the force we feel on our feet, adk 7ppkb3:en4nd (c) any other aspects of gravity you can think of.

Explain how a runner experiences “free fall” ocr9 6xzp,x ,lly3cw9i r “apparent weightlessness” between steps.zr6i,lc,px cyl9 x3w9

The Earth moves faster in its orbit around the Sukw/ fjdr p4+ yo1:vbt;fua5w)n in January than in July. Is the Earth closd;vfwk)w a ob415tu/+jrf: yper 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 relative to the plane of its orbit.]

The mass of Pluto was not m v7wqq9ku0)xknown until it was discovered to have a moon. Explain how this discovery enabled anw0m qv 9uk)q7x estimate of Pluto’s mass.

  The Sun's gravitational-llqj bs9d0 hnqy +g:181vxbl 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 the Earth to the other.]A child sitting 1.10 m from the centlvbgbq0lylx jsq: d8n 91-1+h er 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 xi *xmrul:,77/ip up*(p b6 fprmqur7 1980 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ pulls out of a dive by moving in an arc of radius 6.00 km . What is the plane's acceleration in $g^{\prime}$ 's?    g's

  Calculate the centripetal acceleration of the Earth in its orbit around 31tu5-qq 2ge exanz k*the Sun, and the net force exertkea5 1qu 3-zen 2gxqt*ed on the Earth. What exerts this force on the Earth? Assume that the Earth's orbit is a circle of radius 1.50$ \times 10^{11} \mathrm{~m}$ . [Hint: see the Tables inside the front cover of this book.]    $\times10^{22}$

  A horizontal force of 210 c5: dns zn-dir(w57zj N is exerted on a 2.0-kg discus as it rotates uniformly in a horizontal circ 7rd: z5nj cswn-5iz(dle (at arm’s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is in orbit 400 km from the Earth's surfarn*ay fp:x)/3qbq 7y lce, 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 gravitatlbp*y x7af:/3 )yqn rqional acceleration at the Earth's surface. $\approx$    g

  What is the magnitude o) s cj*my;bo)nwtw)w: f the acceleration of a speck of clay on the edge of a potter’s wheel turning at 45 rpm (revolutions per minute) if the wheel’sbjtwos n:)y;)mc w)w * diameter is 32 cm?   

  A ball on the end of a string is revolved at a uniform ratrj op i u+d c64jxahp;s7x*3/ne in a vertical circle of radius 72.0 cm , as shown in Fig. 5-33. If its speed is 4.00 n ai 7psdp3+ch ;ujx6rxj/4*o $\mathrm{~m} / \mathrm{s}$ 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 wl/; e/ugms cv7hich a 1050-kg car can round a turn of radius 77 sc7v;u g/elm/m on a 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 between the tires a8h ; n(1 rr8 d6m+y(duqifmkh+k fge9wnd the road if a car is to round a level i ef1dqmwy+kk+un9h( ; mh(6f8 grr8dcurve of radius 85 m at a speed of 95km/h ?   

  A device for training astronauts and jet fighter pilots y5 ( )ss9lwbw;. fpbtlr1a2gv) ynx9zis designed to rotate a trainee in a horizontal circle of radius 12.0 m . If the forcesr) 1p nywflysl.) 2zt9(9wb;5x gabv 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.t5m 10y79xbpdsijpn8 0 cm from the axis of a rotating turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable until a rate of 36 rpm is reached and the coin slides off. What is the coefficient of static friction between the 0d8p s in1xjpmybt957 coin and the turntable?   

  At what minimum speed must a roller coaster be tr, mbfau8fo+ e3r1;l fx*,linyaveling when upside down at the top of a+l;3ife a 1oxr,u*n m ,blfy8f 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 thgwoew7f9 i t( 1jpc::he driver) crosses the rounded top of a t (7fp:9wcoeij1: hgwhill (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-diameter Ferris wheelmgqc 34 s8gsf-o47emovefa(jrp 8+4 a need to make for the passengers to fe4gfs oapg- + 3qmvf84os4c8( m7aejerel “weightless” at the topmost point?    rpm

  A bucket of mass 2.00 kg is whirled in a vertical circle of radwmv97+g-17avn agt s njbb*r,ius 1.10 m. At the lowest*1gn- nr wasb9+gv t77, majvb point 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 if a particle 9.00 cm from the axisx)ja zeo;2sg6dabu:wiuhyt * f:,4+m of rotation is to experience an accelerad:e+mhyu )twxs i 42j*zfba aog6:,;ution of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotated in a cyl/e7h c-uqs+gu indrically walled "room." (See Fiucehg7+- us/ qg. 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 d8(zlzmos./y90 mu cv y.vpi4 a circle on a frictionless air-hockey tabletop, and is held in this orbit by a light cord connected to a dangling block (mass m ) through a central hole as shown in Fig. (p9o8vm 4 v .us.yycdz m/l0zi5-
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 weight of ttl2+6qajon*k5+hr lx tlt( 1whe ball which revolves on a string 0.600 m long. In parti xq1* k2(n+ ttl awroltj+65lhcular, 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 travba *k*okg5un *eling 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 masses qz /zo-ki s+cxvfr 6/.$ 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 qugh20+ yus x+evasive maneuver by diving vertically at 310 $\mathrm{~m} / \mathrm{s}$ . If he can withstand an acceleration of 9.0 g 's without blacking out, at what altitude must he begin to pull out of the dive to avoid crashing into the sea?   

  Determine the tangential and centripeh5y5d0r jx( zkt2 q,j5( ckrfmtal components of the net force exerted on the car (by the ground) in mr jyfx55t h0zkq r5(jd2,k( cExample 5-8 when its speed is 15$ \mathrm{~m} / \mathrm{s}$ . The car's mass is 1100 kg . F$_{tan}$ =    F$_R$ =   

  A car at the Indianapolis 500 accelerates uniformly from theox7bx(:drm t ; pit area, going from rest to 320km/h in a semicir :;odx(r7mtxb cular 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 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 k tq wg-;z0cr0m . At a particular instant, wcqtr 00gzk-; its acceleration is 1.05 $\mathrm{~m} / \mathrm{s}^{2}$ , in a direction that makes an angle of 32.0$^{\circ}$ to its direction of motion. Determine its speed (a) at this moment   
(b) 2.00 s later, assuming constant tangential acceleration.   

  Calculate the force of Earth’s gravity on a spacecraft 12,800 kmzay; aba+4j2b (2 Earth radii) above the Earth’s surface if its mass is 1350yj4 za; a+b2ab kg.   

  At the surface of a .*owub57.rw qs)tdj9eep i 0pcertain planet, the gravitational acceleration g has a magnitude of 1j79 eq. 0suiw*r tpo dwp)e5.b2 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 9-f7e f tr7n2qdaksc kt50sy 2due to gravity on the Moon. The Moon's radius is 1.74sq 0cr kanf-y k2 t2s957fdt7e $\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has a radius 1.5 times that of Earth, but has g5o /)yco gz7nthe same mass.cyg /7g)onz5o What is the acceleration due to gravity near its surface?   

  A hypothetical planet has a mass 1.6d, 1sb4 ya; ett jjiaey*c6/n,6 times that of Earth, but the same radius. What is gn*dj,4j 1 e6/ y;t,ybtcaaes i near its surface?   

  Two objects attract each cu6(4dic3m*0 qr 5k x; lxjvdqother gravitationally with a force of 2.5$ \times 10^{-10} \mathrm{~N}$ when they are 0.25 m apart.
Their total mass is 4.0 kg . Find their individual masses. m$_1$ =    m$_2$ =   

  Calculate the effective value of g , the acceleration of gravity, at (a bi97oadb.ikvhs/, j; ) 3200 m dk/ v,i.b7j; sobai9 h   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from km-lfn q)afl; y:yyb+p.acc x0q28r6the Earth’s center to a point outside the Earth where thegravitationmfpyb) ccr8yxqayln6 q kl:f+ 2a0;.-al 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 Sunu)1 o,bhjl-+ko u5 qdb packed into a sphere about 10 km in radius. Estim1 ub5b)odh+k ,jqlu- oate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once was an average stafv 6 *)tjvcrj5r like our Sun but is now in the last stage of its evolution, is the size of our Moon bu *v6fj vj)rt5ct has the mass of our Sun. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronauts feel weightless while oro:t* e5r28fkk od.jez biting in the space shuttle. Your friends respond that they thought gravity was just a lot weaker up there. Convince them and yourself thate*je :.zkdt8for 52k o 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 ofagsq m, l,s6vrg6; t tsbwm-k*r;/a i3 a square of side 0.60 m. Calculate the magnitude and direction ofm- ba ,6/;rttv;k s6awgr,mssgq li*3 the total gravitational force exerted on one sphere by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred years most of the planets line up on the same side of tyd, v; pzh8d: sanpff-5k4 z2s7a m9pnhe Sun. Calculate the total force on ts9z5 -a7 dz nnfkam4y pf:sd v2p,;8phhe Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in a line (Fig. 5-38). The masses are $M_{\mathrm{V}}=0.815 M_{\mathrm{E}}, M_{\mathrm{J}}=318 M_{\mathrm{E}}, \quad M_{\mathrm{S}}=95.1 M_{\mathrm{E}}$ , and their mean distances from the Sun are 108,150,778 , and 1430 million km , respectively. What fraction of the Sun's force on the Earth is this? $F_{Earth-planets}$ =   $\times 10^{17}$ ${F_{Earth-plantes}}$ \ ${F_{Earth-sun}}$=    $\times 10^{-5}$

  Given that the acceleration of gravity at the q q) l,l ;u1 ckk1esz3,8kstl+.xzvunl3rsd 6surface of Mars is 0.38 of what it is on Earth, and that Mars’ radius is 3400 km, determine the m. 3e6quq3zklsu1; ,snkrl ,lvc8ztsd k1)l+x ass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value for+ vnj2bg+bx0t the period of the Earth and its distance from tt+ngb0v+ x 2jbhe Sun. [Note: Compare your answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite moa r4c4p.l-i8+; a6 ji4n sh0kctvuoyzving in a stable circular orbit about the Eavr4;cunp i8i.o c0aalt6jys4h-4 k+ zrth at a height of 3600 km.    $\times 10^3$

  The space shuttle releases a satellite into a //dakmw t:6/byl- bhlvn;r8n circular orbit 650 km above the Earth. How fast mun l8b6t l-b ;dh/nka/mw:r/yv st the shuttle be moving (relative to Earth) when the release occurs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occupants aah ; *xbjegq//re to experience simulated gravity of 0.60 g?b/eg;h/qaj *x 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 foj rfuln /pia( toz6.t:4 uzoz58(rg)yr a satellite to orbit the Earth in a circular “near-Earth” orbit. A “near-Earth” orbit is one at a height ou/ i jzz zfonp5) (46ta.r :tg(8ryulabove 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 v+b;*xw 0xwfjw elocity would a satellite have to be launched from the top of Mt. Everest to be placed in a circular orbit around tf jx0+ wbx;*wwhe Earth?   

  During an Apollo lunar lanqw pc*- qideye5g61uq9 .:kd;(ogbgsding mission, the command module continued to orbit the Moon at an altitude of about 100 km. How long did it take to guqeq.ygp*1: odk(w ;gi-c besgd5q6 9o around the Moon once?    s

  The rings of Saturn are composed of chunks of ice that orbit the pl6io) 0nq0ulb janet. The inner radii0 bo0jqnu )l6us 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 uw, rcek4f+.f5/it x htxj:l:( w nd)m(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 thef5n ( : mxx,c/f ewkw.dr)i:j4+t thul bottom?
(a)    (b)   

  What is the apparent weight 2a x0xmnh);abof a 75-kg astronaut 4200 km from the center of the Earth's Moon in a space vehicle (a) moving at constant veloch2a;x a)m b0nxity,     ----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 l7cb *jzl+dis. b(bs-consists of two stars of equal mass. They are observed to be separated by 360 million km and take 5.7 Earth years to orbit about a point ld7 *b jiczs(-b+.blsmidway between them. What is the mass of each?    $\times 10^{29}$

  What will a spring scale read for the wetbv2cxj); l yno9 g6(might of a 55-kg woman in an elevator that movjxcbnmgyv2 ;t9lo6 )(es
(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 suspended from the ceiling of a a0,u8qo9/daclxox2 6:qfq xin elevator. The cord can withstand a tensio6ialoo2 xuqq 0,x cq:f89xad /n of 220 N and breaks as the elevator accelerates. What was the elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orbits very near the surface of a pw+ u p(+(ty ysgh(rxq/lanet with period T ,/ wh+y+ q((rspt g(yux the density (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 b37 joxy/rcvh fm76-3qys/me the Moon (27.4 d) to determine the period of an artificial satellite orbiting very n7hsb f7/qo3vx3 ymc j6e/y-r mear the Earth’s surface.    s

  The asteroid Icarus, though only a few hundred met 3hg7kmf0cu gm3x,f: eers across, orbits the Sun like the planets. Its period is 410 d. What is its mean distance from theu3x ,m3f0m gfckhge: 7 Sun?    $\times 10^{11}$

  Neptune is an average dista;yl*ug*prk . nnce of 4.5 $\times 10^{9} \mathrm{~km}$ from the Sun. Estimate the length of the Neptunian
year given that the Earth is 1.50 $\times 10^{8} \mathrm{~km}$ from the Sun on the average.    $\times 10^2$

  Halley’s comet orbits the Sun roughly once every 76 years. It comes very clo)x.f7mf4 7j o* zpxtrwse to the surface of the Sun on its clo7 7jwfx tf)pmx*4 orz.sest approach (Fig. 5–39). Estimate the greatest distance of the comet from the Sun. Is it still “in” the Solar System? What planet’s orbit is nearest when it is out there? [Hint: The mean distance s in Kepler’s third law is half the sum of the nearest and farthest distance from the Sun.]    $\times 10^6$

  Our Sun rotates about the center of the Galn8nc:yy*aw:e g 1saxp90i:pjaxy $\left(M_{G} \approx 4 \times 10^{41} \mathrm{~kg}\right)$ 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 bbgf(wf7b xq6 e1ok31x-kuf;mean distance for the four largest moons of Jupiter (those discovered bfg3xbxu1 71k(wq- ofbkb e6f;y 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 Eh 2bshn0i6wu(+y ,nfaarth from the known period and distance of the Moon. 60hh 2y u, san(niw+fb   $\times 10^{24}$

  Determine the mean distance f .c.:m zv(vv/iax,ragrom Jupiter for each of Jupiter’s moons, using Kepler’s third law. Use thz(c. avva/. mr gv,ix:e 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 of many f2i znaae1,jr 5:5f gmxplh75sragments (which some space scientists think came from a planet that once orbieairnlh,z12 j5 5gmp7 :5as xfted 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 describesxr zin8c)rh rk5va8z*.(-b oy7 lko *r an 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). Imag r.ik 5rcz -7*hro nk8ly* )b(8axzvorine that this sun is exactly like our own, that the distance to the band is the same as the Earth-Sun distance (to make the climate temperate), and that the ring rotates quickly enough to produce an apparent gravity of g as on Earth. What will be the period of revolution, this planet's year, in Earth days?   

  Tarzan plans to cross a gorge by swinging in an arc from a hanging vine (lc+a jfxif1 t*g.y 3q(Fig. 5–41). Ii1cq gtf*xy3.j(fa+ l f his arms are capable of exerting a force of 1400 N on the vine, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 80 kg, and the vine is 5.5 m long.   

  How far above the Earth’s surfad. fcyr h,cox3rp. 2t1ce will the acceleration of gravity be half what it is d1 hyf3.crx.oc tr,2pon the surface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spin in a mutual circpjt7 :owz38d74g5otbsz( i phle once every 2.5 s. If we assume their arms are each 0.80 iwo4p:p(gj 7 o587z3 tzth dbsm long and their individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotatrmlk; 1p7gc.bes once per day, the apparent acceleration of gravity at the equator is slightly less than it would be if the Earth didn’ mrcgp1;kl7. bt rotate. Estimate the magnitude of this effect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from ufjc cs *ln*d4hrsd1+nuwku79qq:; 3 the Earth will a spacecraft traveling directly from the Earth to the Moon experil;f*1u w qjd k 93ns*cuhqcnd7u4:r+ sence zero net force because the Earth and Moon pull with equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, but h x2**kamlay6g:b-w n when you stand on a bathroom scale in an elevator, it says your mass is 82 kg. What is the accelerati x ng lhby*w2a-*mak:6on of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consists of a circu2ma j1vbtv;tu2e/ p,zlar 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 (re2b/;vmv2eap ttz1uj , volutions per day) if an effect equal to gravity at the surface of the Earth (1.0 g) is to be felt?$\approx$    $\times 10^3$

  A jet pilot takes his aircraft in a vertical loowvk7;)1j6rlc o0mz w*lkh o 1yi0+ tiip (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 3ydef4,ncy ft u /-l4dthe mass of a planet in terms of its radius r, the acceleration due to g td/ dy4f34fyucl,ne-ravity at its surface and the gravitational constant G.

A plumb bob (a mass m hangincm 3p mi+;vyelv oq564g on a string) is deflected from the vertical by an angl6 ;ly+q mm 3c4o5pievve $\theta$ 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 nzm) eyv,s *)ci 6ms ap;qrs4/coefficient of static friction is 0.30 (wet pavemes6ny ,mmc;pa))svrzei *4 qs/nt), 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 that objects 5,p 68zcv4u uxempph0 h0s)t kat the equator wervph )upx6p040t umech5s,z 8ke apparently weightless?    $\times 10^3$s

  Two equal-mass stars maintain a consdwhfg;v;. i x8tant 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 curki f*(9 )dhm liz* zlxvv0tm9(ve of radius 235 m. A lamp suspendl9 fv0 t)ixv(z hkdzm*(9ilm*ed from the ceiling swings out to an angle of 17.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times as massive as the Earth. Thus, it has been3:e0k xf9m crn 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 wok3 m0c9rxfen:uld 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 deduced the prese qksdx7) u5*z9qqgvbt7am:6 5s 2zaconce 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 clou gmqq675do9 av) asz*:st5b27 cqzuxkds 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 t,3i0mmewayp ,kp vs;+yv(1f v he 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 withm,e0 ms;y a yivvwfp1+( ,3pkvout losing contact with the road at A?

  The Navstar Global Positioning System (GPS) utilizes a group of 24 satellites 8kckklj jz.1e)jr/t: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 eacczt:jkjr/ e)1.klj 8kh 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 Renf-nvfhxm0 x0/t 2vh/.:vtqg a dezvous (NEAR), after traveling 2.1 billion km, is meant to orbit the asteroid Eros at a height of about 15 km . fh/vxxq av: 02g/.f t m0htv-nEros is roughly 40 $\mathrm{~km} \times 6 \mathrm{~km} \times 6 \mathrm{~km}$ . Assume Eros has a density (mass/volume) of about 2.3 $\times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ . (a) What will be the period of NEAR as it orbits Eros? $\approx$    $\times 10^4$sec
(b) If Eros were a sphere with the same mass and density, what would its radius be? $\approx$    $\times 10^3$
(c) What would g be at the surface of this spherical Eros?$\approx$    $\times 10^{-3}$

  You are an astronaut in the space shuttle pursuing a satellite in need of re1 y.o-pv z28qrs:po4okx/ u pupair. You are in a circular o2 yu-zv8o/krso.xp:1p u4 qop rbit of the same radius as the satellite (400 km above the Earth), but 25 km behind it.
(a) How long will it take to overtake the satellite if you reduce your orbital radius by 1.0 km?$\approx$    h
(b) By how much must you reduce your orbital radius to catch up in 7.0 hours?$\approx$    $\times 10^3$

  The comet Hale-Bopp has a period of 3000 years. (a) Whatk eu7lc-gudm2b/ ;03no itde 6 is its mean distance from the Sun?m327otklge0 d n;iudcu6be/-   
(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 i:rl mf/*+(ua/jm*lg vz4 kvv s u:fvs5f you could actually "feel" yourself gravitationally gl/+(rlma/zv s *:um5fuvf*4k svv: jattracted 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 Galaxy (Fig. 5-46k.ptlzkgk ,) 7) at a distance of about 30,000 light-years from the cen., tlkg z7)kpkter $ \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 corner mqivx dz.5-q84hwa c9s of a square 0.50 m on each side. Find the magnitude and direction of the gravitational force on a fifth 1.0-kg mass placed at the midpoint of ta.qvx5-d84 zm wi9 qchhe bottom side of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 r6oh09aop2 o3ylgr4 f9rzt- o 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 experienced by the tip of the 1.5-cm-long sweep second i pq brx:gd+z)+cbt19hab dt91 z+x):gpc+b irqnd on your wrist watch?    $\times 10^{-4}$

  While fishing, you get boredph jgik)1m .tp/0. doh ,u6hvm and start to swing a sinker weight around in a circleut .ih 0dkgjop,mh.m1hp/)6v below you on a 0.25-m piece of fishing line. The weight makes a complete circle every 0.50 s. What is the angle that the fishing line makes with the vertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in a new highway is nk7+7d dccmucl;-e9u+n4dx p pz0lx5c h a.)bdesigned so that a car traveling an5kz+e p0dxlx4u;c7)ml.pnu 7dcc -h9+bdac 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, uk u+.k+rq6fbz4tw g2o3hrif8tilizes 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+ r u o8w643gkz f+kft.hbiq2r 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.$\approx$    $\times 10^2$