Sometimes people say that water is removed from clot* :scs+21qxt/24gqydw j eyeehes in a spin dryer by centrifugal force throwing the water outward. What is wrong with this statemcteq 4ew+/2sg ys qx2 ej:*dy1ent?

Will the acceleration of a car be the same when the car trave, qf,leq2u5qz. 5nsbfa.rx q( ls around a sharp curve at a constant 60 km/h as when it travels around a gentle curve at the same speed? , uq eqqsn.55raxfb 2lfq,.z(Explain.

Suppose a car moves at constant8eiip6 4;pr;. enqrt3(nug gk speed along a hilly road. Where does the car exert the greatest and least forcnne .pe3k6gpig8r ;(uqt ;r4i es 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 of a hill?

Describe all the forces acting fwmqtrb:(m2uu/0 y a .strm;zp 52x8gon a child riding a horse on a merry-go-round. Which of these forces provides the cenyq2r /uf8 bp5(tg ;z2mx:0mrmtu s.watripetal acceleration of the child?

A bucket of water can be w qr-eq2 sc/dfd;ojjf8+d(nih9mct -3 hirled in a vertical circle without the water spilling out, even at the top of the circle when the bucket is upside down. Explain.+jc j-cf8i3q(hfed ;-o qnt s9/r2dmd

How many “accelerators” do you . *+kd, x0grq/hdtmeq z +cy*thave in your car? There are at least three controls in the car which can be used to cause the car to accelerate. What are they? What re*ktq gd/zq c0d. hym,t+*x+ accelerations do they produce?

A child on a sled comes flying over the crest of a small hit4 0j0k*ppk-.hxmc h z .;onetll, as shown in Fig. 5–31. His sled does not lea.0ek;0 *pz4h-tthmp o x.jnckve the ground (he does not achieve “air”), but he feels the normal force between his chest and the sled decrease as he goes over the hill. Explain this decrease using Newton’s second law.

Why do bicycle ridersv x6d0udi,,a zyr3 jn p4*yg,2lr.yoz c*ylj . lean inward when rounding a curve at high speed?

Why do airplanes banu:v /yp;dn9j.xlh5mxk when they turn? How would you compute the banking angle given its speed and radius of the d5hpxnu :x/y.9mjl;v turn?

A girl is whirling a 0zwm/ q/w8)xp yi(loi ball on a string around her head in a horizontal plane. She wants to let go at precisely the right tixi /8mzp(w 0y)oi l/qwme so that the ball will hit a target on the other side of the yard. When should she let go of the string?

Does an apple exert a fo,0wt el;fj* d.:fsv gravitational force on the Earth? If so, how large a force? Cofvo:e.0d,fwj; *tf ls nsider an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what iwxsyr0r, t cx.u2t6x6*6* 3muzzx kbtt is, in what ways would the Moon’s orbit be differr6*3yzxrmwc6txu 2*0z st ktx,.6xu bent?

Which pulls harder gravitationally, the Earth on the Moon, or thewe5d00gz / udtop0j j) Moon on the Ear )0pzdt0jojwd05e u g/th? Which accelerates more?

The Sun's gravitational pull on ty*(yg; a04lv/0ym3 xhy3di uk es x*cthe Earth is much larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: xcy30v (y yidtem;0*s4u 3gahylx*k/Consider the difference in gravitational pull from one side of the Earth to the other.]

Will an object weigh more at the equator or at the poles? What zv4cd l9,,pxfcd6(o s n*uj -ftwo effects are at work? Do they oppose eachn4f x9ju,,v6fl( dops* -z dcc other?

The gravitational force on the Moon due to the Earth is onlyk ,o t1))(ygnvn x/wqt about half the force on th v)t(kxn/qng1tw, yo )e Moon due to the Sun. Why isn’t the Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in its orbit around the -,ekf5 a;v3 3latqnidSun larger or smaller than the centripeta3k,f-q5a vlai;tde3 nl acceleration of the Earth?

Would it require less speed to launch a satellite (a)wu,, lmp 8tk b94oa8se toward the east or (b) toward the wam,plo9b8 ew kust48 ,est? Consider the Earth’s rotation direction.

When will your apparent weight be the greatest, as mea52+ou,koxg hsb sl-k1sured 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 as when you 5skh2,lkgou+ -b1 xs oare on the ground?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long periods in outer spacrpj )(crmor+w l3drc otrd -;- +pn)u0e could be adversely affected by weightlessness. One way to simulate gravity is to shape the spaceshilr(u po+pnr rcj3); d+ cdt0ro-)m rw-p 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 lyut-h fow: :4“free fall” or “apparent weightlessness” between sf: 4-huwty:lo teps.

The Earth moves faster in bn:wq8nl jji,2 sx:4d its orbit around the Sun in January than in July. Is the Earth closer to the Sun in January, or in July? Explain. [Note: This is not much of a factor in producing the seasons — the main faqjd8,s w :xbi4n:2 ljnctor is the tilt of the Earth’s axis relative to the plane of its orbit.]

The mass of Pluto was not know+e zbfj zes*z7n4t-5e,4y slcn until it was discovered to have a moon. Explain how this dissfnyzz5b4 -4lcs,+ t e *zje7ecovery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth is much larger than the Moon's. Yet thb9z-,) sbp-0o mhgxhue Moon's is mainly responsible for the tides. Expz) 0-up9,gbom bhshx- lain. [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 9 6ogr3dka:f 8zrs oq. 1980 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ pulls out of a dive by moving in an arc of radius 6.00 km . What is the plane's acceleration in $g^{\prime}$ 's?    g's

  Calculate the centripetal acceleration of the Earth in its orbit aroun.twgb /d f)cm)d the Sun, and the net force exerted on the Ea)bmf c/.w )gtdrth. 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 a /zqf zz g4ppq.*0gz9us it rotates uniformly in a horizontal cipqgqzzz.* 0u /fz4pg9 rcle (at arm’s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is in orbit 4-s(ns6u. iaj(v unk*uh hb)e300 km from the Earth's surface, and circles the Earth about once every 90 minutes. Find the centripetal accele6svnu-a(*i. huknheu j( bs) 3ration of the space shuttle in its orbit. Express your answer in terms of g , the gravitational acceleration at the Earth's surface. $\approx$    g

  What is the magnitude of the acceleration of g5 +nkvp 3ik+7 tj wm5upy23oh 1nc/cga speck of clay on the edge of a potter’s wheel turning at 45 hnc/3p2tp53v1 yjiwg +7k5n+ kco mgurpm (revolutions per minute) if the wheel’s diameter is 32 cm?   

  A ball on the end of a string is revolved x4zl.b my)*puat a uniform rate in a vertical circle ofub) .4mpz*xy l radius 72.0 cm , as shown in Fig. 5-33. If its speed is 4.00 $\mathrm{~m} / \mathrm{s}$ and its mass is 0.300 kg , calculate the tension in the string when the ball is (a) at the top of its path   
(b) at the bottom of its path.   

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

  What is the maximum speed with which a 1050-kg car can round a turn of q;6alodw y tvj,r6e ;(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 of the cwqr ey;6l6,d(v; aojt ar?   

  How large must the coefficient of static friction be betweeg gxk8 d)u b37rmhj1-wknokn,3 -k4zpn the tires and the road if a car is to round a level curve of radius 85 m at a speed of 95km/h ? -uh38rx k kknz3gm4bnpgj 1-do7k),w   

  A device for training astronauts and jet fighter pxww31 edvcm y3xmenl)gb+* l2 3n: g2rilots is designed to rotate a trainee in a horizontal circle of radius 12.0 m . If the f1 3nylm2bg2xgr3de:+ )ln* xe vm3wwcorce 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 o8g jtwvbcz80 -f variable speed. When the speed of the turntable wgv88jtzb - c0is slowly increased, the coin remains fixed on the turntable until a rate of 36 rpm is reached and the coin slides off. What is the coefficient of static friction between the coin and the turntable?   

  At what minimum speed must a roller coaster be traveling w,kxmdnc 3)+tzhen upside down at the tockn xmzd,t)+3 p 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 roun;6 ,fey- j ,hlfkkmw)ecta:0sded top of a ,heal fetwjks6k, fcm 0)- ;:yhill (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 mie k(0u u9a r 7(vzg33pzwco3,go j)qbbz v4f2bnute would a 15-m-diameter Ferris wheel need to make for the passengers to feel “weightless” at the tzbp7b eovvqjfu(( ,ouz3)k29z33wacrg g 0b4opmost point?    rpm

  A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1.10 m. At n3;pc w0u1e/a-*dy cwom:sk8 l frl/ fthe lowest point of its motion the tension in the rope supporting the bucklwe:aopd /y;u*l31 cc0n- msrwk8 / ffet is 25.0 N.
(a) Find the speed of the bucket. $\approx$   
(b) How fast must the bucket move at the top of the circle so that the rope does not go slack?   

  How fast (in rpm) must a centrifuge rotate if a particle 9.00 cmflm6mxti.s0 3 from the axis of rotation is to experience an a03ls xti 6fmm.cceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnivau ,;pox0f .xdal, people are rotated in a cylindrically walled "room." (See Fig. 5-35.) 0axpo.x fd u,;
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-hockeon+2c(,ifg*:vk rnn,jsi f 8my tabletop, and is held in this orbit by a light cord connected to a dangling block (mass m ) through a central hole as shownjmvnn+2:gorc n,,sk if(f*8i 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 nph n)+,rs.jp eot ignoring the weight of the ball which revolves on a string 0.600 m long. In p+s,prh j) en.particular, find the magnitude of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ , and the angle it makes with the horizontal. [Hint: Set the horizontal component of $\overrightarrow{\mathrm{F}}_{\mathrm{T}}$ equal to m $a_{\mathrm{R}}$ ; also, since there is no vertical motion, what can you say about the vertical component of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ ?] $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ =    the angle =   

  If a curve with a radius of 88 m is perfectly banked f(mg4u* yfd:l*odq by. or 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 massekcbvuno3u4p/ q1u(m2ca vie07i) f5 fs $ 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(5 /n-osutdvv 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 centripetag6 non-;h3ze l6 x*mtil components of the net force exerted on the car (by the ground) in Example 5-8*nzt66-oxm; il ngh3 e 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 0q c zw9e7vlbj:tnbtmbr3*xs5a+8zc4( w lt ; accelerates uniformly from the pit area, going from rxcqzt7ze t 9:bv +*rbab3w j;lls5wc4(t 08nmest to 320km/h in a semicircular arc with a radius of 220 m. Determine the tangential and radial acceleration of the car when it is halfway through the turn, assuming constant tangential acceleration. If the curve were flat, what would the coefficient of static friction have to be between the tires and the road to provide this acceleration with no slipping or skidding? $\approx$   

  A particle revolves in a hor 5nny,6vh5z3a zf pm /2ny(xcfizontal circle of radius 2.90 m . At a particular instant, its acceleration is ,3z5v6 f a cynz52xhp(ny nmf/ 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 om166g 5w yq fl(azh9xlp;idz2f Earth’s gravity on a spacecraft 12,800 km (2 Earth radii) above the E6 pahy zz;9x5l62glw(im 1dq farth’s surface if its mass is 1350 kg.   

  At the surface of a certain planet, the gravitational k;8y5em -vpxm acceleration g has a magnitude of ke my-vm5;x8p12 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 gravityz*2(ii 3txi ,fozqkst58f c;f on the Moon. The Moon's radius is 1.74 8 iozf* ;iz fqs2 ixt,t5c(k3f$\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has a radius 1.5 ti2hw*g ll .*m2y2bmhc j 0h(zydmes that of Earth, but has the same mass. What is th **ml(zh.h yy0gbcdm2jl22whe acceleration due to gravity near its surface?   

  A hypothetical planet has a mass 1.66 times that ofgt1vprac x eg0 + -.qj)-0oz ,nw0sbtd Earth, but the same radius. What is g near its 0-x0r.t)tsgwjga0qp z d ,-vb+ o1ecnsurface?   

  Two objects attract each other gravitationally with tso8)3kvd8pa 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 effectuy o stm90:qk7ive value of g , the acceleration of gravity, at (a) 3200 m osm:u97 tq yk0   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance frd w(i4p/key zq8q,y g0om the Earth’s center to a point outside the Earth where thegravitational acceleration due e qi0g, yd 4w(/zpkqy8to the Earth is $\frac{1}{10}$ of its value atthe Earth’s surface?   $\times10^7$

  A certain neutron star has fivevj aap g6h;0 n,zvtw4(; vi-uh times the mass of our Sun packed into a sphere about 10 km in radiutpznv wi;6va gh( 0aj;v4,h u-s. Estimate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once was an ave) ohk:j1:d51fmim 8a 2fmlkyerage 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. Whfyiek2 d)m fm::oj 51 1kh8lmaat is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronauts feel weightless while orbiting iq:lz +q5 zc6d(gfwj u,n the space shuttle. Your friends respond that they thought gravity was just a lot weaker up there. Convince them and yourself that it isnz(jcd+6zf,w l5uq:qg ’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 locatql + vu:gew3(wed at the corners of a square of side 0.60 m. Calculate : g3luqw(+ vwethe magnitude and direction of the total gravitational force exerted on one sphere by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred years most of the planets line up yn+.ir *yvs,bamdz 7+36 qzf ktq2mu+on the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in a line (F am 2mk+z+fsyz,b3i7*y+nd.t6 qqurvig. 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 accelerai / f83bc:lo6tfyrd3jtion of gravity at the surface of Mars is 0.38 of what it is on Earth, and that Mars’ radius ibj/f:6 ot3fid8cly3r s 3400 km, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun usinljys6vrlyd: -1l1b(5 h).f:dri eleig the known value for the period of the Earth and its distance from the Sun. [Note: Compa1i(dlll:h s yedf5ly :-6.1)jbie vrrre your answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite moving inpnzir( ojm b.h o:0f:2 a stable circular orbit about the Eapzojm.fo02h(nr: i: brth at a height of 3600 km.    $\times 10^3$

  The space shuttle releases a satellite into a circular orbit 650 km abo9tw q0 wks ,ov/symi53ve the Earth. How fast must the shuttle be moving (relative to Earth) when the release occsqv0wi5kw9 3s ,m oy/turs?    $\times 10^3$

  At what rate must a cylindrical spaceshtfo+51;kehit xxmy,,7v(t e (k xkk (dhxl 48oip rotate if occupants are to experience simulated gray+,87tkdtxh 5 kxx (oefe; (i4 o mhkvtkl1,(xvity of 0.60 g? Assume the spaceship’s diameter is 32 m, and give your answer as the time needed for one revolution. (See Question 21, Fig 5–32.)   

  Determine the time it takes for a satellite to orbit the Earth in at.d -ca4rh: ly circular “near-Earth” orbit. A “near-Earth” orbit is one l c-th4ry.ad: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 be launched fr)36. n1yik /zakok/lxk ntzp +om the top of Mt. Everest to be placed in a cirin6)zk pz ok tk1lkn 3+a//yx.cular orbit around the Earth?   

  During an Apollo lunar landing mission, )gfh (9k;c 5wq 3ojgfhe0zd az/1rc3w the command module continued to orbit the Moon at an altitude of about 100 km. How long did it take t j1f9rhog35wq dkzeaw;3( / zfgcch)0o go around the Moon once?    s

  The rings of Saturn are composed of chunks of ice that orbit the0dtptq;x4em 2 planet. The inner radius of the rings is 73,t0 e 2mpqdxt4;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 evsy,4 gc-) upqk;pu+t cho97xgery 15.5 s (see Fig. 5–9). What is the ratio of a person’s apparent weight to her real weight (a) at the top, c9 c; h4ptyos+g)xp- gu,7kuqand (b) at the bottom?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut 4200 km fh4u)jr/(byysv :s/ zdrom the center of the Earth's Moon in a space vehicle (a) moving at consta(sb:ry// )zv4hy du jsnt 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 binarkz)q9x/k6cxz y-star system consists of two stars of equal mass. They are observed to be separateq9z x6z) kk/cxd 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 read for the wedsnr.7e jramiuz7 7:2ight of a 55-kg woman in an elevator that moves s2uarez77im:d.n r j7
(a) upward with constant speed of 6.0m./s   
(b) downward with constant speed of 6.0m./s   
(c) upward with acceleration of 0.33 g,   
(d) downward with acceleration 0.33 g   
(e) in free fall?   

  A 17.0-kg monkey hangs from a cord suspended from the ceiling of athiz0djw, mm0ub h,(: n elevator. The cord can withstand a tension of 220 N and breaks as the elevator accelerates. What was the elevator’s minimum acceleration (magnitude and di uhwztm:,(b00im h dj,rection)? a =   

Show that if a satellite orbits very near the surface of a; jbd 2y5nyr.h planet with period T , the density (mass/volume) of the plane.;5bdh 2yynj rt 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 t* usdu)ib.u z;t(: wckhe period of an ar ;d.zu b()uicwtsu*:k tificial satellite orbiting very near the Earth’s surface.    s

  The asteroid Icarus, though only a few hundred meters across, orbits d2 1a f1z wqo;l6qcqh6the Sun like the planets. Its pewqfz 6ao2q hc11q6 ld;riod is 410 d. What is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average distai.1i mnv :c0lgy/zroihq1s7 0nce of 4.5 $\times 10^{9} \mathrm{~km}$ from the Sun. Estimate the length of the Neptunian
year given that the Earth is 1.50 $\times 10^{8} \mathrm{~km}$ from the Sun on the average.    $\times 10^2$

  Halley’s comet orbits the Sun roughly once every 76 years. It comes very cj c9h2c 8ze7oglose to the surface of the Sun on its closest approach (Fig. 5–39). Estimate the greatest distance of the comet from the Sun. Is it still “in” the Solar System? What planet’s orbit is nearest when it is out there? [Hint: The mean distance s in Kepler’s third law is half the sum of the nearest and farthest distance from the Sun.] h2o79 8jgeccz    $\times 10^6$

  Our Sun rotates aboutv3twx s ,*7swd the 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 forccj-ifl )op w h62a-oxu)1ii0 the four largest moons of Jup6w-)-ilox i2)focuhip ca01j iter (those 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 t( (3rkbfk- gfohe Earth from the known period and distance of the Moon. ko frfkb 3(-g(   $\times 10^{24}$

  Determine the mean digp buhs-1 f+o;stance from Jupiter for each of Jupiter’s moons, using Kepler’s third law.pof+- bg; hs1u Use the distance of Io and the periods given in Table 5–3. Compare to the values in the Table.
$r_{\text {Farip }}^{r}$ =    $\times 10^3$
$r_{\text {Guarymale }}$ =    $\times 10^3$
$r_{\text {Culimio }}$ =    $\times 10^3$

  The asteroid belt between Mars and Jupiter consists of many fragmentsytt4 dvwsoh7 3 h (dg:djkgn:x9wc,;2 (which some space sciy:gt4h wcn j,7d(w3;xv 2tsh k9do dg:entists think came from a planet that once orbited the Sun but was destroyed).
(a) If the center of mass of the asteroid belt (where the planet would have been) is about three times farther from the Sun than the Earth is, how long would it have taken this hypothetical planet to orbit the Sun?$\approx$    y
(b) Can we use these data to deduce the mass of this planet?

  A science-fiction tale describes an artificial "planet" in the g+9i0pw0vtc+sakwm9 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 di s00wi+tvgp9 k9 aw+mcstance (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 csztpu udpfw;0;wno:( 93hdh-kor5. 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 i- z9s0dc kf tw;hodu3.(:rp5hpnu;w os 80 kg, and the vine is 5.5 m long.   

  How far above the Earth’s surfac,0u28ugt ws7eq uhd 7ze will the acceleration of gravity be half what it isz77 ,t2e8gu0 hduuswq on the surface?    $\times 10^6$

  On an ice rink, two skater-etn l)kknewbpj(tw87h 1 a9+ig u7 x)s of equal 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 t7ki+t)hep- g knx7)w9na( 8 lwj e1ubone another?    $\times 10^2$

  Because the Earth rotates once per day, the apparent acceleration of gravity d;nl pzsujy8 7jd( )e,at the equator is slightly less than it would be if the Earth didn’t rotate. Estimate the magnitude of this effect. What fraction ol )n 8(7z,duj;jp sdeyf g is this?    $\times 10^{-1}$

  At what distance from the Earth will a spacecraft traveling directv(/eaybrtb)i eh -))tk 77mi aly from the Earth to the Moon experience zero net force because the Earth and Moon pull with equal and opposite forces )bt7kerbm7-)ya iea/h(t )v i?    $\times 10^8$

  You know your mass is 65 kg, but when you stand on a yxcd. p1t:j)q bathroom scale in an elevator, it says your mass is 82 kg. What is the acceleration of t).j qy:xtdp c1he elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consists of a circular tube that will rotate aboo/*ldjtwg 6re s /7/kkut its center (like a tubular bicycle tire) (Fig.o76kt/sg/ *el/rk jw d 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 qvv+ :drr.m (qv/vq*oloop (Fig. 5–43).
(a) If the jet is moving at a speed of at the lowest point of the loop, determine the minimum radius of the circle so that the centripetal acceleration at the lowest point does not exceed 6.0 g’s.    $\times 10^3$
(b) Calculate the 78-kg pilot’s effective weight (the force with which the seat pushes up on him) at the bottom of the circle    $\times 10^3$
(c) at the top of the circle (assume the same speed).
   $\times 10^3$

Derive a formula for the mass of a planet in terms of its radius r, the acceleraus7/uar 0njs r73st t1tion due to gravity at 07 as3u7rnj1utr sts/its surface and the gravitational constant G.

A plumb bob (a mass m hanging on a ssbjan;arh1w sy1r :. a,(mpw .tring) is deflected from the vertical by an angle ba; ,jhswwn1(1r.sramy .:ap $\theta$ 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 s w9 91pncxcir/peed of If the coefficient of static wc1np9x/ rci9 friction is 0.30 (wet pavement), at what range of speeds can a car safely handle the curve?

  How long would a day be if t-7i-yj t;4lh3 pu-zoqf y 2vezhe Earth were rotating so fast that objects at the equator were apparently weighyq34- 7 j2o y flu--ip;vzhezttless?    $\times 10^3$s

  Two equal-mass stars maintain a constant dis i/0 ipnep lgc2j:1js,yf/r-ctance apart of 8.0 $\times 10^{10} \mathrm{~m} $ and rotate about a point midway between them at a rate of one revolution every 12.6 yr . (a) Why don't the two stars crash into one another due to the gravitational force between them?
(b) What must be the mass of each star?    $\times 10^{26}$

  A train traveling at a constant speed rounds a )r96 pa2*wzgw.nd lmcvf c,r8curve of radius 235 m. A lamp suspended from the ceiling swings out to an angle zla* fnvrg.89cwc 6),wpdm r2of 17.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times qm+c6.fc 5 /u5i h/juotm6qzsas massive as the Earth. Thus, it has been claimed that a person would be crushed by the force of gravity on a planet the size of Jupiter since people can't survive more than a few g 's. Calculate the number of g 's a person would experience at the equator of such a planet. Use the following data for Jupiter: mass 6.jcz c5tu 5mi6u/s+f mqoh/q =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 apt;b)d.p8 f1:nw ni69yn zoz1sx ;jgd n extremely massive core in the distant galaxy M87, so denz x68 nodwjyp;.t:iz1p df bn)9n;s g1se that it could be a black hole (from which no light 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 itcwm6g + a+a6qs 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 w s6aqg6 cma++compare? (Which is largest? Smallest?) Explain. (b) Where would the driver feel heaviest? Lightest? Explain. (c) How fast can the car go without losing contact with the road at A?

  The Navstar Global Positioning System (GPS) uti*(t9p 7i bjm2frqgly0lizes 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 lbg yr70*ijmq p29(ftwithin 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 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 Rendezvo3 .qr,7e4m pfcstmb 3ous (NEAR), after traveling 2.1 billion km, is meant to orbit the astert 3 rmp4,7o.bmfsec 3qoid 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 shuttle pursuing a satellite in need of repairmv, t;as7 psg ,esr8hoki *m0,. You 0m,ss *p; ,h m8igs,kot7arve are in a circular orbit 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) Whatxa a 2lyp531xh is its mean distance from a235y a1xlxphthe 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 act x//k)okkpu4uj w)+h( fi2n wfually "feel" yourself gravitationally attracted to someone near you. Make reasonai4fkw)2+jxku()/f/u woknp h ble assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the Milky Way Ghxz,kqm *8v:.v pgy b-alaxy (Fig. 5-46) at a distance of about 30,000 light-years from the centqy v*vx:pgb.-mz,8h ker $ \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 loc( pg;rrw-e 1rcated at the corners of a square 0.50 m on each side. Find the magnitude and direction of the gravitational force ong1rpc ;r-ew r( 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 orbit*in9//catlg;sf56oi (em vga rvif 67s 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 tip6hv hx i3h*slk3jtc01vz3y e7 of the 1.5-cm-long sweep second hand on your j0h3i331e xlct vy6zhsvhk*7wrist watch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weight arovlm:r j:8/qka +2tk k gy7g*swund in a circle below you on a 0.25-m piece of fishing line. The weight makes a complete circle every 0.50 s. What is the angle that the fishing 8 m:vkl/ ywa72jkqr:+ sg*ktgline makes with the vertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in a ne fs*feh+cd4 v0mxjo.l0pjf4 ;w highway is designed so that a car travelin + 0sjef4h fl; mv0pfoj4c.x*dg at speed $ v_{0}$ can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from $v_{\min }$ to $v_{\max }$ . Derive formulas for $ v_{\min }$ and $ v_{\max } $ as functions of $\mu_{s}, v_{0}$ , and R .

  Amtrak’s high speed train, the Acela, utilizes ti+/fg kz2d5 ik664cen k8n btizk2.qcvlt 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 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 pcn gi6k28 z dfizq /k.v2 tnbkck+64e5assenger 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.$\approx$    $\times 10^2$