Sometimes people say that water is removed from clothes in a spin dr*o*h,j3:(amdnil bov 9lirp9 yer by centrifugal force throwing the water outward. What is wrodv* m*: r 9iloa9(,ojhlib3pnng with this statement?

Will the acceleration of d*yx)(: gyo6eg/dp afa car 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 spee p*(gxy/g): 6dydo eafd? Explain.

Suppose a car moves at con (vb5yq/lgi8 no +dnvi;6fhsv-4vj4dstant speed 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 85+y h/( vl 6n i4-ijd4 gvvsnvd;qobfbetween two hills, (c) on a level stretch near the bottom of a hill?

Describe all the forces acting on a child rg isp y/y.eeyhy*u jm sarv27m,6mgw6 7ap ,5-iding a horse on a merry-go-round. Which of these forces proay5jwegypr s-g ,2 hmpy76esm/,7 avy6mi*u . vides the centripetal acceleration of the child?

A bucket of water can be whirled in a vertical cirj efc +xcbv a07 rg-zv0t736apcle without the water spilling out, even at the top c fxp70r07a6-e3c vvtzba+j g of the circle when the bucket is upside down. Explain.

How many “accelerators” do you have in b( 7q-.bjve 8biyg hrx3,ay/nyour 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 b7q8g nhxia(/, by vy-rb.je3produce?

A child on a sled comes flying over the crest of ap6vev .jxn 1c0ax9e +en .j2nu 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 sn xa n+vxe epv1 jcej0269..nuled decrease as he goes over the hill. Explain this decrease using Newton’s second law.

Why do bicycle riders lean inward whe6rxx:g l h uwree(q) v;(e8q2rn rounding a curve at high speed?

Why do airplanes bank when they turn? How would you compute the,bw;d )b)oe+k w)2z f0uhnnd u banking angle given its speed and radius of tn)d)ubwe+owdn 0 b, hukz;f)2he turn?

A girl is whirling a ball on a string around her head in a hori*wvfo17m hz)kj4mq 4izontal plane. She wants to let go at precisely the right time so that the ball will 4v)mk 4j zqfom*1hi7whit a target on the other side of the yard. When should she let go of the string?

Does an apple exert a gravitationam6-vemj n7zgx5i)6v1 iarg3 sl force on the Earth? If so, how large a force? i67vxsa56gmv j e nzgrmi-3) 1Consider an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were dou02nm +un hpm4qble what it is, in what ways would the Moon’s orbit bpuhnn q420mm+ e different?

Which pulls harder gravitationally, the Earth on the M48 d oucjtfqo+m (v*/zoon, or the Moon on the Earth? Wh d4cqt +o*m(ojfzu/v8 ich accelerates more?

The Sun's gravitational pull on the Earth is much larger than/ 9 lu*jiwqo(tun5h yoz/-2 +b2z fhnd the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hintf/2z/i(o uy z-h jhw9b+5odn2*ntulq: Consider the difference in gravitational pull from one side of the Earth to the other.]

Will an object weigh more at the equator o3mh 6b k:kemiszw(e6+r at the poles? What two effects are at work? Do they oppowhe(bk6m+:mksei z36 se each other?

The gravitational force o 3q/3pylvq+w/b;tqai g eg s--n the Moon due to the Earth is only about half the force on the Moon due to the Sun. Why isn’t the Moon pulled away 3bt/iqwl ;+s- py-e3vq/g aqgfrom the Earth?

Is the centripetal acceleration of Mars in its orbit around the Sun larg)ac 9s t/la)x8f7u meier or smaller than the centripetal acceu8ci7e/) t)m9xs l faaleration of the Earth?

Would it require less speed to launch a satellite (a) toward 1b 6s)g9ciddithe east or (b) toward the west? Consider the Earth’s rotationi)dc1 sgd ib96 direction.

When will your apparent weight be the greates/)sb fyljb z2s jx,0p)i2zal(t, as measured by a scale in a moving elevator: when the elevator ((2xlzb s)b i)j 0,2l/ jpsafyza) accelerates downward, (b) accelerates upward, (c) is in free fall, (d) moves upward at constant speed? In which case would your weight be the least? When would it be the same as when you are on the ground?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long periods in ot/zw 8ozc.cdd-. q) iwuter space could be adversely affected by weightlessness. One way to simulate gravity is to /- di.z.tw8cq owcd)zshape 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 “free fall” or “apparent weightless*c4h7uvjn n3 fness” between *nvuf437 j hncsteps.

The Earth moves faster in its orbsuwk+/pz; 1 -ptxla+- j ,vzpeit around the Sun in January than in July. Is the Earth closer to -j+-1p x,p /+zwaz;psuetl kv 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 known until it was discovered to have a moon. E3yxyl :fg. *cz3srxd1 xplain how this discovery enabled an estimate of yg *fc3:3y1zxs l.r dxPluto’s mass.

  The Sun's gravitational pull on cd. eedc b -;oco*ul.y4nm0 .qthe 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 oneb q nyc0c.coee;do*-mu .dl4. 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 zb tks rc0qy:)z.pq qg 6uo*,so .j+p51980 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 orbi8m *xh b3iao.wt 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 1a38bix .w *hom.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 exertemxhc 9k /ip3h1d on a 2.0-kg discus as it rotates uniformly in a horizontal circle (at arm’s length) of radius 0.90 m. Calculapm/hc31k xi h9te the speed of the discus.   

  Suppose the space shuttle is in orbit 400 km from the Earth's surface, and c: ar (uc1+,t5/ybils-j)jt(piugr b zircles the Earth about once every 90 minutes. Find the centripetal acceleration of the spacej5(b yj ci-) ptturb:g zu+l(1r,a s/i 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 the acceleration of a34m)yzil 85yhdjw. mt speck of clay on the edge of a potter’s wheel turning at 45 rpm (revolutions pe34wjzymh.)tmi dy l 85r 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 v kl*9z0k /ytplertical circle of radius 72.0 cm , as shown in Fig. 5-33. If its speellkpt/*y k0z9d is 4.00 $\mathrm{~m} / \mathrm{s}$ and its mass is 0.300 kg , calculate the tension in the string when the ball is (a) at the top of its path   
(b) at the bottom of its path.   

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

  What is the maximum speed with which a 1050o.pk qit+ e*;x-kg car can round a turn of radius 77 m on a flat road ;i+x.q oktep*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 an a /)pel9o6wz s.flic:ji8t,o *rn+xx d the road if a car is to round a level curve off pxzitlo9en,/wlj i+8s):* cxoa.6 r radius 85 m at a speed of 95km/h ?   

  A device for training astronauts and jet fightegiu :yd8onar x )8or8ycu4.s 3r pilots is designed to rotate a trainee in a horizontal circu8ra :ysogr34 xydciu) .on 88le of radius 12.0 m . If the force felt by the trainee on her back is 7.85 times her own weight, how fast is she rotating?    rev/s    m/s
Express your answer in both $ \mathrm{m} / \mathrm{s}$ and $\mathrm{rev} / \mathrm{s}$ .

  A coin is placed 11.0 cm from the axis of a rotatinn z55mtu.)h rdg turntable of variable speed. When the speed of the turntable is slowly increased, the coinn5)zuhd5.r t m remains fixed on the turntable until a rate of 36 rpm is reached and the coin slides off. What is the coefficient of static friction between the coin and the turntable?   

  At what minimum speed must a roller coaster be traveling when upside down at thq /x1ei. e,zds+b; cwr4uj- pye top of a circquc,+1eb. ei s- jx;yzr/pdw 4le
(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 r8u oaqimapaz-n3sb ,pk/a(1 : ounded top of a hill kqmzi( /,8a31a puap on-sab:(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 wheel need to cf z3e0uhduu8r16 vea l+eo(,.rn 7h wmake for the passengers to feel “weightless” at the topmost point? hu u7eu.8a1fv e0ocwln+d hr,z(6 r3e   rpm

  A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1b76-p1lzo rd j4 xra7f.10 m. At the lowest point of its motion the texaor flb741 d p6r7zj-nsion 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 rot9e oc ar vjp,rhlp3v*jo--32w ate if a particle 9.00 cm from the axis of rotation is to epcvvph-e3 , a rlj -3ojr29wo*xperience an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carni 71jbh2x*q b.-9 8axi-gyenx4 av qnjyval, people are rotated in a cylindrically walled "room." (x*nyhqy9-g a2jae i41xbxb nv 7-qj .8See Fig. 5-35.)
The room radius is 4.6 m , and the rotation frequency is 0.50 revolutions per second when the floor drops out. What is the minimum coefficient of static friction so that the people will not slip down? People on this ride say they were "pressed against the wall." Is there really an outward force pressing them against the wall? If so, what is its source? If not, what is the proper description of their situation (besides "scary")? [Hint: First draw the free-body diagram for a person.]   

A flat puck (mass M ) is rotated in a circle on a frictionlesshvu7loln4jr 2u8yoq4k0 qo i(s2+q a7 air-hockey tabletop, and is held in ro oah4jvoysk4ui2 qn7ul(02+q8 7lq this orbit by a light cord connected to a dangling block (mass m ) through a central hole as shown in Fig. 5-
36. Show that the speed of the puck is given by

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

  Redo Example 5-3 , precisely thiys*g2n*;ju; e8y9*3g z spsrpf y(geis time, by not ignoring the weight of the ball which revolves on a string 0.600 m long. In particular, find the magnf esn*y;p*gri2y;jp * yue3gs9z8 (gsitude 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 traveli5 wu jhw0pr rwpg64r8,ocs* q1ng 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 masse5 osq pse 7l;iivg2i:4s $ 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 bm2v q d fl3bwb5sh6u51y 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 centxg63 gil0ih 1cripetal components of the net force exerted on x 0c3hlg16i githe car (by the ground) in Example 5-8 when its speed is 15$ \mathrm{~m} / \mathrm{s}$ . The car's mass is 1100 kg . F$_{tan}$ =    F$_R$ =   

  A car at the Indianapolis 500 accelerates/6z py)hmr+ la;-zije 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 i zmyia;+- /) hrp6lezjs 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 ofra/(gig/+ :3l/bg2hn r- qde0 hrg jdl radius 2.90 m . At a particular instant, its acceleration is le-+jh 2g/gg/b0hn3raqrr (i dgd :l/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 km (2 Earth radtq sornp8+/8 ;u brke1ii) above the Earth’s surface if its mass is 1350 kg.q 8eon8b ;rupsr1tk +/   

  At the surface of a certain planet, the gravitational acckb,z6y n dcp n1p8w5,eeleration g has a magnitud1yzpp w 68e,kd5 bnn,ce 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 to gravity on the Moon. The Moon's zgmn axad0 8./wd+ b:bradius is 1.7a0b zm wnxa/g d+d.b:84 $\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has hs)gb59mywep; el( boml; a. )a radius 1.5 times that of Earth, but has the same mass. What is the acceleration due to gravity near it)lmel9mhb ypw; ;.age5b)so ( s surface?   

  A hypothetical planet ,jjw qhx jr18k k./20w*iumsr has a mass 1.66 times that of Earth, but the same radius. What is g near its surfacerj /h,0s w8 k*2jrj.xiu 1wkqm?   

  Two objects attract each other gravitationally with a force of 26+,lk eh5iuy qfgr2o;.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 ofn aabsbz7a8wocj ip( ./v(o, 9 g , the acceleration of gravity, at (a) 3200 m(7ipw.an zasc8 vb(a9oo ,/bj    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Eaa6q6l l+9;) tcuzt xdkrth’s center to a point outside the Earth where thegravi6z)udxl9 a+kq t;6ct ltational 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 packed9k(zalz r2v.q du t9q0 into a sphere about 10 km in radius. Estimat 9ql vu.(dz0z2aqkr 9te the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once waw z1zp5 l3f3dvs an average star like our Sun but is now in the last stage of its evolution, is the size of our Moon but has the mass of our Sun. What is the surface gravity on this sta5lz fw1 vzp33dr?    $\times 10^7$

  You are explaining why astronauts feel weightless while orbiting in the space sc4v/d/ 7ybn 9h9a tg-t hfthpcv6i5p4huttle. 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 Earth’s surface in tetchvp h/f4yh9agp96b-nc 5i v4 7/tdtrms of g.   

  Four 9.5-kg spheres are loca-3wh4kb-6 ,bs ,)fdtrq a kughaf( 9frted at the corners of a square of side 0.60 m. Calculate the magnitude and direcw ,-(, )bfk b9r fq6fu4aghrs-dk tah3tion 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 up on the same sidbdwngl;,l9 yyt:-70 nt sjb, vb j,b6we of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four plb jy,:9l,n0dt wb-ljw7vynbs,gb; 6 tanets 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 surface of Mars is 0.38 of what irbr1z: ik)gr;shj 16ut is on Earth, and that Mars’ radius is 3400 km, determine the ma6r uhgr)s ikbr1:; j1zss of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value for the period of the e3 *b8cba.z-lj5*b.cihp xaf Earth and its distance from the Sun. [Note: Compare y .*hp j3bax-acl8bci b.ze5f*our answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite moving in a stable circular orbit aboumqb+9:hz8( p8gp8 p (hmq ij (f+,wdprl5mrkk t the Earth at a f8hr5mlp wjhri ,( m:km+g8(p (q pb+k8 d9pqzheight of 3600 km.    $\times 10^3$

  The space shuttle releases a satellite -m- hfazz,ab) into a circular orbit 650 km above the Earth. How fast must the shuttle be moving (relative to Earth) when the release occurs --, ma)zfhabz?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occupant)f cgsz: 8lkm3pe7oj *s are to experience simulated gravity of 0.60 g? Assume the spaceship’s diameter is 32 m, and give your answer as the time needed for one revolution. (See l s7o)f:k8egpcz jm3* Question 21, Fig 5–32.)   

  Determine the time it takes for a satellite to oo,evu*x v tc uf54yte+i2-d v.zvr44nrbit the Earth in a circular “near-Earth” orbit. A “near-Earth” orbit is 4ty-+oc trn 4v 25vuve.fu*,xdi4 zveone 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 /b am7 7slnc0osatellite have to be launched from the top of Mt. Everest to be pla7 a sl7mnc0/boced in a circular orbit around the Earth?   

  During an Apollo lunar landing mission, the command module continufeqq3p+b 6 h(bed to orbit the Moon at an altitude of about 100 km. How long did it take to go around the Moon once?hbqf+ e6 3qpb(    s

  The rings of Saturn are composed of chunks of ice that orbit the planet. The inhm *rw h0v:ajk/2g b3lner radius of the * g02lwv :rh3ha b/mjkrings is 73,000 $\mathrm{~km}$ , while the outer radius is 170,000$ \mathrm{~km}$ . Find the period of an orbiting chunk of ice at the inner radius and the period of a chunk at the outer radius. Compare your numbers with Saturn's mean rotation period of 10 hours and 39 minutes. The mass of Saturn is 5.7 $\times 10^{26} \mathrm{~kg}$ .


$T_{\text {irner }} $ =   
$T_{\text {outer }} $ =   

  A Ferris wheel 24.0 m in diameter rotates once every 15.5 s (see Ft(d6r r7naagshg g,,) ig. 5–9). What is the ratio of a person’s apparent weight to her real weight (a) at the top, an tr gaadgn)6s 7g,,rh(d (b) at the bottom?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut 4200 km from the center of the -g6obwtmm12+uk njc0Eartgt+wum2ko6b j1- n0 cmh's Moon in a space vehicle (a) moving at constant velocity,     ----待选择----towards the Moonaway from the Moon 
(b) accelerating toward the Moon at 2.9 $\mathrm{~m} / \mathrm{s}^{2}$ ?     ----待选择----towards the Moonaway from the Moon 

State the "direction" in each case.

  Suppose that a binary-star systelmaka,velb 0-+n h/6im consists of two stars of equal mass. They are +,lv06 m blena- iah/kobserved to be separated by 360 million km and take 5.7 Earth years to orbit about a point midway between them. What is the mass of each?    $\times 10^{29}$

  What will a spring scale reazhr 0.9 erkzw7d for the weight of a 55-kg woman in an elevator that mz z w7e9kr0hr.oves
(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 ceir qd kzk4bl .uzb biu4ji*6173ling of an elevator. The cord can withstand a tension of 220 N and breaks as the elevator accelerates. What was the elevator’s minimum acceleratio4*jlubzr ziqi 6k3db.k7u4 b 1n (magnitude and direction)? a =   

Show that if a satellite orh xxe1n+vis5+nr 2fzm) z7 u)gakj,y1 bits very near the surface of a planet with period T , the density (mass/volume) e)xz7ug51+ v jrn)f,iah2 x+mkyn1z sof the planet is $\rho=m / V=3 \pi / G T^{2}$ .
(b) Estimate the density of the Earth, given that a satellite near the surface orbits with a period of about 85 min .

  Use Kepler’s laws and the period of the Moon (27.4 d) to det;8hlu)8. mkqxznt8ifg 1 1p3ydgit 5vermine the period of an artificial satzd l 88 5thu8 3i 11p)kxygi.ntvf;gqmellite orbiting very near the Earth’s surface.    s

  The asteroid Icarus, though only a few hundred -as/v9fojff 0y ;, tr7vjrmb+q4vwr8 meters across, orbits the Sun like the planets. Its period is 410 d. What is its mean distance from ;,mwsv7+ rjy0o 4fr-afj t/vb vf8r9qthe Sun?    $\times 10^{11}$

  Neptune is an average dd2/i ksh o6hn,istance 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 rou fwv( jdu,:z1kghly once every 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 thewufk1 vd(, :zj nearest and farthest distance from the Sun.]    $\times 10^6$

  Our Sun rotates about the center of the Galaxy hsu zdxhvt;ib7,+ s5mb 2a)h3$\left(M_{G} \approx 4 \times 10^{41} \mathrm{~kg}\right)$ 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 flh z99 ,,zaapuonvzx315b 3qj our largest moons of Jupiter (th9 zb ,9 l xuv1jaz3q3z,opn5ahose discovered by Galileo in 1609).
(a) Determine the mass of Jupiter using the data for Io.$M_{\substack{\text { Jupiter- } \\ \text { Io }}}$ =    $\times 10^{27}$
(b) Determine the mass of Jupiter using data for each of the other three moons. Are the results consistent?
$M_{\substack{\text { Jupiter- } \\ \text { Europa }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Ganymede }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Callisto }}}$ =    $\times 10^{27}$

  Determine the mass of the Earth from the known period and distance of the Moonw x qht g.mm2x/r.8yd9. mwx8d .xrm9q 2ytgh /.   $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jupiter’s mxhez)tf 2 tbp,9)g)o doons, using Kepler’s third law. Use the distance of Io and the periods give hg,f)b2o9e td)tpx)zn 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 fr-*3l.et9r.ytqp+ be is* mjuxagments (which some space scientists think came from a planet that once ot.l9-xe*isymr+.jp 3 qb* teurbited 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 0tcx n9)yflp(nr hd05the form of a band completely encircling a sun (Fig. 5-40). The inhabitants live on the inside surface (where it is always noon). Imaginen0dhxr)nyfplt ( 09 5c 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 vi2mol7e9xso19whzt(xvq9 p fp(oa r/ +ne (Fig. 5–41). If his arms are capable of exerting a force of 1400 N on the vine, what is the maximum speed he can tolerate at t tz9fr7xvo9 pwe+ /hl9xsm o (1oap(2qhe 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 acceleryb - 2uuch4yem ow/or24t(m0pation of gravity be half what it is on the s o4tourcywbp /( mm-2e420yuh urface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and scqayqfwj 7,uq am,4u1 a cb+0/dj62lppin in a mutual circle once every 2.5 s. If we assume their arms are each 0.80 m long and their individual masse qc qjj71/mpc, 4uu0,f+ylqb2a d aw6as are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates once per day, tcc1egpta*hw-: tt q-/1 5 vkdehe apparent acceleration of gravity at the equator is slightly less than it would be if the Earth didn’t rotate. Estimate the magnitude of this effect. What fraction of g is this?e1t -hg t/-a*q1ec 5pkwvdct:    $\times 10^{-1}$

  At what distance from the Earth will a spacece*p zq9: th21cnzi9clraft traveling directly from the Earth to the Moon experience zero net force because the 9 ne2* ch z19ztl:cpqiEarth and Moon pull with equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, but when you stand on a bathroom scale inzvmablq6np03-g x8;4cgl eb + an elevator, it says your mas63m c +;04 bpxlgzlv qbe8-gans is 82 kg. What is the acceleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consists of a circular tube thatfq;ud)8j+f z xgh rou t.6vd2: will rotate about its center (like a tubular bicycle tiro d :)dhfu2vg8t q;f ju+.xrz6e) (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 in a vertical loop (Fig. 5–43). ns 4 :c w6620 pmtxm)/bwa+nl,xbmd wn
(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 th :i /y5*wzae j t.y)ozc-7zujne mass of a planet in terms of its radius r, the acceleration due to gravity at its surface a /a o.zyz)i: y-zjujne5t *w7cnd the gravitational constant G.

A plumb bob (a mass m hanging on a string) is deflected fec*g5b/az av)eed 0/crom the vertical by an anglgavce)eed5*b / c/0aze $\theta$ 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 coeffb+*mv0i *uam xicient of static friction is 0.30 (wet pavement), at what raum*xm+ bi v*a0nge of speeds can a car safely handle the curve?

  How long would a day be if the Earth were rotating so fast/.s5zhpoez 0ix*ulp zs :7 (-ln4m kzd that objects at the equator were apparently weightlesls5 d 4sz.u* n zepzh0/:kmx 7lpiz(o-s?    $\times 10^3$s

  Two equal-mass stars maintain a constant distance agi *9mlq2c b *zj1)hd.1cguqepart of 8.0 $\times 10^{10} \mathrm{~m} $ and rotate about a point midway between them at a rate of one revolution every 12.6 yr . (a) Why don't the two stars crash into one another due to the gravitational force between them?
(b) What must be the mass of each star?    $\times 10^{26}$

  A train traveling at a constant speed rounds a curve ozgw;xvyj.s*y9bu v/ 6f radius 235 m. A lamp suspended from the ceil9 vzj6.y yux;*sbvg/wing 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 been cybj 0 ej*bc63dki8s65civ)nvh;6 sxd laimed that a person would be crushed by the force of gravity on a planet the size of Jupiter ihke6bj cdd3bv*;iyn s 0j65c)vx s68since 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 Space Telescope deduced the presence of an extremg cv)pun/dq qq/o0/lz7(cf1h ely 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 measur)qzvc g0ld(chp7u// on1/qfqing 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:m3df6x ))o5 odrm pbn as it traverses the hill and valley shown in Fig. 5–45. Both the hill and valdx35) bon6m rfmdp ):oley have a radius of curvature R. (a) How do the normal forces acting on the car at A, B, and C compare? (Which is largest? Smallest?) Explain. (b) Where would the driver feel heaviest? Lightest? Explain. (c) How fast can the car go without losing contact with the road at A?

  The Navstar Global Positioning Systnhz *xyo8azk627h ;w xem (GPS) utilizes a group of 24 satellites 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 z6yo*xzwx h n2hk;8 a7few centimeters. The satellite orbits are distributed evenly around the Earth, with four satellites in each of six orbits, allowing continuous navigational “fixes.” The satellites orbit at an altitude of approximately 11,000 nautical miles [1 nautical ]. (a) Determine the speed of each satellite.    $\times 10^3$
(b) Determine the period of each satellite. =    hours

  The Near Earth Asteroid Rendezvo (.,giagz5vkcv ok6kb2 c;. dhus (NEAR), after traveling 2.1 billion km, is meant to orbit the asteroid Eros at a height of about 15 km .g.bic6 z5;c kv2k(gok v.d,ha 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 spaaj3wk33v + lqmce shuttle pursuing a satellite in need of repair. You are in a circular orbit of the same radius as the satellite (40wm3+ l jvk3a3q0 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 perio(6i;kk ui3xg81h m flbd of 3000 years. (a) What is its mean distance from the Sun?(6b ifxhg3lm 8i kk1;u   
(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 cou,j hzn/qb8bv)ko)3h hrx+ gh/ ld actually "feel" yourself gravitationally attracted to someone near you. Make reasonable assumptionsh qg3+x)rbnzhv/ohb ,kj/ h)8, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of . xfwg;jaw1x)the Milky Way Galaxy (Fig. 5-46) at a distance of about 30,000 light-years from the centegwx ;w) fa1.xjr $ \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 squarex5yq13 j m*re1tdoa6)antp- j 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 the bottom side o15n* ox3 tajjremdt1y qpa-)6f the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 kg b0//qk.e der5ed9aoe 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 second5f*oulk)a1f 8 *dcv wk 8 lw5kkv)oufc*ad f*1hand on your wrist watch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weight aro(q94cewop9 s qund in a circle below you on a 0.25-m piec(qp9q w49eoc se 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 designed sk:k wu7qez :b7y1q vr,8ldts, o that a car traveling at speeb7tdvqku 7,r:1ezk s,q 8lw:yd $ 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 trau5n.*xhndu3i in, the Acela, utilizes tilt of the cars when negotiating curves. The angle of tilt is adjusted so that the main force exerted on the passengers, to provide the centripetal acceleration, is the normal force. The passengers experience less friction force against the seat, thus feeling more co*3 5uhxinnd. umfortable. 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.$\approx$    $\times 10^2$