Sometimes people say that water isj;bh c(:n6aem ly1 m4o removed from clothes in a spin dryer by centrifugal force throwing the watyb6eao :(1n j4;mchml er outward. What is wrong with this statement?

Will the acceleration of a b9vmsf 0:-/w. tjx*+6z+botap) irk pkxcc w:car be the 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 sam/c f:x9.vx6 * bzkiotp0)t w+pckb-:ms +rjawe speed? Explain.

Suppose a car moves at constant speed along a hillyoz ig8 gfmh1o/s:eh7 azq 7u;1 road. 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)q;i7gz78m1 fsu ga:h e1o/hz o on a level stretch near the bottom of a hill?

Describe all the forces ac1ezl-cxt qb4, ting on a child riding a horse on a merry-go-round. Which of these forces provides e-xtb41 clz,qthe centripetal acceleration of the child?

A bucket of water can dl6a,c vg v4vm1 5)prtbe whirled in a vertical circle without the water spilling out, even at the top of the circle )cpdm l1 t va4rg5,vv6when the bucket is upside down. Explain.

How many “accelerators”q5x4y ozx gj43vsz2z+ do you have in your car? There are at least three controls in the car which can be used to cause the car to aczsz v4 og+z3y254xqxjcelerate. What are they? What accelerations do they produce?

A child on a sled comes flying over the crest of a small hill, as shown iq81tmz * h0hkcn Fig. 5–31. His sled does not leave the ground (he does not achieve “air”), but he feels the normal force between his chest aq8cm 10 hzh*ktnd the sled decrease as he goes over the hill. Explain this decrease using Newton’s second law.

Why do bicycle riders lean inwi h5.9 x 71augrxfqx9j/gc q/uard when rounding a curve at high speed?

Why do airplanes bank when they turn? How would you compute the bankl) j9w9d,zi2 i0 nehfling angle given itwdz )ihnl ,el029j9i fs speed and radius of the turn?

A girl is whirling a ball on at( 5 i0g88vfi-wznf4g) jgsxj string around her head 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 shf8fjgnsvj0 x(g5i)g 8w 4-tziould she let go of the string?

Does an apple exert a gravita* vikpcw)wp (sd)f3 4e0ph8awtional force on the Earth? If so, how large a force? Conside( spw3 kv*appw)e)cd04 h fiw8r an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, in what ways would the Moon :i4wml s .wniix5qno7+m c)3y7lxj (c’s orbit be difmws.)i4cj:(lqmi 5o37yxlx+nn c 7wiferent?

Which pulls harder gravitatj bsag ez;ly *q*xp((k,8oyq )ionally, the Earth on the Moon, or the Moon on the Earth? Which accelerates moreq z 8 ;gklxj(,oqpy)ysb*(ae*?

The Sun's gravitational pullrgy gwn5 z7i51 on the Earth is much larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider th7 gy1 zw5ngri5e 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?v e:g.hhl4e,-fsce e6 What two effects are at work? Do they lese-6.h,v4 cef ehg:oppose each other?

The gravitational force on the Moon due to the Earth is only about h c4l7o2s* cx+qo wl-koalf the force on the Moon due to the Sun. Why isn’t the Moon pc7qxl soo*k lc4-wo+ 2ulled away from the Earth?

Is the centripetal acceleration o dm3zp bd/52p an-l5j)7 ruzitf Mars in its orbit around the Sun larger or smmni )z -23t5zl bd7ru5dpjap /aller than the centripetal acceleration of the Earth?

Would it require less speed to launch a satellite (a) toward the east or sk 1.t-wkh:yuku-z qvu-2q7 w(b) towaquztws u :kkukw- - q-1h7.yv2rd the west? Consider the Earth’s rotation direction.

When will your apparent weight be the greatest, as measured by a scale vb+1zei +mn r.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 t+ mb.reiz1+v nhe same as when you are on the ground?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long periodsp),ym2o1tv)y +ir +ef h8r y-h5szdgb in outer space could be adversely affected by weightlessness. One way to simulate gravity is to shape the spaceship like a cylindrical shell that rotates, with the astronauts walking on the inside surface (Fig. 5–32). Explain how this simulates gravity. Consider (a) how objects fall, (b) thzds m- fghry5)o+i) 1r vy+h,y82ptbee force we feel on our feet, and (c) any other aspects of gravity you can think of.

Explain how a runner expe0zxjxvh3c+a9 o34- g ht4ecad riences “free fall” or “apparent weightlessness” between stepe9x 4hc3a+j ctxvzad0o4gh3-s.

The Earth moves faster in its orbit around the Sugz. 9lpu/l 6w:dr ;nwen in January than in July. Is the Earth closer to the Su pegllnwz6d ru/:w9.; n 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 discov6b n9- wl3 ewkwwb5h)p;7pw wvet +u1ef mhy91ered to have a moon. Explain h 59; ) hly+bmkww-w1eptne3hp 9 vw6bww1u 7feow this discovery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Ear-eow:qifd.-njfn* umqq 1:u1 th is much larger than the Moon's. Yet the Moon's is mainly responsible for the tidesfdnq e*jw q f- :uuo1miq.:-1n. 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 traveling 198t t;ed5ya.n*0wov b4 s0 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 orbithx8c.2 sz h7np;.htl c around the Sun, and the net force exerted on the Earth. What exerts this force h csz.hx8c l;72.pthnon 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 210e38x: f qisse39ci 1md N is exerted on a 2.0-kg discus as it rotates uniformly in a horizontal circle (at arm’s length) of radius 0.90 m. Calculate the speed of the did xcee13msisqf3 9 i8:scus.   

  Suppose the space shuttle is in orbit 400 kmgny2/n m2n* :x3y6lmn 1gucjc from the Earth's surface, and circles the Earth about once every 90 minutes. Find the centripetal acceleration of the space shuttle iu/nnc 1263m:yxcg2m* gjnl ynn 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 onmj- u8 9sv0tyl the edge of a potter’s wheel turning at 45 rpm (revolutions per minute) if the wheel’s diametm9-vly0ujs t 8er is 32 cm?   

  A ball on the end of a string is revolved at a uniform rate in *m6 e tzfctu57a vertical circle of radius 72.0 cm , as shown in Fig. 5-33. If its speed is 4tmfc5 ze *u6t7.00 $\mathrm{~m} / \mathrm{s}$ 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 tu d.5p:1xdxs wfofj -k/rn of radius 77 m on a flat road if the coefficient of static friction between tires and road is 0.80? Is this result independent of the mass of the car? :fox5d-1 .wxjfsd p/k  

  How large must the coefficient of static frictionvb,6pjpuj-0n be between the tires and the road if a car is to round a level curve of radius n6bpv ju-,pj085 m at a speed of 95km/h ?   

  A device for training astronauts and jet fighter pilots is designed n5 )mxn- zapr8jb0sr2to 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 farn-0p zabx852r jm )snst 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 roy h8z( gl,u,*zb.x pksz.k1y1frmj7 i tating turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable until a rate of 36 rpm is reached and the coin slides off. What is the coefficient of static friction between th,,hsp8uklz yfz* b rz1 .g( 7k.mx1iyje coin and the turntable?   

  At what minimum speed must a roller coaster be traveling w:ggw /cq x+*ountqw92h vb,:mhen upside down at the top *g:,vg:uwx t b+c2/omqnw9h qof 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 5r z:/1ga,fvsba4o gt rounded top of a hgza:s1 /,avrgot b4f5 ill (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 minute would a 15-m-diameter Ferris wheel needh/ je7hs/ grgl vgv5v 5aqb),5 to make for the passengers to feel “weightless” at the topmost pgvv,5/r ge5)ah/jqh 7g lbvs5oint?    rpm

  A bucket of mass 2.00 kg is whirled in a vertical circle of radius -1p uc+w6cqwipqe+f:xu2:q a 1.10 m. At the lowest point of its motion the tension in cc 2up+au wp:qex-iw1qf q+6: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) mu)vf1a trt,y1s *actj* st a centrifuge rotate if a particle 9.00 cm from the axis of rotation is to experience an accelerat* tfavtr1t1)sy ac,j *ion of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, uwe ,se,4 f08:wusp.toqet1s people are rotated in a cylindrically walled "room." (Sfs,,s0tow4e. s uw:uq81tee pee 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 ) c,a7, n w-uy.kjwm0rt is rotated in a circle on a frictionless air-hockey tabletop, and is held in this orbit by a light cord connected to a dang awunw,0k- t jr,m.yc7ling 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 ofudigne8+yl /53-tzzd the ball which revol/ -8dt d3n5gzluzey+ives on a string 0.600 m long. In particular, find the magnitude of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ , and the angle it makes with the horizontal. [Hint: Set the horizontal component of $\overrightarrow{\mathrm{F}}_{\mathrm{T}}$ equal to m $a_{\mathrm{R}}$ ; also, since there is no vertical motion, what can you say about the vertical component of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ ?] $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ =    the angle =   

  If a curve with a radius of 88 m is perfectly banked foy;w fp+sdn p8v 26/eyur 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 v evxq7dg+x)* $ m_{1}$ and $m_{2}$ , are connected to each other and to a central post by cords as shown in Fig. 5-37. They rotate about the post at a frequency f (revolutions per second) on a frictionless horizontal surface at distances $r_{1}$ and $r_{2}$ from the post. Derive an algebraic expression for the tension in each segment of the cord.

  A pilot performs an evasive maneuver by diving vertically at 31b xfzk5 0e4y:6og6ox8hl *wfi0 $\mathrm{~m} / \mathrm{s}$ . 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 thtr1oe5sc .smc7 kepbn4 d;,n/ fsnv(2e net force exerted on d(/vns ncs pte 71m4nkrb5;e 2,oscf.the car (by the 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, nkm(:)4. ijmfm)zr0 nq tzdbwo1,y 37rapzo1going 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 constant tangential acceleration. If the curve were flat, what would the coefficient of static friction haf 14),kiabznz(rj om3 q)tp1zm7d w.0 m:ronyve to be between the tires and the road to provide this acceleration with no slipping or skidding? $\approx$   

  A particle revolves in a horizontal circl2c.1w9s hygv8cm*q8nzu.u aqe of radius 2.90 m . At a particular instant, it v1u.c8z9 g.8n2wq*cau hsqmy s acceleration 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 awq hq8rj +(5n+gqs3zo spacecraft 12,800 km (2 Earth radii) above the Earth’s surface if its mass is 1350 wq+ n og+r53(qq8hszj kg.   

  At the surface of a certain planet, the gravitational accelee wtu.1cxy)/wi q4lq (+ ;jzpgration g has a magnitude ou /gz lwyc(p xqj q;.i1+4e)wtf 12 m/s$^2$ A 21.0-kg brass ball is transported to this planet. What is (a) the mass of the brass ball on the Earth and on the planet   
(b) the weight of the brass ball on the Earth and on the planet? $W_{Earth}$   $W_{planet}$  

  Calculate the acceleration due to grav bs9ly.+:aktzfx9 6of bi mknt1 0n3i/p f38rdity on the Moon. The Moon's radius is 1.74 if+mk.da/l t3f z syrfn90no9x kp b68i:31bt $\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 ovf)i++so l2tjf Earth, but has the same mass. What is the acceleration due to gravity neft+os +l)2 ijvar its surface?   

  A hypothetical planet has a mass 1.66 times that of Earth- /yre3 lja u8uamx+af g*;usl iv1.3m, but the same radius. What is g nv+*fls a gxjye8u1u 3.r m;a 3l-ai/umear its surface?   

  Two objects attract each other gravitationally with a fo8.,q vaclb* xyrce 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 accel7dypm-tu5udmc 5 5k 1dmgq43;yx.d iheration of gravity, at (a) 3200 m 5;mdg- 4ci 5dd7m. hpd3kum5x1yutq y   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside the Earth 5nkzus496 rs;od(g j mwhere thegravdu6o 5(sg4n9r;zj s mkitational 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 ou hbmc9 -kakaa1a4b5v(r Sun packed into a sphere about 10 km in radius. Estimate the surface gravity on bm 54 1k9ak va(abac-hthis monster.    $\times10^12$

  A typical white-dwarf star, which once was an average star like our Su w;alf xd2 ie4yt4u:e;n but is now in the last stage of its evolution, is the xd ;;2 4uf te:lay4ewisize of our Moon but has the mass of our Sun. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronauts feel weightless whifthpfg+ 9lcih 7:/8tzm3j c;e le orbiting in the space shuttle. Your friends respond that they thought gravity was just a lot weaker up there. Convince them and yourself that it isn’t so by calculating the acc;ti c9mgf:7+8zh/3fl hcep jt eleration of gravity 250 km above the Earth’s surface in terms of g.   

  Four 9.5-kg spheres are lv-l+l5au l)o: 7ujp(s p*c ceyocated at the corners of a square of side 0.60 m. Calculate the magnitude and direu ppllj*5ce(7l-: ac+souvy)ction 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 on the same side of thmjmq2*efqn j(dt 8u:c45p h+v e Sun. Calculate the total force on the Earth due huv:nc2qj*(ej mp 5tfmq 84+d 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 accell)zv jcl+sed5. y1ixz u3,q2 heration of gravity at the surface of Mars is 0.38 of what it is on Earth, and that Mars’ radius is 3400 km, determine the mass of Mars. ,5.i dqluz21hs j3y)vx c lez+   $\times 10^{23}$

  Determine the mass of the Sun using the known value for l zg:zci.b w, *o/liojr:(wm; the period of the Earth and its distance from the Sun. [Note: Compare your answer joc*;i o(:mz.zlwlrbw/ g:,i to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a sateg 5qjqb2/ dxz;llite moving in a stable circular orbit about the Earth at a height o5/jxd;qbz g2 qf 3600 km.    $\times 10^3$

  The space shuttle releases a satellite into a circular orbit 650 km above the Ea b13ha0x pfz3 c/e0fljrth. How fast must the shuttle be moving (relative to Earth) when thfchp 3fez 00x /1a3jble release occurs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotatg*b ;nc(oiqq,e if occupants are to experience simulated gravity of 0.60 g? Assume the spaceship’s diameter is 32 m, and give your answer as the ti(n q *gqc,;boime needed for one revolution. (See Question 21, Fig 5–32.)   

  Determine the time it takes for a satell0a*qd 2 h)c)nwl gid/4vfako-ite to orbit the Earth in a circular “near-Earth” orbit. A “near-Earth” orbit is one at a height above taln0kq4*d 2d f/ v)c-hgw)aoihe 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 velo uzyc25.7 +mob:r-8h votrk *pqsyg:zcity would a satellite have to be launched from the top of Mt. Everest to be placed in a circular orbit a2y-r:t zo *m:s c85+.gz7oqbykr vuh pround the Earth?   

  During an Apollo lunar landing mission, the command module continued to o m.f gw4pi;1hgrbit the Moon at an w.gmp4fg1hi; altitude of about 100 km. How long did it take to go around the Moon once?    s

  The rings of Saturn are composed of chunks of ice that orbit th1kemy(n+wuyv48)yc k+oa54 prczp l/ e planet. The inner radiuyky wym e z1orupv8 lcn4 /(+5kcap+)4s of the rings is 73,000 $\mathrm{~km}$ , while the outer radius is 170,000$ \mathrm{~km}$ . Find the period of an orbiting chunk of ice at the inner radius and the period of a chunk at the outer radius. Compare your numbers with Saturn's mean rotation period of 10 hours and 39 minutes. The mass of Saturn is 5.7 $\times 10^{26} \mathrm{~kg}$ .


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

  A Ferris wheel 24.0 m in diameter rotate0vp/nsto a -s*8+uud o3ir8j es once every 15.5 s (see Fig. 5–9). What is the ratio of a person’s apparent weight to h*0p/vj 8uui- o3sdnto+r 8aeser real weight (a) at the top, and (b) at the bottom?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut 4200 km from the center of thrlvbo.w.ub/y n ,egljdf+69 +e n fb+9 o uj/ gl.v,.lry+bewd6Earth's Moon in a space vehicle (a) moving at constant velocity,     ----待选择----towards the Moonaway from the Moon 
(b) accelerating toward the Moon at 2.9 $\mathrm{~m} / \mathrm{s}^{2}$ ?     ----待选择----towards the Moonaway from the Moon 

State the "direction" in each case.

  Suppose that a binary-star system consists n yyjj+ 6j.7e2kzag kb- w9*bpof two stars of equal mass. They are observed to be separated by 360 million km and take 5b jp+*2- kyeg7j n zkja9by.6w.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 ra5gsj+q j34p read for the weight of a 55-kg woman in an elevator tha+aq5j4sp3r gjt moves
(a) upward with constant speed of 6.0m./s   
(b) downward with constant speed of 6.0m./s   
(c) upward with acceleration of 0.33 g,   
(d) downward with acceleration 0.33 g   
(e) in free fall?   

  A 17.0-kg monkey hangs from a cord suspended from the ceiling of an elevat :. tdre,phkdr 1ag.lzcu/q; 8:z:-aixwk3jr or. The cord can withstand a tension of 220 N and breaks as the elevator accelerates. What was the elevator’s minimum aclr h qgaera-3 tp.d. ;1 , 8jkd/:wr:iczxk:zuceleration (magnitude and direction)? a =   

Show that if a satellite orbits very near the surface of a pl5fr *zc(hs ch3anet with period T , the density (mass/volume) of the pl5rcczs 3 *hf(hanet 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 o+iec x1p3n 3dlf the Moon (27.4 d) to determine the period of an artificial satellite orbiting very near the Earth’s surfacp xnd 3el3c+i1e.    s

  The asteroid Icarus, though only a few hun iuze34g t2u3adred meters across, orbits the Sun like the planets. Its period is 410 d. What is its mean d234u uezg3i atistance from the Sun?    $\times 10^{11}$

  Neptune is an average distance offp6 sgg*iks,.dk c- u/ 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 cl1e4q-m*5mnvcfhls), : fyu yrose 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)crvfnymmhu 5 *eq fys-l,1:4 is half the sum of the nearest and farthest distance from the Sun.]    $\times 10^6$

  Our Sun rotates about the center of the Galdvfq d, d4+/ulaxy $\left(M_{G} \approx 4 \times 10^{41} \mathrm{~kg}\right)$ at a distance of about 3 $\times 10^{4}$ light-years $\left(1 \mathrm{ly}=3 \times 10^{8} \mathrm{~m} / \mathrm{s} \times 3.16 \times 10^{7} \mathrm{~s} / \mathrm{y} \times 1 y\right)$ . What is the period of our orbital motion about the center of the Galaxy? $\approx$   $\times 10^8$

  Table 5–3 gives the mass, period, and mean distance for the four largest moo6.a l6lbfh0gi8 c v5zbns of Jupiter (those discovered by Galileo ib 6i6lg b.zvcfl 08ha5n 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 periodmx3s x9y n;usqp5(e *s and distance of the Moon. ny xss (9;s5* pm3quex   $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jupiter’s moons, usingnnyy9g9wq()7gc*bgzk d8 h8ez)tia ) Kepler’s third law. Use the distance of Io and the periods given in Table 5–3. Compare to the values i)k zq89(8w yzb)h9gneit) nc7ygd*ga n 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 f.mt70 f) hu6mjh;je meragments (which some space scjhu0ee)mm7j t m ;h.6fientists 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 artificia3uucj +ro6 0)zto,gk tl "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 6o)juot0z3 kgt, cru+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 1hm23x 4heuku a hanging vine (Fig. 5–41). If his arms are cax3e ku41hm hu2pable of exerting a force of 1400 N on the vine, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 80 kg, and the vine is 5.5 m long.   

  How far above the Earth’s surface will the acceleration of gravity be half wh-dt9j-uta j y;at it is on the surfaaj 9-y-ttu jd;ce?    $\times 10^6$

  On an ice rink, two skan)*u0bene6q)cb 4jyg ;i z al/ters of equal mass grab hands and spin in a mutual circle once every 2.5 s. If we assume their arms are each 0.8u0 ) eqj;g4n*izbanc l)ye6b/ 0 m long and their individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates once per day, the apparent acceleration of gravity a967tvn+6 l9k tcyy9uex ycw2ht the equator is slightly less than it would be if the Earth didn’t rotate. Es7y 6yh9c9y 6ku9 +nttcxwle2vtimate the magnitude of this effect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from the 03a*7cq xgv,4 awc xqkEarth will a spacecraft traveling directly from the Earth to the Moon experic,k*v xwxg3cqq0a 74aence zero net force because the Earth and Moon pull with equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, but when yoc.vje1ld8uy2 gpqfgp6 3fi: 3u stand on a bathroom scale in an elevator, it says your mass is 82 kg. What is the acceleration of the elevator, and in which dire:6egvd3.3yq28pfl i1 jfugp cction?     ----待选择----upwartddown 

  A projected space station consists of a circular tube that will rotate agb 4f y+k6qd+cbout 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 (revolutions per day) if an effect equal to gravity at the surface of the Earth (1.0 g) is to be fe fgk4++d6c qbylt?$\approx$    $\times 10^3$

  A jet pilot takes his aircraft in a vertical md, xcn2km;7o w9ay ;aghjwq *ag9;f9loop (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 ac tf492k6fm7vnmi ds,v2 jd/z pceleration due 2 tk9mivddp f4n 6vm2fj,7s/z to gravity at its surface and the gravitational constant G.

A plumb bob (a mass u)z*b 9bk6u zpzc2d1km hanging on a string) is deflected from the vertical by ankb *)z1 zpzub9u62dck angle $\theta$ 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 blr efk*tn // mac,9.lcanked for a design speed of If the coefficient of static friction is 0.30 (wet pavement), at what *t,/lc elcmk9/ ra.nf range of speeds can a car safely handle the curve?

  How long would a day be if the Ear2g 0b ud zb+-t)lpkk sowx-z:44-atdtth were rotating so fast that objects at the equator bk4t2)4txz dp0z +ugtob - kda:s-w-l were apparently weightless?    $\times 10^3$s

  Two equal-mass stars maintain a conshpn -ch85dv +o;r/ukf cj1wpe20yhqd2m2p4jtant distance apart of 8.0 $\times 10^{10} \mathrm{~m} $ and rotate about a point midway between them at a rate of one revolution every 12.6 yr . (a) Why don't the two stars crash into one another due to the gravitational force between them?
(b) What must be the mass of each star?    $\times 10^{26}$

  A train traveling at a constant speed a0 mwnha1mt /4rounds a curve of radius 235 m. A lamp suspended from the ceiling swings out to an angle 4tm1aw/0mnah of 17.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times as massive as the Earth. Thus, it has been 8 :)q5xj oyk:8rmr rjjclaimed 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 mjrj:)r8q5 k8r:xojy following data for Jupiter: mass =1.9 $\times 10^{27} \mathrm{~kg}$ , equatorial radius =7.1$ \times 10^{4} \mathrm{~km}$ , rotation period =9 $\mathrm{hr} 55 \mathrm{~min}$ . Take the centripetal acceleration into account.    g's

  Astronomers using the Hubble Space Telescope deduced the presence of7b 9h9m )pmlzi u*fuj0 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 measu0b)lu 7 mujhmi9f*p9zring 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 spee h8r3 e;sw9c pzv4nue1d 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?) Expu r 9zn;h3p vs8c14eewlain. (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 2mw8jzagis pv:i es5h:y 73x9( 4 satellites orbiting the Earth. Using “triangulation” and signals transmitted by these satellites, the position opi:398wxgs hve(5am i y j:s7zf a receiver on the Earth can be determined to within an accuracy of a few centimeters. The satellite orbits are distributed evenly around the Earth, with four satellites in each of six orbits, allowing continuous navigational “fixes.” The satellites orbit at an altitude of approximately 11,000 nautical miles [1 nautical ]. (a) Determine the speed of each satellite.    $\times 10^3$
(b) Determine the period of each satellite. =    hours

  The Near Earth Asteroid Rendezvous (NEAR), after traveling 2.1 billion km, i-,a jwaz,jjtu ; 2e+wqs meant to orbit the asteroid Eros at a height ofa j+ zjtq,ue- ,j;aw2w 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 z 1.5trg vo9 1ozrr3yp /4d-qkenx+j mrepair. You are in a circular orbit of the same radius as the satellitep z vdt5j. kx4r +y1qo3 9z-ngo1r/emr (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 ik-tr)((k h skziu5b43000 years. (a) What is its mean distance from tz4kk(-i)thik b ru5( she 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 n8o4y3szz9(kcso wnfkd9 4q m e j7nt+8eed to be if you could actually "feel" yourself gravitationally attracted to someone near you. 8 sq+onz7z4fk 89ye( wodjtm ckns349Make reasonable assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the zy6 ckc,s0 u3di4e7fhpwdte: o ak*;/Milky Way Galaxy (Fig. 5-46) at a distance ;0a7dewedo6zkc y:tfs*u /h 4 ,k 3picof 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 locaqll4ypkxj fj *5 0x(/wted at the corners of a square 0.50 m on each side. Find the magnitude and direction of the gravitational force on a fift*lfjw y40j/q5xpk(x lh 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 o,ov5iu:aje f *rbits the Earth $ \left(\operatorname{mass}=6.0 \times 10^{24} \mathrm{~kg}\right)$ and has a period of 6200 s . Find (a) the magnitude of the Earth's gravitational force on the satellite, $\approx$    $\times 10^4$
(b) the altitude of the satellite.    $\times 10^5$

  What is the acceleration experienced by the tip of the 1.5-cm-long s3 3oqnwmd-7ssweep second hand on your w osnm3q3dw7s- rist watch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weight around gfyujumhy e*l1i9 (3 ax5l4;b in a circl 59hilugab1;34 ej l(* mfyxuye 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 the vertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in a new highway in+(ltxp st *2zs designed so that a car traveling at sp*psztt2 xnl(+eed $ 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 Ac 3/tlwlj -d8ycela, 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 (approximated3cl8-ty jlw/ ly ). (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.$\approx$    $\times 10^2$