Sometimes people say that water is remo 2zy5 lxy6m:vpved from clothes in a spin dryer by centrifugal p myx6z2vyl5 :force 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 sha+ud 473 f,vdrizi xhx4rp curve at a constant 60 km/h as when it travels aroui3d4zrf 7 vx4+uh id,xnd a gentle curve at the same speed? Explain.

Suppose a car moves at constant speed along a hilly road. k5:6+w rsscdlc46vdwu i7 lcp06 rpr -Where does the car exert the greatest and least forces on the road: (a) at the top of a hill, (b) at a dip between two hills, (c) on a level stretw skdi rc7dv4r: 0l6p+crcuws 65pl6-ch near the bottom of a hill?

Describe all the forces acting on a child riding a horse on a merry go fwtx.60 hm6 m:b3-i pvcmksw9ib27-go-round. Which of these forces prxwb c 6o7i6gfwips3.-mhbmm:v0t2k9 ovides the centripetal acceleration of the child?

A bucket of water can be whirled in a verticaqbwt d cm-plv-c;8 ov -+l(hp(l circle without the water spilling out, even at the tod8to p(mwclq- ;+- p vb-h(lcvp of the circle when the bucket is upside down. Explain.

How many “accelerators” do you have in y/w kq vy-fu j57ceigy*;1f:u rour car? There are at least three controls in the car which can be used to cause the car to yeg j u1-c :f7iw*5;kf/quyrvaccelerate. What are they? What accelerations do they produce?

A child on a sled comes flying ov tt,n ,v788hdi jxkbihv pn5:q ):wsj4er the crest of a small hill, as shown in Fig. 5–31. His sled does not leave the ground (he does not achieve “air”), but he feels the normal force between his chest and the sled decrease as he goes over the hill. Explain this decrease using Newtn8ditn,: ):kjh7q hb viw8xvp45jst,on’s second law.

Why do bicycle riders lean inward w 067jku1. i cmla+srw+xzkap:hen rounding a curve at high speed?

Why do airplanes bank when ,n vk;i/uxfx a *qxg6,ht2 apit6 7cc1they turn? How would you compute the banking angle given its speexntk2ct x g;v,a1h6cxq*7f6 au ,ip /id and radius of the turn?

A girl is whirling a ball on a string around her heac ejd-7 6ah.sko5dbb- d in a horizontal plane. She wants to let go at precisely the right time so that the ball will hit a target on the other side of the yard. When shoud d7cjb6esb-.h k-5 oald she let go of the string?

Does an apple exert a gravitationabu 8j/e 4lnvpo e5r.u3ufo38 fl force on the Earth? If so, how large a force? 85/f8olunje uf4v3e ru 3 bo.pConsider an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it 8bmy xv0/fi c4 zy6dw8is, in what ways would the Moon’s orbit be diffic mb4 0w zx8/ydf8yv6erent?

Which pulls harder gravitationally, the Earth on the Moon, or the 9 edcu y-h-b;tMoon on the Earth? Which accelerate uh;eytdc 9--bs more?

The Sun's gravitational pull on5b- 4rlbik k/s the Earth is much larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider thb5l-i s 4kbkr/e difference in gravitational pull from one side of the Earth to the other.]

Will an object weigh mole2omsf, q vy7 hkb5308wu*tdre at the equator or at the poles? What two effects are at work? Do they q ,eb3uw0vlkhyst 7 52m*o d8foppose each other?

The gravitational force on the Mooa khdk9vm-+i9n due to the Earth is only about half the force on the Moon due to the Sun. Why isn’t the Moon pulled away from the Eart ikk 9vdh9m+a-h?

Is the centripetal accel vik /b-c:6hmperation of Mars in its orbit around the Sun larger or smaller than the centripetal acceleration of the Ea6ip:v k-b/ chmrth?

Would it require less speed to l7b5g.gp.tfrw aunch a satellite (a) toward the east or (b) toward the west? Consider the Earr p7 gg5.tbfw.th’s rotation direction.

When will your apparent weight be the greatesf9( y093lkrzb gjuy;lt, as measured by a scale in a moving elevator: when the elevator (a) accelerates downward, (b) accelerates upward, (c) is in free fall, (d) moves upward at constant speed? In which case would your weight be the least? When would it bf9 90zgj(lr bul;ky3ye 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 adversel- u at,it(o+azy affected by weightlessness. One way to simulate gravity i, - utaiz+ota(s 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 experiences “free fall” or “apparent weightle/w)fjek.7;f yup-(/ls carf w ssness” between stepur)s p(j7e fww ffkyc-;/ la/.s.

The Earth moves faster in its orbit around the Sun in January ,3n*owd4r 8g9b4r 8yzu(yqpgg qeib8 than in July. Is the Earth c 4rz3de gb,ybiq8ogwy 8r(98pu4g qn*loser to the Sun in January, or in July? Explain. [Note: This is not much of a factor in producing the seasons — the main factor is the tilt of the Earth’s axis relative to the plane of its orbit.]

The mass of Pluto was not known until it was dis zpcthh0d,1 4mcovered to have a moon. Explain how this discovery ,tch zd4 m10phenabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the g5k1 uytqoc 9l3bn72h( hzzz 3vk/ of1Earth is much larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explai1 3bzn1k39z qhf7ly zvu(o/ khtog52 cn. [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 kf5i:z:g u( qgoy :b op1hg0 xjrl+*n: 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 k,l4u65yaw yarlzc67 in its orbit around the Sun, and the net force exerted on the Earth. Wha ula6 y5la z,yw4k7cr6t 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 au,62 k:qku6d:zzyo sf 2.0-kg discus as it rotates uniformly in a horizontal circle (at arm’ q dkfuk26,oyzu::zs6 s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is in 2,sg lmupo+0;vmb2-7m ou csh orbit 400 km from the Earth's surface, and circles the Earth about once every g mpbm c02 +ms-h2;uouvsl,o790 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 the e c m.ph5gxbp/o6:bj x-dge of a potter’s wheel turning at 45 rpm (revolutions per minute) ifpj6: /pxbgm- chx .5bo the wheel’s diameter is 32 cm?   

  A ball on the end of a string is revolved at a uniform + 62hqwsit-ebpen2r8ym ,b5s rate in a vertical circle of radius 72.0 cm , as shown inw i8 s,5hp6e sen+2 yqb2-brtm 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 can9prq7gz28o j0xz/k astz; o1.g wy;pr round a turn of radius 77 m on a flat road if the coefficient of static friction b ;pg1 j8/oqatp2xz.os;r rz 9z7y0gkwetween 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 tiyu43s* xd5lciyi6 f,ares and the road if a car is to round a level curve of radiai5sy4x uld*if, y3c6us 85 m at a speed of 95km/h ?   

  A device for training astronauts an v,wy :7n8+tlwwbrl w2d jet fighter pilots is designed to rotate a trainee in a horizontal circle of radius 12.0 m . If the force felt by the train+tv8wr,7lwwbw :l2 y nee 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 tuo.wu.oqtov./ av52ia rntable 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 .o 2.aa q5u.twi vvoo/the coefficient of static friction between the coin and the turntable?   

  At what minimum speed u*nno)ifidz** h( w/hmust a roller coaster be traveling when upside down at the top of fhhwi***/ i(dnzo )nu 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 (includvod pt*h6r6 )aing the driver) crosses the rounded top of a hill (r6d*p6v) rao htadius =95$ \mathrm{~m}$ ) at 22 $\mathrm{~m} / \mathrm{s}$ . Determine
(a) the normal force exerted by the road on the car,   
(b) the normal force exerted by the car on the 72-kg driver   
(c) the car speed at which the normal force on the driver equals zero.   

  How many revolutions per minute would a 15-m-diq34ya,l;e.am+bh ic,xd ) qjy5p )6r nrhpev1ameter Ferris wheel need to make for the passengers to feel “weightless” at tn,eri p6jhme5q)ap.41 vxhdab;lq3r yc+,)y he topmost point?    rpm

  A bucket of mass 2.00 kg ipvuz / 113mi1ndedbx9 s whirled in a vertical circle of radius 1.10 m. At the lowest point of its motion the tension in the ropp1d dxm/n1ez93v 1ub ie 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 9.0y6*e0okl 5v4 oz0nbob 0 cm from the axis of rotation is to experience an acceleration vo00obz*knyl 4o e56b of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotpch3e ky36 av03vc 8izated in a cylindrically walled "room." (See6c33evhvpa yi c3 z80k Fig. 5-35.)
The room radius is 4.6 m , and the rotation frequency is 0.50 revolutions per second when the floor drops out. What is the minimum coefficient of static friction so that the people will not slip down? People on this ride say they were "pressed against the wall." Is there really an outward force pressing them against the wall? If so, what is its source? If not, what is the proper description of their situation (besides "scary")? [Hint: First draw the free-body diagram for a person.]   

A flat puck (mass M ) is rotated in a circle on a frictionless air-h34 fzx,pt 7bcsejk0-pockey tabletop, s3- ,pztk407xbfp jecand 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 weight of the ball nji.9n+a p j;0hflvdx o(0.mu which revolves on a string 0.600 m long. In particular,jd(n 0iv +;fauph9m. xn 0.jlo 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 bankek4j32v x kwku,d 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 masses ko78;;qvma964cl9eu8 n:r sgasnxie$ 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 a)y9r4oq379l rcct ,ajjgrg v3 t 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 t5ch:bkn5h-1j zy6ajv+-wi u rz;4tre he net force exerted on the car (by the; ne1j4 h-wv z a z-tyui5+r5jk6c:bhr ground) in Example 5-8 when its speed is 15$ \mathrm{~m} / \mathrm{s}$ . The car's mass is 1100 kg . F$_{tan}$ =    F$_R$ =   

  A car at the Indianapolis 500 accelerates uniformly from the pit area, going f u(maorc h*g2:rom 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 constant tangential acceleration. If the curve were flat, what would the co uc( :omgra*2hefficient 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 r4i)5 sfl w8gbpscr-1i l :f(bxadius 2.90 m . At a particular instant, its acceleration islw4i)cb(gs 1b5:lfi- pr f 8sx 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 spacec,d,* n mffm5wirv6) loo;rgedu( lm56raft 12,800 km (2 Earth radii) above the Earth’s surface if its masi( ermvl ,5m;6 fwfugnrmlo5 d6d)* o,s is 1350 kg.   

  At the surface of a certain planet, thec6. rui, p ;jw.p 9aeeq7yzwl/ gravitational acceleration g has a magnitude of 12 m.jwy/aeu ,rec7 ;ppz6wql. i9/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 gravilh txi+7*8 mpjty on the Moon. The Moon's radius is 1.74il7*tp +m8 hjx $\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 tha* w9i)5yzyib j1 i5kaot of Earth, but has the same mass. What is the acceyizbiy5)* kio9wj15 a leration due to gravity near its surface?   

  A hypothetical planet hasn hhz9z7p-v a dw5+suq,b/; cz a mass 1.66 times that of Earth, but the same radius. Whatqv /b5;wzzp79u ,-hahcd sn+ z is g near its surface?   

  Two objects attract each other gravitationally with a force of situ fy o1mf a*4j *pt)a6v61u 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 , theo s:6evle m r6;c,syf5 acceleration of gravity, at (a) 3200 m f;s :r y, 5cs6lmeve6o   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance j -l w7cf,m7r0hy1mfqfrom the Earth’s center to a point outside the Earth where thegravi0,q1hr- fmcwmfjyl7 7 tational 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 packed into a sphere amj+g gqyi5xljj; yl3/vf* a,x4q ;b0cbout 10 km in radius.j4b*a+yl ;c0m;, iylg3xf5jg qvqx j/ Estimate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once was an averag*j s)hc9l4 c/sxpf6eve star like our Sun but is now in the last stage of its evolution, is the size of our Moon but has the mass of our Sun. What is the surface gravity e s/v*l4)fsp cjc9hx6 on this star?    $\times 10^7$

  You are explaining why astronauts feel weightless whilex3k*zee)a 3h26ewu ug;n ssr; orbiting in the space shuttle. Your friends respond that they thought gravitykx see6w*s3au;hnz2)eru ;3g 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 l,xam:ra/5c m,t eh -lxocated at the corners of a square of side 0.60 m. Calculate the magnitude and ae-/rx5xhalmt,,c m: 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 on9 e: jlsjqspkl7+xty y*o(s;9 the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets arkos s9e+*y 97jjpx ytl:(l q;se 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.*g9i h6(qkqvmxq .kn0( zhflzg3w9+u2q .bdi38 of what it is on Earth, and that Mars’ radinzdqfzqxi k(w0hu.m3*9(kh6+q gb.q2l9 vgius is 3400 km, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value for the periow b+i0j oep0o;d of the Earth and its distance from the Sun. [Note: Compare your answer to tha j+e 0;opo0ibwt obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satelotc:8 kb*t70kfly; zdlite moving in a stable circular orbit about the Earth a8;oktb0td :l7 ykzcf* t a height of 3600 km.    $\times 10^3$

  The space shuttle releases a s i6iak.ec4pztg .n)jgr9, vv7atellite into a circular orbit 650 km above the Earth. How fast must the shuttle be moving (relative to Earth) whrg 4ceka.9.vnpvg i, tj7)z 6ien the release occurs?    $\times 10^3$

  At what rate must a cylindrical sa :ped r2ct9ws1frqa8h+ 1b:tpaceship rotate if occupants are to experience simulated gravity of 0.60 g? Assume the spaceship’s diameter iscphabf91e: t d2r18saw tr:+q 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 a circula./5 i h;+vspb uiz2df63mu vyl1 pa*nzr “near-Earth” orbit. A “near-Earth” orbit is one at a height above the surface of the Earth which is very small compared m3i+pvl2p5ay bs d n1hzzuf ;/vi6*. uto the radius of the Earth. Does your result depend on the mass of the satellite?    s ~    min

  At what horizontal velocity would a satellite havej534u3sj4j93 3 h hhu u 8gg(gfrgxrja to be launched from the top of Mt. Everest to be placed in a circular orbit arxg5338hjgu gj4 hu(rs hj934 ru3jafgound the Earth?   

  During an Apollo lunar landing mission, the command modumw8eu(r9chd cui-k3;le continued to orbit the Moon at an altitude of about 100 km. 9(c8 chmw-k iuud e3;rHow long did it take to go around the Moon once?    s

  The rings of Saturn are composed of/+efm4hr*b sq chunks of ice that orbit the planet. The inner radius of the rin4/bhe qf+r m*sgs is 73,000 $\mathrm{~km}$ , while the outer radius is 170,000$ \mathrm{~km}$ . Find the period of an orbiting chunk of ice at the inner radius and the period of a chunk at the outer radius. Compare your numbers with Saturn's mean rotation period of 10 hours and 39 minutes. The mass of Saturn is 5.7 $\times 10^{26} \mathrm{~kg}$ .


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

  A Ferris wheel 24.0 m in diameter rotates once every 15.5 jy6kh o:ws j7m -1cw25vkj /hjs (see Fig. 5–9). What is the ratio of a person’s apparent weight to her real weight (jy1ckjo 6s: -j5w/ hk 7hjv2mwa) at the top, and (b) at the bottom?
(a)    (b)   

  What is the apparent weo6 lrbkpxhy;woi*y286 3, ocnk oto0 xm;9yl( ight of a 75-kg astronaut 4200 km from the center of the Earth's Moon in a sr(k xh2 , 3o6pibko 0mt6;yyooxc8l nw l9oy;*pace 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-sta* fdy+.m htlsdham810r system consists of two stars of equal mass. They are observed to be separated by 360 million km and take 5.7 Earth years to orbit about a point midway between them. Whfy +s*mh da0.d8 m1lhtat is the mass of each?    $\times 10^{29}$

  What will a spring scale rx ,0tfgv,hv o jfwr;4g4 xr82-gusa h:ead for the weight of a 55-kg woman in an elevator that mogj ax2g,4rux,-w:rs go v vfhh04;8ft ves
(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 th415au ;tujd1od4u*uiruwjf 2 e ceiling of an elevator. The cord can withstand a tension of 220 N and breaks as the elevator accelerates. What w2j u1uud jwu r15*4i;oaf4tud as the elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orbits very near the7xz +:,prsuwg/ r2na3hts2 ek surface of a planet with period T , the density (mass/volume) of the planet i : sh2/r ,w3uxn 7ra+pktsge2zs $\rho=m / V=3 \pi / G T^{2}$ .
(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 pez5jt6+tdg y*q riod of the Moon (27.4 d) to determine the period of an artificial satellite orbiting very near the Ea6z* 5+tdgjqt yrth’s surface.    s

  The asteroid Icarus, though only a fujbnkp b-)67new hundred meters across, orbits the Sun like the planets. Its period is 410 d. Wh6-unb)nb 7kpjat is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average distance of 4 it*re gcd: hu;u:q4 +pecn/c9.5 $\times 10^{9} \mathrm{~km}$ 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 commxo;+- pup ei1/8jmqxes very close to the surface of the Sun on its closest approach (Fig. 5–39). Estimate the greatest distance of the comet from the Sun. Is it still “in” the Solar System? What planet’s orbit is nearest when it is out there? [Hint: The mean distance s in Kepler’s third law is half the sum of the nearest and farthest distance from the Sun.] m+1-mjx/8 ;p up oqxie    $\times 10^6$

  Our Sun rotates about the 86ghw)fkn*q 1fjvc sw2 t0c.h 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 di ny1rg05*zxl;k 5 ryika2.g0.w wyaiastance for the four largest moons of Jupiter (ty ywra gi.52a .yrl ax*n01wk;0gzik5hose 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 t lre)ta z99vih/lg9-okru2r3+djma1 5d, e lshe known period and distance of the Moon. +ljear k19ivzmg9le ddu,/2rts-a9 35r h) ol   $\times 10^{24}$

  Determine the mean distance from Jupiter for e yi/wb 8o; yahvs(lgca;c8.ql4r4w )xach of Jupiter’s moons, using Kepler’s third law. Use the distance of Iqowrv4c 4l/;8h (.;yxais)g8l ac bwyo 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 m: huank j 8ub/d2j5 zl,/,tboMars and Jupiter consists of many fragments (which some space scientists think came from a planet that once orbited the Sun j kl5a o,,:2thn//jdmub z8ubbut 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 artifici-3gderu ; ,vq( ww4vzial "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). 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 enough to produce an apparent gravity of g as on Earth. What will be the period of revolution, this planet's year, in ru4vq( 3e w-,wizvg;dEarth days?   

  Tarzan plans to cross a gorge by swing x:91r v83xzyb:c6hs kig0*iv9w auxd ing in an arc from a hanging vine (Fig. 5–41). If his arms are capable of exerting a force of 1400 N on the vine, what is the maximum speed he can tolerate at the lowest point of his swing? His massu x0x1 9rvb96i*zdvi8a hwcsy :3kx:g is 80 kg, and the vine is 5.5 m long.   

  How far above the Earth’s surface will the acceleration of gravity be half ugm83 ov gr4vhlq:gld d mc;*+nb4i41what i: l+ md14dlg;8oih* u gcbgm3nv 4qv4rt is on the surface?    $\times 10^6$

  On an ice rink, two skaters of equal mass gra cbc 8zap23p;jb hands and spin in a mutual circle once every 2.5 s. If we assume their arms are each 0.80 m long and bcc;ppz2 3aj8 their individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates once per da 2mok:94gamxkbp* 6wxy, the apparent acceleration of gravity at the equator is slightly less than it would be if the Earth didn’t rotate. Estimate the magnitude of this effect. What fraction of g is thisxo*4gmb96axk wpkm2: ?    $\times 10^{-1}$

  At what distance from the Earth will a spacecraft traveling directly fr 3kzluxm2 4il+om the Earth to the Moon experience zero net force because the Earth and Moon pull with equal and opposxm2 il3kul4+zite forces?    $\times 10^8$

  You know your mass is 65 kg, but when you stand on a bathroom scale in an el,m 1fgeo/v 8vesxoug+* j88gk evator, it says your mass is 82 kg. What is the accelereogosg*8 18k,ve vg8 xju/fm +ation of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consists of a circu- , 0eoy7inptglar tube that will rotate about its center (like a tubular bicycle tire) (Figo- pn,eyg0t7i . 5–42). The circle formed by the tube has a diameter of about 1.1 km. What must be the rotation speed (revolutions per day) if an effect equal to gravity at the surface of the Earth (1.0 g) is to be felt?$\approx$    $\times 10^3$

  A jet pilot takes his aircraft in a vertical loop (Fig. 5 jozn 88,8fbvc na.wjof qw669–43).
(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 o z(.f;8jc jt strp/*7ji(iqnj f a planet in terms of its radius r, the acceleration due to gravity at its surface and the gravitational cont. (;jpn 8jzs(tjiic/ frj7*qstant G.

A plumb bob (a mass m hanging on a sairtu6:;2.zk,og bb xtring) is deflected from the vertical by an angle gbx.iz k;:t6ba oru,2$\theta$ 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 bankuvr6w(: 5ypb ved for a design speed of If the coefficient of static friction is 0.30 (wet pavement), at what range of speed (by6 5vvup:wrs can a car safely handle the curve?

  How long would a day be if the Earth were rotating so fast that obr5le)d: kcmkv 8ele8o k85 /bgjects at the equator were apparently weightllgb:deek5k5morv kc88/8 l)eess?    $\times 10^3$s

  Two equal-mass stars maintain a constant distance apart oi+vv 0e4 :bxxpfs1: cx 6uza70pf vmj58h1kicf 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 y bp1rnghpn )8x+d3r0-zjl 3bh+q:zv a curve of radius 235 m. A lamp suspended from the ceiling sb3pn+rz3lx hjb8 )1grph dv:0-yq+ z nwings out to an angle of 17.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times as massive as the Earth. Thus, it has be6y02skygxi8-rf*gj c)ugjm+ en 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 thg-yi2syjxcg 08)jgr mf k6+ *ue 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 qw0u 4m( 2dhzwdvee23m mbt6 1Telescope deduced the presence of an extremely massive core in the distant galaxy M87, 2 v0(tmzdhwueed m6b4wq2m13so dense 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 consty 5 7w:f nrpfz4qt.7qmant speed as it 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 cr5fz f7m:.4ptq w qyn7ar go without losing contact with the road at A?

  The Navstar Global Positioning System (GPS) utilizes anrf(v s(iey2 1 group of 24 satellites orbiting the Earth. Using “triangulation” and signals transmitted by these satellites, the position of a receiver on the Earth can be determined to within an accuracy of a few centimeters. The satellite orbits are distributed evenly around the Earth, with four satellites in each of six orbits, allowi2(ienv rfsy 1(ng 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 (NEb59fvbtxq7 fk0zg1n,k bqq7 ;AR), after traveling 2.1 billion km, is meant to orbit the asteroid Eros at a height of about 15 km . Eros is roughlyxzqnb, ; 71q 9f5b0qb fgtv7kk 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 spafm.oj 6d 6m*ngce shuttle pursuing a satellite in need of repair. You are in a circular orbit of the same radius as the satellite (400 km above the Earth), but 25 kmfo 66*m jnm.dg 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 is k8-14* hkh4i c xaqduqmji( mgqi1(x-*ij6jqits mean distance from the Sdi -m4ug6ii(1cqa8mh-4* qjqk1 x x*j ki qj(hun?   
(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 youg6xjrz z)h p+8edp9 4 ckfj.-c could actually "feel" yourself gravitationally attract jzg9j p4+hxc-rdf )c.ke86zped 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. 5aym5sw*c zrv4 c8 ciq35;,(tshz ,aqn-46) at a distance of about 30,000 lig(nv5hat,mcr 5q z,icsz; c wa8y3q4s*ht-years from the center $ \left(1 \mathrm{ly}=9.5 \times 10^{15} \mathrm{~m}\right)$ . If it takes about 200 million years to make one rotation, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun $\left(2 \times 10^{30} \mathrm{~kg}\right)$ , how many stars would there be in our Galaxy?$\approx$    $\times 10^{11}$

  Four 1.0-kg masses are located at the corners of a square ooz6,p6,)vymk zf*uf 0.50 m on each side. Find the magnitude and direction of the gravitational force on a fifth 1, 6oz6o upk ,y)fzvfm*.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 nj. g8tn7 o1i:8khqe5 ysc /mdorbits 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 experienck0bl,b+ e*uu-vf6 gx wr w6:oged by the tip of the 1.5-cm-long sweep second hand on your wrw:6 -kl+e0b*go,uv g6rfuxb wist watch?    $\times 10^{-4}$

  While fishing, you get bored and u c0q7t ikts +mr5kc o p4ehu0ka5(9r2start 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 fishing line makes with thaeu i rpcck5r2k( mtu05 ots70+q4 9khe vertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius ytt /og;jh 86zR in a new highway is designed so that a car travelingy o8htz 6gjt/; 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,e1 n44kxqka otm666u,qn)v u3bk 4kf t utilizes tilt of the cars when negotiating curves. The angle of tilt is adjusted so that the main force exerted on the passengers, to provide the centripetal acceleration, is the normal force. The passengers experience less friction force against the seat, thus feeling more comfortable. Consider an Acela train that rounds a curve with a radius of 620 m at a speed of (approximately ). (a) Calculate the friction force needed on a train passenger of mass 75 kg if the track is not banked and the trai)6mkxat6 k1n3tq 4,k onfk6 v4 euq4ubn 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.$\approx$    $\times 10^2$