Sometimes people say that water is removed from clo b76td+ yq t ;/pbp:s,jp+viaothes in a spin dryer by centrp:;t+qao p,/bbj +tdy 7 6svpiifugal force throwing the water outward. What is wrong with this statement?

Will the acceleration of a car be t08o: tqaco nw3he same when the car travels around a sharp curve at a constant 60 km/h as when it travels around a gentle curve at the same speedoo:3t 08 ancqw? Explain.

Suppose a car moves at constant speed along a hilly road. Where does the cyfs-;- nstpp x5ee43p ar 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 l -4step psfp3xe-yn;5evel stretch near the bottom of a hill?

Describe all the forces acting on a child riding a horse on a merry-go-r/veo)vh;d jh: ound. Which of these forces provides the centripetal acceleration of tde;ovh/ v):jh he child?

A bucket of water can be whirled in a vertic*4 fc z6 g0bxwwc(rkv1al circle without the water spilling out, even at the top of the circle when the 06( cwr*kzb1gcvw 4xf bucket is upside down. Explain.

How many “accelerators” do you have in your car? Tbl,:vcq +dln (n -1tvqhere are at least three controls in the car which can be used to cause the car to accelerate. What are they? What acceler1 nnqd:t b(lqc+-vvl, ations do they produce?

A child on a sled comes flying over the crest of a small hqm4 y hkbx979kill, as shown in Fig. 5–31. His sled does not leave the ground (mkk9 yb94xq h7he does not achieve “air”), but he feels the normal force between his chest and the sled decrease as he goes over the hill. Explain this decrease using Newton’s second law.

Why do bicycle riders lean inward when rounding a curve a 1oxli4fzta:9)pm )sgt high speed?

Why do airplanes bank when they turn? How would you compute the bane,nf..s duf,tb yz 1q b6nj zc(:vo16aking angle given its speed an(1.,c6yaetunn1.z,:v bof fqdb s z6jd radius of the turn?

A girl is whirling a ball on a string around her head in a horizon y2.oqhkca,1 qtal plane. She wants to let go at precisely the right time so that the ball will hit a targaqqk2o. y,1c het on the other side of the yard. When should she let go of the string?

Does an apple exert a gravitational force on the Earth? If so, how l/0kgs p s(1ogxarge a force? Consider an appo1xs gkgp(s0/le
(a) attached to a tree, and (b) falling.

If the Earth’s mass were doubl i j-kljeo.a92e what it is, in what ways would the Moon’s orbit be dif9okja 2.e-ijlferent?

Which pulls harder gravitationally, the Earth ;q;1t9;w(lp;ic av iecf9j;rhf vi : e7n )cpmon the Moon, or the Moon on the Earth? Whichv;; j;; a91cit nlphmrefqc: 7c9 f;( )ivewpi accelerates more?

The Sun's gravitational *l rg 3g0e -xobzb fx478zz)cnpull on the Earth is much larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from one side of the Earth tn0o4 lzgbzg)x7b z r- xc3ef8*o the other.]

Will an object weigh more at the equator or atg 3.o (m15cwp qbznzq3 the poles? What two effects are at work? Do they oppose .on1w 3qm53 g(cqb pzzeach other?

The gravitational force on the Moon due to the Earth is only aboutw 1wm)76ao *bd1f5-zj;jnksi (khxz f half the force on the Moon due to the Sun. 1n1x( zm;65k fdsw wbhi)kj zj of7a*-Why isn’t the Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in its orbit around the Sun larger dfgf;ii5ld5 0 or smaller than the centripetal acceleration of the Earth?;dfi 5dlgi0 5f

Would it require less speed t qwzz5j/e;sbwqk o-:5 o launch a satellite (a) toward the east or (b) toward the west?5zobeqk- z;5 w:qsw /j Consider the Earth’s rotation direction.

When will your apparent weight be the great xeg:40w1tehe -fv9tf est, 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 f e91-tw4 evhge0xf:tconstant speed? In which case would your weight be the least? When would it be the same as when you are on the ground?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long periods in outer space could behak srdw 4b3hh,hcj iw1zfd-7mr +)m9-hj +w( adversely affected by weighr++h)j9db zw -hh- dwafw7,k13hh r cjm(4s mitlessness. 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 other aspects of gravity you can think of.

Explain how a runner expetq1vtm8 ;fi juikjri7io e ,2)i- 0(wbriences “free fall” or “apparent weightlessness” between stepsi8t ii;j0f jv1)i7o-mk(q ei2r ,tubw .

The Earth moves faster in its orbit around the Sun in January than in wrq zo;,bj5k ,c0 e*gnJuly. 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 facqrzw b,0*;5cgo,je nktor is the tilt of the Earth’s axis relative to the plane of its orbit.]

The mass of Pluto was not known until it was discovered to haibm d nkho95boa.+5 xy* n) vb9hoe2 8xgop37kve a moon. Explain how this discovery enabled an estimate of Pl*5hd87b vax3i)go nen 9moo o+92byb .k5hkxputo’s mass.

  The Sun's gravitational pull on the Earth is 5k0m, 82qqs yg psf8 ro) f0g)ofx9gyxmuch larger than the Moon's. Yet the Moon's is mainly responsible for the tidess,5oq8oyx9g80) y)fm rkfqpf2x s 0gg. 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 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 travelingp s3z:aqo2 (yo 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 aenbc/d5g9 70fis9: t 1izc (dengd1kzcceleration of the Earth in its orbit around the Sun, and the net force exerted on thei7:d/gnz1 s9id1d0 k(bge5z9ct ne c f 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 ej4 i. pi9bp/vxxerted on a 2.0-kg discus as it rotates uniformly in a horizontal circle (at ar .b9i vxpj/ip4m’s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is in orbit 400 km from the Earth's surface, and aetl7 :pxl5eba29* 7 jvscvt:circle7c9tla7:2:vv*a5bsxtejp l es the Earth about once every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of g , the gravitational acceleration at the Earth's surface. $\approx$    g

  What is the magnitude of the acceleration of a speck of clay on thel9v0 t y9* lfa+c-6dhqvvrd e. edge of a potter’s wheel turning at 45 rpm (revolutions per minute) if the wheel’s diameter is 32 tdhvqvy*v9ac0 lr- 6efd+ l9.cm?   

  A ball on the end of a s:+zwqggr0.k *z v8x carr5y(atring is revolved at a uniform rate in a vertical circle of ra8+yg:r.( v cazw ak rrzxq5*g0dius 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 speedey/ba 939d.la y6mb 4lzt jb1wqh(qu. oa r3t0 with which a 1050-kg car can round a turn of radius 77 m on a flat road if the coefficient of lboh9a t3z l3r19uyt q/0. 6awjbma(bye 4qd.static friction between tires and road is 0.80? Is this result independent of the mass of the car?   

  How large must the coefficient of static friction be between the tifqobb a6bzl., cple8k9p0 d-ag/ t3x ,res and the road if a car is to round a level cur-z b 0,lbfb/eao. g8xad3qclp6 tk9 ,pve of radius 85 m at a speed of 95km/h ?   

  A device for training astronauts and jet fighter pilots ir(0r6ttshg, cs 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 s,s6( gr0h rctthe 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 fr88t9kbgi nb x08hr sc/om the axis of a rotating turntable of variable speed. When the speed of the turntable is slowly increased, thebkbr9/88ngh0 xit8cs 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 coaste,7qqk5qw 8cmjvcq+h7 r be traveling when upside down at the tock75 hj mv,+qqqwq78 cp 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 (includ+5 ykx8 kc,a x2omx hnj **dxg,fc9r-ding the driver) crosses the rounded top of a hill (radgxx-+*cdr2do,5 fa 9*xj ,8yknhcm xkius =95$ \mathrm{~m}$ ) at 22 $\mathrm{~m} / \mathrm{s}$ . Determine
(a) the normal force exerted by the road on the car,   
(b) the normal force exerted by the car on the 72-kg driver   
(c) the car speed at which the normal force on the driver equals zero.   

  How many revolutions per minute would w qwu3aoy*;xe7omb 4qwo6+ jkzl5/ /na 15-m-diameter Ferris wheel need to make for the passengersz 6*+yqu /nlkqb;4wma/jow wxo e 735o to feel “weightless” at the topmost point?    rpm

  A bucket of mass 2.00 kg is whirled in di ,tiz 3-a9fua vertical circle of radius 1.10 m. At the lowest point- 93itda fiu,z of its motion the tension in the rope supporting the bucket is 25.0 N.
(a) Find the speed of the bucket. $\approx$   
(b) How fast must the bucket move at the top of the circle so that the rope does not go slack?   

  How fast (in rpm) must a centrifuge rotate if a particle 9.00 cm from the axiwyxs ky.x(by,rp 3ip lsta3 1 p.+nx0,s of rotation is toy3sa .(yt 3+,pbsx,xlnk xp 0.iwp1r y experience an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotated in a cylindrically wajchv xl-0yxyiss)6:,lled "room." (See F: )jvyxy-,xl6 hcis0sig. 5-35.)
The room radius is 4.6 m , and the rotation frequency is 0.50 revolutions per second when the floor drops out. What is the minimum coefficient of static friction so that the people will not slip down? People on this ride say they were "pressed against the wall." Is there really an outward force pressing them against the wall? If so, what is its source? If not, what is the proper description of their situation (besides "scary")? [Hint: First draw the free-body diagram for a person.]   

A flat puck (mass M ) ;yj 5xx t zpq(1m;y7zqis rotated in a circle on a frictionless air-hockey tabletop, and is held in this orbit by a light cord connected to a dangling block (mass m ) through a cemqx ;y( 1y ;xqzt7z5pjntral 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 ignh 2e0 1,a5 xgqj7 pmfnqkz-6hnoring the weight of the ball which revolves on a string 0.600 m long. In particular, fqknhpjhngz 1 5m26 e-0a7,x qfind 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 perfecs)1oz1+yrd k 8iwwub(tly 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 masse6((;ko p ey:y0fcixqis $ 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 diving7oj0mwmwqeu 6rf 41tr m+zj+6 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 centripetal components of thg()n n2blri -qe net force exerted on the car (by the ground) in Example 5-8 when its sp(g-n 2 ql)ibnreed 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 area, goingj cwb6gtwtae08v6 ,jrg /+qg (j8,isl.s;du c 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 through the turn, assuming conw,g(tgr0c8c.dle ;j+bg uqjij aw ts68v,s/6 stant 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 h1 yq;e7m8zdg9rt; tm eun.;rs orizontal circle of radius 2.90 m . At a particular instant, its accele t;yudtz.s9 rmm;gq7ee r8 n1;ration is 1.05 $\mathrm{~m} / \mathrm{s}^{2}$ , in a direction that makes an angle of 32.0$^{\circ}$ to its direction of motion. Determine its speed (a) at this moment   
(b) 2.00 s later, assuming constant tangential acceleration.   

  Calculate the force of Earth’s gravity on a spacecraft rn4 jn7e0u-bsq dk,g012,800 km (2 Earth radii) above the Earth’s surface if its mass nkub 7 dsne00-g qrj,4is 1350 kg.   

  At the surface of a certain planet, the gravitational acceleration g has a mag0.i xow48 ;1vjxyquse6: nqfb nitude of 1j;xwxqoi4 nq16sy 0 e: vb.uf82 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 acceleratic1lyu-4m :x o+ 1tgplfon due to gravity on the Moon. The Moon's radius is 1.74 f l m 11g+xctyupo:l-4$\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has a radius 1.5 times that hyatkn8r .ps/x+4m /tof Earth, but has the same mass. Whaty / + trkmsnp.a8tx4h/ is the acceleration due to gravity near its surface?   

  A hypothetical planet .+tlfkx(f 1zp has a mass 1.66 times that of Earth, but the same radius. What is g near its( tpz+flf1. xk surface?   

  Two objects attract each other gravid08+izt d ag2gtationally with a force of 2.5$ \times 10^{-10} \mathrm{~N}$ when they are 0.25 m apart.
Their total mass is 4.0 kg . Find their individual masses. m$_1$ =    m$_2$ =   

  Calculate the effective value of g , the acceleration of gravity, at (a)b h(7ngz ;ln+s 3200 m ( n7zsnlh; b+g   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside the Earth where yovaan: -k4t9 *j r1wrthegravitational ac* nrj 194tw:-vayoarkceleration 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-uhian7x y7a2 jmv7g+ packed into a sphere about 10 km ijm-hayiu77v2 g7 xan+n radius. Estimate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once was an aver+0m(xdb-uc *fpk n;hq age 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 x+pdk( 0;hq -cmnf *ubour Sun. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronauts feel weig g; g2zhvy,ip+htless while orbiting in the space shuttle. Your friends respond th gvzp; y+g,2ihat they thought gravity was just a lot weaker up there. Convince them and yourself that it isn’t so by calculating the acceleration of gravity 250 km above the Earth’s surface in terms of g.   

  Four 9.5-kg spheres are located at the corners of a square of side 0w/o. x9uu 8xkz.60 m. Calculate the magnitude and direction of the total gravitational fo 8ku9xw.z o/uxrce 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 uj/d js6g8.0e *4av a)bz u-b6yrdd qbwi9+uvdp on the same side of the Sun. Calculate the total d)rgj6z/da db- q.ued+yj bivu8 6 4*w9as0vbforce 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 o*4fji k;i ywi*f Mars is 0.38 of what it is iyjw;4*ik* if on Earth, and that Mars’ radius is 3400 km, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value for thrj*vqg9- 7t mmjdkgc -(3o- dpe period of the Earth and its distance from the Sun. -7ctdoj-r 93gkjmdpg*qvm- ([Note: Compare your answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite mj23: -xtrn4ska1, qw-5tah zezrx s+g d*9qnm oving in a stable circular orbit about the Earth a,a3:- 4azme*s-r q5nngs k2 1ttx+q9rjxhz dwt a height of 3600 km.    $\times 10^3$

  The space shuttle releases a satellite into a circular f8gu2n6yu0)dz+jhpbw ,ras 2 orbit 650 km above the Earth w ha8ds)2br z26uj0p,+yf ngu. How fast must the shuttle be moving (relative to Earth) when the release occurs?    $\times 10^3$

  At what rate must a +)8 ydk1b8 k.wn :m0gbjbvtp ccylindrical spaceship rotate if occupants are to experience simulated gravity of 0.60 g? Assume the sp )g b18kk8wmcbn:b pjyvtd.0+aceship’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 takes44sld7*+ ti ghz9cb5fd9q.yu fphr9v for a satellite to orbit the Earth in a circular “near-Earth” orbit. A “near-Earth” orbit is one at a height above the surface of the Earth which is very small compared to s c qbyrv*7.f59hzh 9g4fp+9ilud td4 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 from the top .h8+ ldk(:pw o bvpop2of Mt. Everest w( pkpp+d hb.8o2vol :to be placed in a circular orbit around the Earth?   

  During an Apollo lunar landing mission, the command module continwv +.vo/pi/oaued to orbit the Moon at an altitude of about 100 km. How long did it take to go arouv.//+ioop awvnd the Moon once?    s

  The rings of Saturn are compf ch,- pgcr )-h4 n0i+,srffrgosed of chunks of ice that orbit the planet. The inner radius of trn crih-ff ,pf+ rg )40,g-shche rings is 73,000 $\mathrm{~km}$ , while the outer radius is 170,000$ \mathrm{~km}$ . Find the period of an orbiting chunk of ice at the inner radius and the period of a chunk at the outer radius. Compare your numbers with Saturn's mean rotation period of 10 hours and 39 minutes. The mass of Saturn is 5.7 $\times 10^{26} \mathrm{~kg}$ .


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

  A Ferris wheel 24.0 m in diameteey0r 7 8tt8oiu ozfv91r rotates once every 15.5 s (see Fig. 5–9). What is the ratio ofu v0oi7 ft1e9zt8roy8 a 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 h84 -3xyaojjuuh. r 1mkm from the center of the Earth's Moon in a space vehicle (a) moving 3y.uja-h8uo4rj1 xh m 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 conh s*pg8.39xbbsg ll/4xbmnl 7 sists of two stars of equal mass. They are observ47l 8n3x l.b9bsh*m/b xsggpled to be separated by 360 million km and take 5.7 Earth years to orbit about a point midway between them. What is the mass of each?    $\times 10^{29}$

  What will a spring scale re7c -vflf40q++1*u blssqxzuuad for the weight of a 55-kg woman in an elevator that moves us +s4 xu1qb-7* +vfuzqlflc0
(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 an elevator. Th.p ,ww;uficccg 77n;ze cord can withstand a tension of 220 N and breaks as the elevator accelerates. Whatfpc;iw7c n.wu, ;cz7 g was the elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orbits vu kk1 oo;bb*8.ha nqi7 6vo6oxery near the surface of a planet with period T , the density (mass/volume) of the o6hnkoaxo.*v qo 1 6i;k7ub8 bplanet 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 determiyu e,o lk0p5w7j;og)lne the period of an artificial satelli7 l0pego yuokj)w;l ,5te orbiting very near the Earth’s surface.    s

  The asteroid Icarus, though only a few hundred meters across, orbits the Sun )a 5drzy3(s u8nr4vzclik5rd) nsy(r8uaz vz4c3 e the planets. Its period is 410 d. What is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average distance of*mhq a*6lwwnz6o3kx tf93 , sy 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 clob69wou2 5i6sufap js*d /g*gose 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 st56puuog9d2sij/of a*g wb s *6ill “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 ce3 +tx3ky6rg dsnter 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, periowih/ ::uen hx -ya(ca*d, and mean distance for the four largest moons of Jupiter (thuh/-h*: waiacynx:e(ose discovered by Galileo in 1609).
(a) Determine the mass of Jupiter using the data for Io.$M_{\substack{\text { Jupiter- } \\ \text { Io }}}$ =    $\times 10^{27}$
(b) Determine the mass of Jupiter using data for each of the other three moons. Are the results consistent?
$M_{\substack{\text { Jupiter- } \\ \text { Europa }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Ganymede }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Callisto }}}$ =    $\times 10^{27}$

  Determine the mass of the Earth from the known period and distx-xk6 21pu3kslb)kqbance of the Moon. 1kqpl- 2)xxku6b3 b ks   $\times 10^{24}$

  Determine the mean distance from Jupite 0l ps nvy)s5rphrm0cfb13s4 ,r for each of Jupiter’s moons, using Kepler’s third law. Use the distance of Io and )ss0,05r sfrb4pvph3mn1cl y 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 Mardm5 pnda79e:b nae.;);uxzfv s and Jupiter consists of many fragments (which some space scientists thin:fd un 7;ae; e9.zap d5x)nbmvk 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, g15zfmtao 5q an artificial "planet" in the form of a band completely encircling a sun (Fig. 5-40). The inhabitants live on the inside surface (where it is always noon). Imagia tm,5zqo5 g1fne that this sun is exactly like our own, that the distance to the band is the same as the Earth-Sun distance (to make the climate temperate), and that the ring rotates quickly enough to produce an apparent gravity of g as on Earth. What will be the period of revolution, this planet's year, in Earth days?   

  Tarzan plans to cross a gorge by swinging in an arc from a hanging vine (Fig zap ewwo6 1t/niu8pl*3t +1nc. 5–41). If his arms are capable of exerting a force of 1400 N on the vine, what is the maximw/1183e *n wuznltctp +6pa oium 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 willh stglv:3,-gt the acceleration of gravity be half what it is onglhs-gtt,: v3 the surface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spin in a mutual circlg: xiab lh)g002bpt.ot9x: gwe 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 on 0:g)0bh .:tbgxpo la 2wx9tige another?    $\times 10^2$

  Because the Earth rotates once per day, the apparent acceleration o(xz*xzf0 fp6*su yqji , 5zs:uf gravity at the equator is slightly less than it would be if the Earth didn’t rotate. Estimate th0q ji*u z*zfyfx(6x,z :sspu5e magnitude of this effect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from the Earth wn6(x b86ud35zecb dvw ill a spacecraft traveling directly from the Earth to the Moon experience zero net force because the Earth and Moon pubdbv56ec6 3wxd8nz u (ll with equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, but when v:ttlyi5v1 /lep o (r9you stand on a bathroom scale in an elevator, it says your mass is 82 kg. What is the acv 59:ir1voy(e t /ptllceleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space statc16)z 3z9pwm,okn2q x dej4 mtion consists 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 (revnzte36 p)zd x2j 4,ocmwk1m 9qolutions 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 g2qog zh81du*vertical loop (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 aan slvwb3:hmt p10o -5*kfy+t cceleration due to gravity aa vt5h+ n*3mk1-w0syob :fptlt its surface and the gravitational constant G.

A plumb bob (a mass m hang 0u17e9ym+gg pby;mpl y7vna1 ing on a string) is deflected from the vertical by an vm;a0blgpy+uym 97 y17np1geangle $\theta$ 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 desig.xzkvhnq :v 4:n speed of If the coefficient of static friction is 0.30 (wet pavement), at what range of speeds can a car safely handle the x4:n vvkqz:.h curve?

  How long would a day be if trc bpaw0. 6o8ehe Earth were rotating so fast that objects at the equator were apparent.0woa8c r 6pebly weightless?    $\times 10^3$s

  Two equal-mass stars maintain a constant distancejd4+fq o(n u6u 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/a 3 sd3w)eymh t*)jmn;kksh 4/ka7s e speed rounds a curve of radius 235 m. A lamp suspended from the ceiling swings out to an ansjm)*3mk kwts)k/enaahh;s 4y7e3/d gle of 17.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times as mas ql k.c/o0yru0r4m9a lne a,8usive 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 sin9raeua/l8m ,or4n. 0kuq cly0ce people can't survive more than a few g 's. Calculate the number of g 's a person would experience at the equator of such a planet. Use the following data for Jupiter: mass =1.9 $\times 10^{27} \mathrm{~kg}$ , equatorial radius =7.1$ \times 10^{4} \mathrm{~km}$ , rotation period =9 $\mathrm{hr} 55 \mathrm{~min}$ . Take the centripetal acceleration into account.    g's

  Astronomers using the Hubble Space ub-gzu mp+ +v6Telescope 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 the speed of gas clouds orbiu+ gzupmb6 v+-ting the core to be 780 $\mathrm{~km} / \mathrm{s}$ at a distance of 60 lightyears $ \left(5.7 \times 10^{17} \mathrm{~m}\right)$ from the core. Deduce the mass of the core, and compare it to the mass of our Sun. v =    $\times 10^{39}$solar masses =    $\times 10^9$

A car maintains a constant speed as it traj8 xe 1)xw1dk5m78eapt+wc hmverses 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, atm +jwk1cp 188ex) 7w5m hxeadnd 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 System (GPS) utilizes a group of 24 ) i+3 zl6) genng*zxx xnou3-w*x p7hbsatellites 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 accuz gx) *-hixxg olnu3 z6 e*pxwn)nb+37racy 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 Rendezvous (NEAR), after trd:u 2xlz0gjgatn : :eo*jt; )kaveling 2.1 billion km, is meant to orbit the asteroid Eros at a height of about 15 km . Eros isxuaz;gneod 0 tjg: l*)tj2 ::k 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 pureh;ah6j i(zfv6 *x1yvsuing a satellite in need of repair. You are in a circular orbit 6(vvha6ex; hy *z1jfiof 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 perizrdy mf ;.f(,7;u ddtsod of 3000 years. (a) What is its mean distance from the Sy7fr.dtdz f;u ;m (sd,un?   
(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 wojm)wblp9 e2iqd:.69 m0mtiyv uld need to be if you could actually "feel" yourself gravitationallybiv 09dl emimpy.69wt2q):mj attracted 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) aoyu 0( ac1-ps a7:yzgat a distan a7c(opygu-1z0 aay:s ce of about 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 squao:vk xojg1 h4/z ny in0nj((x;re 0.50 m on each side. Find the magnitude and direction of the gravitational force on a fifth 1.0-kg mass plak;izvn jhj(n1 o:nx0xgoy 4/(ced 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 Eartu gcetw e/n3)vs: 8zq-*imroz m9mj(+h $ \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 experienced2- /2;fu8qxhlnar ywlka .e9a/.ib1v pefh (d by the tip of the 1.5-cm-long sweep second har 2a.h /f1l9d;vabw/2.ue(xl-k8fh yqi pea nnd on your wrist watch?    $\times 10^{-4}$

  While fishing, you get bore8 y9cd /u9hj lrm2qy(ad and start to swing a sinker weight 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 that the fishi qc haud2( yrjly989/mng line makes with the vertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in a new highway is designed so tad/x0+xffati :f+8uc5 on8 s ghat a car traveling at sa /u8n:o+sidcx0g+8 f 5txfaf peed $ 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 t7:70zzm s kp8qj z1kfhi2r -mthe 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:zf s07mmj7p-2t 8qkkhiz rz 1 (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.$\approx$    $\times 10^2$