Sometimes people say that water is removed from clothes in a spinz8sex)iw-5fz dryer by centrifugal )xzfzew-s58 iforce throwing the water outward. What is wrong with this statement?

Will the acceleration of a car be the same when the car travels around a sha2d t, mi-n*timrp cmn* id t,t-2imurve at a constant 60 km/h as when it travels around a gentle curve at the same speed? Explain.

Suppose a car moves at constant speed alongu kgftzl 4 jjeu58am;f :)/lx+ a hilly road. Where does the car exert the greatest and least forces on the road: (a) at the top of a hillex j; m:4ff)uk8lt5 ujgla z+/, (b) at a dip between two hills, (c) on a level stretch near the bottom of a hill?

Describe all the forces acting on a cs/czlm5: l;l khild riding a horse on a merry-go-round. Whi slml l;kc/5:zch of these forces provides the centripetal acceleration of the child?

A bucket of water can be whirled in a q86:1 uq)dr;y i f,cigf0+oik6l ltkz vertical circle without the water spilling out, even 6u1 qcki,f i:k) ;l d0qz6gitrloy+f8at the top of the circle when the bucket is upside down. Explain.

How many “acceleratoqe4 fhn21r(e +crxx.ers” do you have in your car? There are at least three controls in the car f 4 r (2e+qx.cenexhr1which can be used to cause the car to accelerate. What are they? What accelerations do they produce?

A child on a sled comes flying over the crest of a small hill, as shown ij)r b/n(r7t ees jag 2,c6u-wyn Fig. 5–31. His sled does not leave the ground (he does not achieve “air”), but he feels the normal force betweu6)wjyen,- cgte2 b /r7(srjaen 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 inwardna (lg* a(gkr2 when rounding a curve at high speed?

Why do airplanes bank w40y;y ai9t pxa,t,iizhen they turn? How would you compute the banking angle given it t9a t,yiz ,pxa04yii;s speed and radius of the turn?

A girl is whirling a ball on a string around her head in a horizonta+xxqg cjro8 (1l plane. She wants to let go at precisely the right time so t x8g+q1x cro(jhat 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 gravitationahg s4.-;ff6opq*5 x w 1 iwe ;ayh+rm9ffqw3kml force on the Earth? If so, how large a force? Considexprmsfyw 9qf gf .3;;ief+ -*khhqo5amw641wr an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, in what ways wo hd )vsukq *ksdo:2 +,sh0m/yb; ;whfauld the Moon’s orbit be different2kao) wsvyb;dqhs +0s:/um;d,kf h h *?

Which pulls harder gravitationally, the Earth on the Moon, or thxrg r9evh fdd-4*-9cge Moon on the Earth? Whicde- * rh9r-dfgc4gv x9h accelerates more?

The Sun's gravitational pull on the Earth is much larger than the Moo+pekj.a2owokig5 u..y o;k:dn's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in grae.i2 jo.p: + .ko5udow;kg ykavitational pull from one side of the Earth to the other.]

Will an object weigh more at the equatvp 49b neekb ,y/leg.jep7e7wm ,:t6hor or at the poles? What two effects are at work? Do they oppose each other? p7k/ 6wnjee79:b ,gp4bhee.yt ml,ve

The gravitational force on the Moon due to the Earth is only about half the ft/w-*h928stgrrfl n iorce on the Moon due to the Sun. Why isn’t the Mot8s9*n if2t wrl/h g-ron pulled away from the Earth?

Is the centripetal acc,-wetva5x*c celeration of Mars in its orbit around the Sun larger or smaller thaa*5exc vc w-,tn the centripetal acceleration of the Earth?

Would it require less speed to launc:dl2x h1pb h rnma,20ph a satellite (a) toward the east or (b) toward the west? Consider the Earth’s rotation dir: l0aphb 1 2mxpr,hnd2ection.

When will your apparyl.k40fmx w rjtw16 qf/if 3;c(mt3qoent weight 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 leaf4lr 1 cj oxm0 f(mit3.qy6/qtkw;wf3st? 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 adverseb;mn (l x c+yn9me2rf4ly 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 simulates gravity. Consider (a) how objects fall, (b) the force we feel on our feet, and (c) any or(e; yf94x mnlb+m c2nther aspects of gravity you can think of.

Explain how a runner experiences “free fall” or “apparen.j7v2n,fn w1 7pmp oynt weightlessness” between steps7,v 7 fyw21onn. pjmpn.

The Earth moves faster in its orbit around d 4h):)fmtm-hg:ybdz*nefa6the Sun in January than in July. Is the Earth closer to the Sun in January, or in July? Explain. [Note: This is not much of a factor in producing the seasons — the main factor is the tilt of the Earth’s axis re:gn)* dehh) af4zmfybtm 6 -:dlative to the plane of its orbit.]

The mass of Pluto was not known until it was discovered to have a moon. Es (z jd1.itj:hxplain how this discovery enabled an estimate of : (hi1.tj jszdPluto’s mass.

  The Sun's gravitational pull on the Earth is much largez.9zitgk1 k .sr 8hwe0nt2(el mi cpk: q+1gr0r 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 center of a merry-go-round moves withh.sl krrzwi1008+n :2emqe gk9(itt1kg cz p. 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 198mp3zab n+oy6i x;u,/h 3nwb)w0 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 accelera jltqdmvc.9ch9 4b2l 7tion of the Earth in its orbit around the Sun, and the net force exerted on the Earth. What exerts this fortc dj9lb.27 ml9vq4h cce 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 exerte 6 svlk4vfxi8-d on a 2.0-kg discus as it rotates uniformly in a horizontal circle (at arm4 vl -8kfxsi6v’s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is in orbit 400 km from the Earf1us;yz 0xws e 1/ej-;wy.xo uth's surface, and circles the Earth about once evfujso/ 1sew;z;-w.y eyxu 1x 0ery 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of g , the gravitational acceleration at the Earth's surface. $\approx$    g

  What is the magnitudxe f;a xltg -+/ceqr1pn5u+ ( s49zs6swtabf;e of the acceleration of a speck of clay on the edge of a potter’s wheel turning at 45 rpm (revolutiog/ ; 1;l+f6esn(fx-4 r5t qes az bputa+9scwxns per minute) if the wheel’s diameter is 32 cm?   

  A ball on the end of a(lmnv isqh/9x . noqu x6h+51x string is revolved at a uniform rate in a vertical circle of radius 72.0 cm , as shown in Fig. 5-33. If its speed is 4.9qou 1i.lmx+ v/hnqhxsn 6x 5(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 car can -1k1gos , xtoxround a turn of radius 77 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 ks-oxxg 1, to1of the car?   

  How large must the coefficient of static friction be betwext3m6tfjr k1 v4 qoy548x/zrw en the tires and the road if a car is to round a level curve of radius 85 m at a speed of z x rw8f14 ttxr 3j/o5ym4kvq695km/h ?   

  A device for training astront sk s1i5ky4py85ozv qy/*x 5jauts and jet fighter pilots is designed to rotate a trainee in a horizontal 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 rot y 4/8ki*p555 oyvjzytskqxs1ating?    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 rotatingwx:h0wq cm 0-bsp/t.c lxf6h7 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:s/ h.t0 xq0-w fw h6xm7cpclb between the coin and the turntable?   

  At what minimum speed must a roller coaster be traveling when upsidef uy fco 9umhws 3ac7u190h1+j down at the top of a cij yc+30 ow uhusu1f97c 9f1hmarcle
(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 2n : txr j4(dz /+8xppx6flxltof a/6x xlrjtxn4+8( pz:l p 2dtxf hill (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 lj 17z1ycb+ r7cq zq/hminute would a 15-m-diameter Ferris wheel need to make for the passengl hrqj z+y/bc71z1cq7ers to feel “weightless” at the topmost point?    rpm

  A bucket of mass 2.00 kg is whirled inpfi8g jq66 )jww,tl)g a vertical circle of radius 1.10 m. At the lowest point of its motion the tension in the rope supportij,pggwf) 6i)l t q8j6wng 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 partiq+u)jggu k5,3k8z tbo3i6g4erqkr ql he6n,6 cle 9.00 cm from the axis of rotation is to experience an acceleration of q63b u3k un8gqike+ hr6e 4l6tzo,qrgjkg5,) 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotatedw;d)mx/a j4hu in a cylindrically walled "room." (See Fig. 5-3a uw)mx/dj; h45.)
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 xc9 ,9eq0jen bx(w-zd2n *qigin a circle on a frictionless air-hockey tabletopznxbd n *eijc( x,-we q0q2g99, 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. 5-
36. Show that the speed of the puck is given by

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

  Redo Example 5-3 , precisely this time, by not ignoring the weigh ,9xpe6pu0 )z ghc sok5+kwhu0t of the ball which revolves on a sc)e65,wkou px gh0uk 09+ hpzstring 0.600 m long. In particular, find the magnitude of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ , and the angle it makes with the horizontal. [Hint: Set the horizontal component of $\overrightarrow{\mathrm{F}}_{\mathrm{T}}$ equal to m $a_{\mathrm{R}}$ ; also, since there is no vertical motion, what can you say about the vertical component of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ ?] $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ =    the angle =   

  If a curve with a radius of 88 m is perfxyp/so m9qk1* ectly banked for a car traveling 75 $\mathrm{~km} / \mathrm{h}$ , what must be the coefficient of static friction for a car not to skid when traveling at 95 $\mathrm{~km} / \mathrm{h}$ ?   

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

Two blocks, of massej,8wm 6az hc(us $ 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 vertically at h(h,i f 7zjtcg8hwe;6310 $\mathrm{~m} / \mathrm{s}$ . If he can withstand an acceleration of 9.0 g 's without blacking out, at what altitude must he begin to pull out of the dive to avoid crashing into the sea?   

  Determine the tangential and centripetal c6a4np *g3t/ :wfhkfz domponents of the net force exerted on the car (by the ground) in Example 5-8 when its speed is 1z4:f pg/nwhd6af t k*35$ \mathrm{~m} / \mathrm{s}$ . The car's mass is 1100 kg . F$_{tan}$ =    F$_R$ =   

  A car at the Indianapolis 500 accelerates uniki( 9rh(m8 va(4mymmw (0l imvformly from the pit area, going from rest to 320km/h in a semicircular arc with a radius of 220 m. Determine the tangential and radial acceleration of the car when it is halfway thr(4am 0im 8mhm((9vmil(vy w krough 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 pb8c a5da(.zex1wsc9ww eau42articular instant, its asaw.uew1a 8bc9x 42(ac edz 5wcceleration 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 spacecra,xmgy; hw+ 9 t2zl,nsuft 12,800 km (2 Earth radii) above wnyxg,ht 2 ;mul9 z,+sthe Earth’s surface if its mass is 1350 kg.   

  At the surface of a certain planet, the gravitatif h 4y0yn ut9i6dt37anonal acceleration g has a magnitude of 12 m/s aiu7 tyn fy0dt3h4n96$^2$ 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 radius is kcyp 9xmz8:x6 1.74c:9x 6zp8m yxk $\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has a rxs+c sa )2khx ,d)1ggvadius 1.5 times that of Earth, but has the same mass. What is the acceleration due to gravity near it)+d 1chsv2gxg)k ,asxs surface?   

  A hypothetical planet hag9zpmky v b 8 7b-9(lu z*n8;ztlxt9kz4w;hcxs a mass 1.66 times that of Earth, but the same radius. What is g near its surfl vpz wbz9-x9 *nt( c8h zy;98gxzbkl7u;t mk4ace?   

  Two objects attract each other gravitationally with a force of 2- +naxgvquc9eh sv t/ohd19;:.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 kbtm w* ;+wj1jvalue of g , the acceleration of gravity, at (a) 3200 m w1b tkwm ;*j+j   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside thexrp )g i)d3zyfh 6yd31 Earth where theg6ydd xp13 gizf)3) hyrravitational 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 (f,a 2bdf3cgzpacked into a sphere abobc3(d,aff gz 2ut 10 km in radius. Estimate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once was an average star like our Sun bhem;.3 fef.nl ut is now in the last stage of its evolution, is ;ml h ef.fn3.ethe 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 weightless while f5rb8s*c6 ghv g b)m*vcvwo5t+, l y/xorbiting in the space shuttle. Your friends respond that they thought gravity was just a lot weaker up * y56g+f cl/c5, *tmx) hvs bvrowgb8vthere. 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 corners2tg4lg 9z-;dqd1dcex of a square of side 0.60 m. Calculate the magd94ez2d 1 -dtgqx;lcgnitude and direction of the total gravitational force exerted on one sphere by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred years most of the5 e, 5asjrfgu1r 9mtxdd. z*,f planets line up on the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming al.rx gta5ms zd5u* r9efd 1j,f,l 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 of Mars is 0.38 ofy5,ej 7f:yih,lt cm -v what it is on Earth, and that m7e,yl-ht f,vj: c5 iyMars’ radius is 3400 km, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the*rqixxh t66drm5gfw2p i /a * 58(dodq Sun using the known value for the period of the Earth and its distance from the Sun. [Note: Compare your answer to that obtained using Keplex6h dm wtr*5(*f6x8qo arid2g /5 iqdpr’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite moving in a stable circular orbi d02hw:4kp cima3 f2nct about the Earth at a height ah4dc:mfcn2ik3w 02p of 3600 km.    $\times 10^3$

  The space shuttle releasesyb 7py:l odht)x3 3b11owsz h9 a satellite into a circular orbit 650 km above the Earth. How fast must the shuttle be moving (relative to Earth) when the release occurs? 7bx hhsl w 19z3bt)dpy3 1:yoo   $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occupants aru5xn. - rjj4( gbgoyff;/mj 3me to experience simulated gravity of 0.60 g? Assume the spaceship’s diameter is 32 m, and give your4j/nm (ymb 3;rfo.gjgj5u f-x answer as the time needed for one revolution. (See Question 21, Fig 5–32.)   

  Determine the time it takes for a satellite to orbit the Earth in a cvi-p-pw m 8vt8ircular “near-Earth” orbit. p8mtp--8iwv v 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 twus6c0dsi9eo up dlar;4/64oiy 5fx 1o be launched from the top of Mt. Evla5ei sxw60fu1u6yd cdsr ; o4/ ip9o4erest to be placed in a circular orbit around the Earth?   

  During an Apollo lunar landing mission, the**slw37 f4tn7iqe0caj2z .ei (xbl vf command module continued to orbit the Moon at an altitude of about 100 km. How long di**70l3lifz.ebn4ija t x cvf7se 2w(qd it take to go around the Moon once?    s

  The rings of Saturn are composed of chunks of 2 yyq+ p5;zz73gp edbrice that orbit the planet. The inner radg; 7bpq e y2z5drzp3y+ius 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 mc5 1op7.nj2zel8 ovtc in diameter rotates once every 15.5 s (see Fig. 5–9). What is the ratio of a7vpojlt1.e n zo52 cc8 person’s apparent weight to her real weight (a) at the top, and (b) at the bottom?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut 4200 km from the center of b-5xg1 ficy,g*oj 9tbthe Earth's Moon i bg,f1cj y*-b5oxgt 9in 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-w wi.dz/ gb rx81)jq5byu- (umstar system consists of two stars of equal mass. They are observed to be separated by 360 mu/mg- r1 5. 8jxb( ywbqiu)wzdillion km and take 5.7 Earth years to orbit about a point midway between them. What is the mass of each?    $\times 10^{29}$

  What will a spring scale read for tb kf7r +d0ooa3iq v:,.gv qp.4s rq+kche weight of a 55-kg woman in an elevator that mo:qg a r0i v.p+bqqr,vfk3 dc.7o+sk4oves
(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 qly k**3as5z-t/vwjnn/m .j5 7k er xisuspended from the ceiling of an elevator. The cord can withstand a tension of 220 N an-z 7m*kr5 v3/s5tqk.njxn* a/w ilej yd 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 planet with pqop n0e1z(+8:ll4gnqcbm .h hr1w;re eriod T , the density (mass/volume) 1blhlc 1wmhq8e o4eg0 n;+nz .r(r:qpof 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 determine thp(vlol7hde 2 1e period of an artificial satellite orbitie7l 1ovl2(hp dng very near the Earth’s surface.    s

  The asteroid Icarus, though only ac:7hev d 50fyp few hundred meters across, orbits the Sun like the planets. Its period is 410 d. What is its mean distef:0dc p5 v7yhance from the Sun?    $\times 10^{11}$

  Neptune is an average distand ( 3pj7 l(;oxufjv*xmce 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 Sc wy* f r)b.sqfuh5/a+un roughly 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 sumsbc /yf) 5+.uwrqahf* of the nearest and farthest distance from the Sun.]    $\times 10^6$

  Our Sun rotates about the center of the Galax5au3w , -/q rc:dl8/vvf ighwxy $\left(M_{G} \approx 4 \times 10^{41} \mathrm{~kg}\right)$ 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 mf2. fx krbv,:0:zmivehzn:i(ean distance for the four largest moons of Jupiter (those discovered by Galileo in h 2fm:rvizkx.: n(fib,e0v: z 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 Mooszbp r be e*t2;s2fzyt/:*da- 5*bmlyn. 2y2pt lzs/ 5*d;asbey:b f* t*z-merb   $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jupiter’s moons, usingmt/q cbnn z7sk+ 1vi1ga8 za3+zu*1n :zj,ubj Kepler’s third law. Use the distance of Io and the periods given in Table 5–3. Compare to the valu,+ k j+bat7g8cz1mnvsbj131uqn i z* /zua:z nes 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 coru1itxg u z) a8+sisj)fn.1 *rnsists of many fragments (which some space scientists think ca *x+ n.trszfu)8i 1igsr)uaj1 me from a planet that once orbited the Sun but was destroyed).
(a) If the center of mass of the asteroid belt (where the planet would have been) is about three times farther from the Sun than the Earth is, how long would it have taken this hypothetical planet to orbit the Sun?$\approx$    y
(b) Can we use these data to deduce the mass of this planet?

  A science-fiction tale describes an artificial "planet"7qzx))ztrh5w7nuy jp8nw03 p 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). 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), and that the ring rotates quickly enough7wp tp n z05jyhz8r3qwn ))7ux 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 gor x23o hvd:qg.jge by swinging in an arc from a hanging vine (Fig. 5–41). If his arms are capable of exerting a force of 1400 N on to xq:2d v3.gjhhe vine, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 80 kg, and the vine is 5.5 m long.   

  How far above the Earth’s surface will the acceleration ofl(x fc/o*-z to gravity be half what it is on the /lzf*x(- cootsurface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and s2yr5s5xkg yi;l8jg / spin in a mutual circle once every 2.5 s. If we assume their arms are each 0.kxily y/gr g;8ss2 5j580 m long and their individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates once per day, the apparent acceleration of 5si0 han+ev/cgravity at the equa+s ea0hi/ 5nvctor 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?    $\times 10^{-1}$

  At what distance from the Earth will a spacecraft traveling directly fro p 1u9ds8hw(1 ualj.9t b)hbqjm the Earth to the Moon experience zero net force because the Earth and Moon pull with ej98)ubh(1pqwb. l huad19 jst qual and opposite forces?    $\times 10^8$

  You know your mass is 65 kari7necz1s37,1xr8 3xybsj e g, but when you stand on a bathroom scale in an elevator, it says your mass is 82 kg. What is the acceleration of the elevator, and in whichze1r7isarecbxj y7, 8xn13s3 direction?     ----待选择----upwartddown 

  A projected space station consists of a circular tube that will roq)gqd4kzmv7) cesd.2 ckb5b1 tate 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 1qd.7czv q4k5c md)bke2)gbsmust 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*jhbab(3ky9y n egh)4–43).
(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 accelera*po jzo*9h .** gam*uk zt8a:x wpi/gation due to gravity 9pxat *.zhm * **:ogz8woa ugk*ji/ap at its surface and the gravitational constant G.

A plumb bob (a mass m hanging on a st vomb o7pf2)y9ring) is deflected from the vertical by an ang mopv o9yfb)27le $\theta$ 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 67eem+tjnsvfdeo /g 9 nbuo7 ,f660w 3x/ m is banked for a design speed of If the coefficient of static friction is 0.30 (wet pavement), at6dtmo6 fe+en e // u,vn3g7s0jb 9xwof what range of speeds can a car safely handle the curve?

  How long would a day be if thx/:cr-;ynt dief2x +x e Earth were rotating so fast that objects at the eq;2r /xyt-fedix+:cnx uator were apparently weightless?    $\times 10^3$s

  Two equal-mass stars maintain a constant distance acchoy9ka-.,hde : cx; f6d5g ppart 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 cx+m5xvp 3cxr4onstant speed rounds a curve of radius 235 m. A lamp suspended from the ceiling swings out to an angle of 1prm +x vcx345x7.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times as massive as the Earth. Thus, it has been claimed y/)o r2m97;1rp nz ci-u-n1ra bjoreqthat 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 would experience at the equator/-1noy27r)qm1o a-j 9rnzbu;prr cie 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 extremelnn8 n cxguu;1/y massive core in the distant galaxy M87, so dense that it could be a black hole (from which no l1 n;8u /nnxucgight escapes). They did this by measuring the speed of gas clouds orbiting the core to be 780 $\mathrm{~km} / \mathrm{s}$ at a distance of 60 lightyears $ \left(5.7 \times 10^{17} \mathrm{~m}\right)$ from the core. Deduce the mass of the core, and compare it to the mass of our Sun. v =    $\times 10^{39}$solar masses =    $\times 10^9$

A car maintains a constant speed as i5 wv r6ncx, lof4jam6/ai 4iwl0rrit0xb //4at traverses the hill and valley shown in Fig. 5–45. Both the hill and valley have a radius of curvature R. (a) How do the normal forces acting on the car at A, B, and C compare? (Which is largest? Smallest?) Explain. (b) Where would the driver feel heaviest? Lightest? Explain. (c) How fast can the car go without losing contact with the 4jfm ii 0lwa5irb4 6 6//cowr4aanxt0v /x, rlroad at A?

  The Navstar Global Positioning System (GPS) utilizes a group of 24 gm6/n*u,oqz isatellites 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 satelliu/ig zo*6,mqn tes 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 Rendezvous (NEAR), after traveling 2.1 billion kycjo(3dfk +gdg 5y:6v ti0x;rm, is meant to orbit the asteroid Eros at a height of about 15 km . Eros +6dv;y:xytji 0r3o 5d(gcgkf 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 satel+qakjh 1 bdv,.:dx7qe lite in need of repair. You are in a circular orb. ,d1bj:7ek h+ vdxqqait 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) What b- u5.5ks *ejl ooy;pbis its mean distance from the Sun? bej5ol;yubo*5 .-k s p  
(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 w k9hulimcd 9(r m-r2*1kzgj/hould need to be if you could actually "feel" yourself gravitationally attracted to someone near you. Make reasonable assump 9mudrg(r hkchi-k9ml/jz2* 1tions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the Milky Way Galaxy (Fig. 5-46) at 9a) wr rnu :wnd1:67qr;waph zh-y)yza distance of about 30,000 light-y r:ywh;andu yr ah))zzqp 7r1:w-9w6nears 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 ll7d 2k.nm8mirh gh.h: ocated at the corners of a square 0.50 m on each side. Find the magnitudkhhg.hmril8nm7. 2 d: e 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 or -qpg/ pt+gygsrz5d51bits 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 sv,vfl89;ggepx m/7vl w 8azm*econd hand on your wr;xv w e8vvflm/*l7z89ga p g,mist watch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker westz0)6f g /fzoi92u nxight around in a circle below you on a 0.25-m piece of fishing line. The weight makes a complete circle every 0.50 s. What is the angle tfgixns69f0u t)z2 oz/ hat the fishing line makes with the vertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in a ns2uc.u 1n iig,ew highway is designed so that a car traveling at speg,1su.i uc2ined $ 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 cars when negotiati7t3nm ) xe:2qeq9ojgwng curves. The angle of tilt is adjusted so that the main force exerted on the passengers, to provide the centripetal acceleration, is the normal force. The passengers experience less friction force against the seat, thus feeling more comfortable. Consider an Acela train that rounds a curve with a radius of 620 m at a speed of (approximately ). (a) Calculate the friction force needed on a train passenger oftj9)2w:xq7o3gq em en 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.$\approx$    $\times 10^2$