Sometimes people say that water is r,:ravog u v4q-emoved from clothes in a spin dryer by centrifugal force throwing the water ou:-,4 u vgarvqotward. What is wrong with this statement?

Will the acceleration of a car be the same when the car travec fopc8 5n 4hhl6h5da5ls around a sharp curve at a constant 60 km/h as when it travels around a gentle curve at tl64h5a5h5 cd8 fno cphhe same speed? Explain.

Suppose a car moves at constant speed along a hilly road. Wh wh0n-vzng1n0 ere 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 o0zhgn v0n -1nwf a hill?

Describe all the forces acting on a child riding a h*v-8bd 7wo k* w(1 xgu/w)nzh srn f6iwg0ir:horse on a merry-go-round. Which of these forces pro:6 wrwwbn8hks)i (w* zi1h7gv/d u gr0o*nfx -vides the centripetal acceleration of the child?

A bucket of water can be whirled in a vertical circle wi,8 z7ios4mgr)0tb c0y gqyj8kthout the water spilling out, even at the top07og kiy4mbg8 y0jzt)8cq, rs of the circle when the bucket is upside down. Explain.

How many “accelerators” q,5m4atkjf8pp8 6ab jdo you 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 themjj, 4qftpk p86 8baa5y? What accelerations do they produce?

A child on a sled comes flyiny;0 dxab3gk-+* kwko+tq 6 xvug over the crest of a small hill, as shown in Fig. 5–31. His sled does not leave the ground (he does not achieve “air”), but he feels the normal force between his chest and the sled decrease as he goes over the hill. Explain this decrease usinx- d +;aqwvok0t*g3+ k6 ykxubg Newton’s second law.

Why do bicycle riders lean inward when rounding a cuc b.+ 7nl9xklrrve at high speed?

Why do airplanes bank when they turn? How would6/p/bwca4jabw .e)a h you compute the banking angle given itcwh /abw)b6a pe 4/j.as speed and radius of the turn?

A girl is whirling a ball on a string around her head in a horizontal plane. Sha 27zecdx( r op73* v0z)dsrope 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 pp )z(7a*xzr7 0dr dce3os2v othe string?

Does an apple exert a gravitational force on the Earth? If so, how large a5h ucf8m1 fcgi d;f92s forceff c c9s251idugf;m8 h? Consider an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, in wh;q gg0jz9 js+sat ways would the Moon’s orbit be dg0sjs9zj g;+q ifferent?

Which pulls harder gravitationally, the Earth on the M+fv+mafq9nhz- p/:l2 npkv8 foon, or the Moon on the Earthn:f+fl8-vv2 qf h9z/pn ka+mp? Which accelerates more?

The Sun's gravitational pu) e4s;:bj wubi3 et5gyll 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 oe5:y gejs43tb b)uwi; f the Earth to the other.]

Will an object weigh more at the equator or at the poles? What two effects are aip1hg1)6.xk;uder0 mdvad 0n t work? Do they oppose each othev;eah u pd01d.6kgd n0)im1 rxr?

The gravitational force on the Moon due to the Earth )1 ey pq5an0r )mwqzgq-qw2-h is only about half the force on the Moon due to the Sun. Why iqwgmzrh2pn0q qq)-e -a w)1y 5sn’t the Moon pulled away from the Earth?

Is the centripetal acceleration of Mars /jxlrj9wpyg: ph-s9t f*sn*9;8dc6ykaqr. iin its orbit around the Sun larger or smallerrshw8i. j;y9n9*pcj6r a9 tk/x *pg: dflsyq - than the centripetal acceleration of the Earth?

Would it require less speed t rs bbcuxxm;4 mw9)z5*o launch a satellite (a) toward the east or (b) toward the west? Consider the Earth’s rotation direc;9 4crxxbbuz5smwm * )tion.

When will your apparent z6tksj 3 1sb86s ikv8obt7 m5mweight be the greatest, as measured by a scale in a moving elevator: when the elevator (a) 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 68btj ktmozss 6 8km3v51bis7 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 adxa6 2)j3ehm4emoh zcm)1 ck)s versely affected by weightlessness. One way to simulate gravity is to shape the spaceship like a cylindrical shell that rotates, with the astronauts walking on the inside surface (Fig. 5–32). Explain how this simuh13)sech 6mz k xj emcm2)oa4)lates 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 “appa6-/c l+v-jwgy4 twiies ud 7.qrent weightlessness” between stepc4g-uj/ .yw qd7i- itvsw+el6s.

The Earth moves faster in its orbit around the Sun in kz78t o11v 8ondgkt+ qJanuary 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 relative to thed1ntg+k 1o q z8o8k7vt plane of its orbit.]

The mass of Pluto was not known until it was discoveredje rna0* wepn +8y qd8;ah)6xioy*vl+ to have a moon. Explain how this di;ne+j 8l*+)ya0qr8pd *oiyvx aw n6ehscovery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth is much larger than the Moon'efj(c hrbt 1-4k1rr+ ul l( g(1zhgm/rs. Yet the Moon's is mainly responsible for the tides. Explainr gkt r gj/h1llf(hm br(u-erz4 1(c+1. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]A child sitting 1.10 m from the center of a merry-go-round moves with a speed of 1.25 $\mathrm{~m} / \mathrm{s}$ . Calculate (a) the centripetal acceleration of the child,   
(b) the net horizontal force exerted on the child ( mass =25.0 $\mathrm{~kg}$)   

  A jet plane traveling ,8s:wh qkum o-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 acce i*x3)u7 qr2i2en(xu jqwi7cznwo7 q4 leration of the Earth in its orbit around the Sun, and the net(7 77u4i3jn wco)xw2niiezxq qq*r 2u force exerted 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 N is exerted on a 2.0-kg discus as it rotq( xn4 0a;ojf*uh57)l dpgkoyates uniformly in a horizont;la7 o y()uk0 of4hqdnpg5jx* al circle (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 thef12er*ppxde- .tw)yj Earth's surface, and circles the Earth about once every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms ot*j p1)ew-fxe ydp2.rf g , the gravitational acceleration at the Earth's surface. $\approx$    g

  What is the magnitude of the acceleration of a speck of clay on the edge of a at5g+vnd17 p+ qzfqb) potter’s wheel turning at 45 rpm (revolutions per minute) if the wheel’s diametaq++d) v 1tzp7q ng5fber is 32 cm?   

  A ball on the end of a (d0 knf0 /d dqoy(xmu;string is revolved at a uniform rate in a vertical circle of radius 72.0 cm , as shown in Fig. 5-33. If its spe;/f0mxnyo0 k(du(q dded 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 1050-kg 89:wwpva )fi zcar can round a turn of radius 77 m on a flat road if the coefficient of static friction between tires and road :f8 9v)aiz pwwis 0.80? Is this result independent of the mass of the car?   

  How large must the coefficient of static friction be bec8*xt8 xrlf94gdy mm6tween the tires and the road if a car is to round a level curve of radius 85 m at a 8m 49tl6r8x*m f dcxgyspeed of 95km/h ?   

  A device for training astronauts and jet fighter pilots is designed to rotateh jjstiwp(3e n6-( t3z a trainee in a horizo36 p3zt( htjsij(wn-e ntal circle 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 rotating turntable y+hvq57*lg:g4 hi c hwof 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 frictiq h5*+ :gcighy74vhlwon between the coin and the turntable?   

  At what minimum speed must a roller coaster be travelt ww,u+qqy(2y96diuymli7 2hing when upside down at the top myqhw,+9ti(iuu2d7 6y2ylqw 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) crosses the rounded top g2c4t9axcq v dq g)qa-/xk;n8of a d;ta xq )k8 c2a9qn4-/g gvxqchill (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 mink*fiom: /fw 3zqs g(th/elq yc/0,:luge v3k-ute would a 15-m-diameter Ferris wheel need to make for the passengers to feel “weightless” kuv mwfkhis*0etcgl/ 3y- ::o3, fle(qz/q g/at the topmost point?    rpm

  A bucket of mass 2.00 kg is wnf i, 6 o3vvmjlm0+09jf6shuwhirled in a vertical circle of radius 1.10 m. At the lowes6jwlij00vf 96fh,o mv u +nm3st 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 r q+jm h -*y8bp2dunfrq w58h*t0n5 tbzotate if a particle 9.00 cm from the axis of rotation is to 05-2f5h d8rn*qtyt 8uw b*qm +pjzhn bexperience an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people m fja*;5a,ack are rotated in a cylindrically walled "room." (See Fig. 5-35.)a*,k af;ac5 mj
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 taabjs w2aa:hzc-pa ilh3** f5- bletop, and is held in this orbit by a light cord connected to a dangling block (mass m ) through a central ho z:h-jca3- aa f2la*wipb* hs5le as shown in Fig. 5-
36. Show that the speed of the puck is given by

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

  Redo Example 5-3 , precisely this time, by not ignorin)x- unv 0ymjo3g the weight of the ball which revolves on a string 0.600 m long. In particular, find the magnitudnyx)v03 uo-j me 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; pbw6az zl .7f 5msfw,6y*cdnhh j2+c a car traveling 75 $\mathrm{~km} / \mathrm{h}$ , what must be the coefficient of static friction for a car not to skid when traveling at 95 $\mathrm{~km} / \mathrm{h}$ ?   

  A 1200$-\mathrm{kg}$ car rounds a curve of radius 67 m banked at an angle of 12$^{\circ}$ . If the car is traveling at 95$ \mathrm{~km} / \mathrm{h}$ , will a friction force be required? If so, how much and in what direction? $\approx$    down the plane

Two blocks, of masses n2obyq4ten 65 $ m_{1}$ and $m_{2}$ , are connected to each other and to a central post by cords as shown in Fig. 5-37. They rotate about the post at a frequency f (revolutions per second) on a frictionless horizontal surface at distances $r_{1}$ and $r_{2}$ from the post. Derive an algebraic expression for the tension in each segment of the cord.

  A pilot performs an evasive maneuver by diving vertic d27uv *utp)yp71lz zeally 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 a9rx+-9n hp teund centripetal components of the net force exerted on the car (by the groh9e rpx-9 +untund) 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 uniformly from the pit arelqx/(nkp ey w)(t7db)a, going from rest to 320km/h in a semicircular arc with a radius of 220 m. Determine the tangentia nq/by(tp w7) (l)xedkl 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 m . At a particular itoy5nrk0 h j/rb() ka7nstant, its acceleration is 1 r(0rh75 )k/boj kanty.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 d0yvt4ozw : o412,800 km (2 Earth radii) above the Earth’s surfvo o tdwyz404:ace if its mass is 1350 kg.   

  At the surface of a certain planet, the gravitational acce.,af,ju biehs scmyc2+l * g;+wyk,.t leration g has a magnituuc*w,ya.kgice,bj+2;+ lmf t y., hssde 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.. rkb( nil-rr6 The Moon's radius is 1.74b- rr( klrn.6i $\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has a radiqd7f,- fzfrh+ 1g7r uaus 1.5 times that of Earth, but has the same mass. What is the acceleration due to gravity near its surf f7rg-qu 1 fd7,+hazrface?   

  A hypothetical planet has a mass 1.66 times that of Ea3dlznxc2 jv7i:7fg/ b rth, but the same radius. What is g near its surface? gi/72lfdcz :xjb n73v  

  Two objects attract each other gravitaij *.up(h7q 0kp7y 2a+ qzzipationally 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 ofyu2giuz4k. t3 gravity, at (a) 3200 mk3ziy.u42u g t    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from t 1322y+ srtcos+dydu/t; lezyhe Earth’s center to a point outside the Earth where thegravitational acceleration 3y y2 1+dut+le;so/styzd cr 2due to the Earth is $\frac{1}{10}$ of its value atthe Earth’s surface?   $\times10^7$

  A certain neutron star has five uc9/1t ca,kpjtimes the mass of our Sun packed into a sphere about 10 km in radius.c 1kcuj,/p9 ta Estimate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once was an average sq 6a49kjexxbkmw g ey: 11;,86cwyiek tar like our Sun but is now in the last stage of its evolution, isw 1we4mqk1k;ei:6 yy,xjxk 98 bc e6ag the size of our Moon but has the mass of our Sun. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronauts feel weightlessc 8ff5v2kork7(xjbu 5 while orbiting in the space shuttle. Your friends respond that they thoug kcbxv528jk5o7ruf( fht gravity was just a lot weaker up there. Convince them and yourself that it isn’t so by calculating the acceleration of gravity 250 km above the Earth’s surface in terms of g.   

  Four 9.5-kg spheres are located at the corners bc :nsp/ 0)woh*l*i byof a square of side 0.60 m. Calculate the magnitude and direction of the total gravitational force e*hyi l) w:cos/ 0bp*bnxerted 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 onuj0b i 5m;pp/(zah m-5hha d us2y.o mg9zs68i the same side of the Sun. Cal6sm8 dazhua -oi5 j;( 2m5/pbp.i0z g mushy9hculate the total force on the 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 surface bs3 g*7yzb9 s 7a+lhunof Mars is 0.38 of what it is on Earth, and that Mars’ radius is 3400 km, determine the masl shny77abg+3z9b *u ss of Mars.    $\times 10^{23}$

  Determine the mass of the Sun uzu.2s*oh zok+sing the known value for the period of the Earth and i+.skz zuo2*oh ts distance from the Sun. [Note: Compare your answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite moving in a stable c;d-l2 j-geb dgircular orbit about the Earth at a height of 2g;dlebj-dg - 3600 km.    $\times 10^3$

  The space shuttle releases a satellite into a k s3, pm+*fo/ed5mgz n x*;rjwcircular orbit 650 km above the Earth. How fast mm5+;e gzs,jn *wxfkpdm*ro/ 3ust the shuttle be moving (relative to Earth) when the release occurs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occup5(+u8 pluq qseants are to experience simulated gravity of 0.60 g? Assume the spaceship’s diameter is 3u8 elq5u+(p sq2 m, and give your answer as the time needed for one revolution. (See Question 21, Fig 5–32.)   

  Determine the time it takes for a satellite to orbit the Ea 7)vjs8;qfwr9dmj+bl rth in a circular “near-Earth” orbit.wjqlmj7) ;9sv +frd8 b 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?    s ~    min

  At what horizontal velocity would a satellite have to 2 d5d v-r z-6qlg6js:c,lvykxbe launched from the top of Mt. Everest6vjlzvq-rlcgdsd2:-yx6 k 5 , to be placed in a circular orbit around the Earth?   

  During an Apollo lunar landing mission, the command module continued to ori4 gi9t *n/(iwmja3fi wo/qa 8bit the q 3i*ni8a/wmta /fi4o j(g 9iwMoon 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 chunks of ice 4uw; j.oau *pxthat orbit the planet. The inner radius of the rings ipuwjx;.*u4 ao s 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 1f;)yv sazw5 l kib;l175.5 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) a;b7k5 lsv ;wfiazy l1)t the bottom?
(a)    (b)   

  What is the apparent weight of a 75-kg asbf55qs w)l1jqgus/ aj(pp41qvy:(jn f(vmf9tronaut 4200 km from the center of the Earth's Moon f v/q1 qms(p9swlfg4ja5b)j1nfj( qy p ( :vu5in a space vehicle (a) moving at constant velocity,     ----待选择----towards the Moonaway from the Moon 
(b) accelerating toward the Moon at 2.9 $\mathrm{~m} / \mathrm{s}^{2}$ ?     ----待选择----towards the Moonaway from the Moon 

State the "direction" in each case.

  Suppose that a binary-star system consi2:ni8i x0uqw8usb j p/sts 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 midway between them. What is the suw:j 88pxi b n/0q2iumass of each?    $\times 10^{29}$

  What will a spring scale read for thp e;, bil/oo)qe weight of a 55-kg woman in an elevator thao,iq/;boepl )t moves
(a) upward with constant speed of 6.0m./s   
(b) downward with constant speed of 6.0m./s   
(c) upward with acceleration of 0.33 g,   
(d) downward with acceleration 0.33 g   
(e) in free fall?   

  A 17.0-kg monkey hangs from a cord suspended from the ceixb.njhm-k)/gyuale y0 8( x b+o a2:njling of an elevator. The cord can withstand a tension of 220 N and breaks as the elevator accelerates. What w j2 oa 8)lg+hy/y.(njbx-bmnue:kx0 aas the elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite-v jy0 c1tpyda9t 5/iw orbits very near the surface of a planet with period T , the density (mass/volume) of the py 19wty/5jct -0vdapilanet 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 determine the pe) ixn p 9ao.e:)ahy;guo:ag( sriod of an artificial satellite orbiting very near the Earth’s surface. es pi: a)ghag o9yx :ona;).(u   s

  The asteroid Icarus, though only a few hundred meters across, b6x7 8 y.qn/li36z t tqo9msclorbits the Sun like the planets. Its period is 41cq9.z/l m t36oy b 68t7qnxisl0 d. What is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average distamqf,k58c:z kgchs3 .;j(tiynnce 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. 6e j-.k+jelggc0* ptomkuu/8It 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 jt*ekck8el0+gmg./jp -6u uo 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 r rc,ioqo7/6r center of the Galaxy $\left(M_{G} \approx 4 \times 10^{41} \mathrm{~kg}\right)$ at a distance of about 3 $\times 10^{4}$ light-years $\left(1 \mathrm{ly}=3 \times 10^{8} \mathrm{~m} / \mathrm{s} \times 3.16 \times 10^{7} \mathrm{~s} / \mathrm{y} \times 1 y\right)$ . What is the period of our orbital motion about the center of the Galaxy? $\approx$   $\times 10^8$

  Table 5–3 gives the mass, period, and mean distance for the f0ht pl3i 6(be6r op*7:vjcirnour largest moons of Jupiter (those discoveribj r66v*phr t0peo3nc7i l:( ed 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 kp af:a-.neag6)5s wnm:qv l s+nown period and distance of the Moon. w:p-gaa5qe sn :6.) + fmanlvs   $\times 10^{24}$

  Determine the mean distance from Jupiter for each of J3avx,r 9qrc,r 85kwkzbgn9 a 5upiter’s moons, using Kepler’s third law. Use the d5kk,ncbr,53gwra 9rax vq9 8zistance 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 consistq(n-6/dp k-h-sgedydd 2xj 1ts of many fragments (which some space scientists think came from a planet that on2t-yd-jds6(/nq dg 1 kxpdhe-ce 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" i5;b(zxbr, p pnsi3vid bm:f8. n the form of a band completely encircling a sun (Fig. 5-40). The inhabitants live on the inside surface (where it is always noon). Imagine that this sun is exactly like our own, that the distance to the band is the same as the Earth-Sun distance (to make the climate temperate( xbrb85 d nfipib,z v3.s:;mp), 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 cons3( v2x+ vbfro l7(an arc from a hanging vine (Fig. 5–41). If his arms are capable of exerting a force of 1400 N on the vine, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 80 kg(xn 7 c2vs+f(l o3bovr, and the vine is 5.5 m long.   

  How far above the Earth’s surface will the accelerati00, r r(unfcnuon of gravity be half what it is on the surfacenru,c00 rfn(u?    $\times 10^6$

  On an ice rink, two skaters of eq 903jkez i)t x)x3y* vmbj/sliual mass grab hands and spin in a mutual circle once every 2.5 s. If we assume their arms are each 0.80 m long and their individual masses are 60.0 kg, how hard are they pulling on one an l9t i*kxj b30jyix)esm)/z3vother?    $\times 10^2$

  Because the Earth rotates once per day, th7saf9uqhmg w98shjjw4 w8r*td 4h .c;e apparent acceleration of gravity at the equator is slightly less than it would be if the Earth didn’t 9dgtr4 jahwqwwm7h; s f8h *j4uc.98 srotate. Estimate the magnitude of this effect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from the Earth will a spacecraftw- 3gb9qbtma4 traveling directly from the Earth to the Moon experience zero net force because the Earth and Moon pull with equal and opposite fob qa b3w-t49gmrces?    $\times 10^8$

  You know your mass is 65 kg, but when you stand o njmczt5-*x1*oe l4hqn a bathroom scale in an elevator, it says your mass is 82 kg. What is the acceleration c -1q*zlt*h xjn5ome4 of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consivc w )nizq-1v4sts of 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 (revol4v ivn c-)q1zwutions 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 (Fi y h,.qye1(ho)xumf n/g. 5–43).
(a) If the jet is moving at a speed of at the lowest point of the loop, determine the minimum radius of the circle so that the centripetal acceleration at the lowest point does not exceed 6.0 g’s.    $\times 10^3$
(b) Calculate the 78-kg pilot’s effective weight (the force with which the seat pushes up on him) at the bottom of the circle    $\times 10^3$
(c) at the top of the circle (assume the same speed).
   $\times 10^3$

Derive a formula for the mass of a planet in terms of its radius r, the my/ wde-+ 9 nxgwiuoh6(2c1kx acceleration due to gravity atkx92 wd-ey h1ogi+ u/c(wxn6m its surface and the gravitational constant G.

A plumb bob (a mass m hanging on a string) is rho-hwq01g 7xc l lq:qt/36xn deflected from the vertical by an an7ow03ch qrxq ln/tq6-:1xgl hgle $\theta$ 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 banke,doabsps;, aa lh.,;uzzv (-id for a design speed of If the coefficient of static friction is 0.30 (wet pavement), a-.sau(,z p vsh, ,z;ba;lao dit what range of speeds can a car safely handle the curve?

  How long would a day berz0/7 igjl ch7 if the Earth were rotating so fast that objects at the equator were apparentlc0 7ljrgi/ zh7y weightless?    $\times 10^3$s

  Two equal-mass stars maintain arwo x)/.f 9taw)t4jqq constant distance apart of 8.0 $\times 10^{10} \mathrm{~m} $ and rotate about a point midway between them at a rate of one revolution every 12.6 yr . (a) Why don't the two stars crash into one another due to the gravitational force between them?
(b) What must be the mass of each star?    $\times 10^{26}$

  A train traveling at a cons0 1qzutn86 1js 7zo d*3abtajutant speed rounds a curve of radius 235 m. A lamp suspended from the ceiling swings out tq30n1 87zujtujabat* z6dso1 o 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 claime)4:h n,lt 2mqm (tfqt88yykjqd 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 t (4, qq8qyn t:8)mmj2fkhylt 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 Hubbl,n ( :acbmd6sje Space 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 t :,ancdb m(s6jhe 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 sp6ru2,7 g /w .yg- fethe4c8so 2cothnpeed as it traverses the hill and valley shown in Fig. 5–45. Both the hill and valley have arh2y g .6 p74nefetwgo-/sct o28cu ,h 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 Systemt(w*lmi cege 63p +.am (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 few centimeters. The satellite orbits are distributed evenly around the Earth, with four satellites in each of six orbits, allowing continuous navigational “fixes.” The satellites orbit at an altitude of approximam*w ic+g epe.t3 m(la6tely 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 R,j nbimf*r b pc++ee3/endezvous (NEAR), after traveling 2.1 billion km, is meanm ei++3r/b*cefpb,jn t to orbit the asteroid Eros at a height of about 15 km . Eros is roughly 40 $\mathrm{~km} \times 6 \mathrm{~km} \times 6 \mathrm{~km}$ . Assume Eros has a density (mass/volume) of about 2.3 $\times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ . (a) What will be the period of NEAR as it orbits Eros? $\approx$    $\times 10^4$sec
(b) If Eros were a sphere with the same mass and density, what would its radius be? $\approx$    $\times 10^3$
(c) What would g be at the surface of this spherical Eros?$\approx$    $\times 10^{-3}$

  You are an astronaut in the space shuttley93ury, ptyd3 pursuing a satellite in need of repair. You are in a circular orbit of the same radius as the satelliy tydu3,p39yr te (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 h c,jg o;em)g3,cps9dh tx rzt8)1t j;uas a period of 3000 years. (a) What is its mean distance from s ,9)mo;zetj1 ; rhcx,dgctujg)8p t3 the Sun?   
(b) At its closest approach, the comet is about 1 A.U. from the Sun ( from Earth to the Sun). What is the farthest distance?   
(c) What is the ratio of the speed at the closest point to the speed at the farthest point? [Hint: Use Kepler’s second law and estimate areas by a triangle (as in Fig. 5–29, but smaller distance travelled; see also Hint for Problem 59).]   

  Estimate what the value of G would need to be if you could actuaya.lv5u 5ptwv .vp03rlly "feel" yourself gravitationally att3 v l..p0yvuwp55avt rracted 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. 5-46) ckem.yxnb / ,axwei8)v3da-84vxq my7vi1p . at a distance of aby xeyp/ a8i,bmk8icnew .7dvaxvm3 x)-1q .v4 out 30,000 light-years from the center $ \left(1 \mathrm{ly}=9.5 \times 10^{15} \mathrm{~m}\right)$ . If it takes about 200 million years to make one rotation, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun $\left(2 \times 10^{30} \mathrm{~kg}\right)$ , how many stars would there be in our Galaxy?$\approx$    $\times 10^{11}$

  Four 1.0-kg masses are located at the corners of a square 0.5vb/h2o qt5c9 )bwnrb-*m( iuo 0 m on each side. Find the ob*q/n9w-r(u5c m oivb2) thb magnitude and direction of the gravitational force on a fifth 1.0-kg mass placed at the midpoint of the bottom side of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 kg orbits the Earth vi-x2n le7+hp$ \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 r kul5qly ut5lu,cf8 bz3;(e2 sweep second hand on your wre3;8 ( ll5 uyzf5ul,rckubt2 qist watch?    $\times 10^{-4}$

  While fishing, you get bored andzwfar*76em k0 start to swing a sinker weight around in a circle below youk wme 0zr76*fa 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 designed so tht(a60ricc1 9:tmiragat a car traveling at speed1iac (:c 69rgitamrt 0 $ 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 the ca;0zf* l1oor/g 8u6x gte qfo/b,j(tmars 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 seatfeo; *zg mo a60rj,tu bfo1 /lg/8(qtx, 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.$\approx$    $\times 10^2$