Sometimes people say that water is removed from clothes in a spin dryer by cengu,g o( n7*yvp 7apugjwh1)pe37* mkxtrifugal forcw)7 m 7ge(p ao7npkx3u hg,*vp*y 1gjue throwing the water outward. What is wrong with this statement?

Will the acceleration of a car be the s4f1gljy nsz1)5ip cp)ame when the car travels around a sharp curve at a constant 60 km/h as when i1fp nyg1jz5li4) ) psct travels around a gentle curve at the same speed? Explain.

Suppose a car moves at constant speed along a hilly road. Ww6 q+;wf gbozv-hq+j;here does the car exert the greatest and least forces on the road: (a) at the top of a z gfvwjq h q;;o+w+6b-hill, (b) at a dip between two hills, (c) on a level stretch near the bottom of a hill?

Describe all the forces acting on b b3gutu52 ol:a child riding a horse on a merry-go-round. Which of thesg53:ubobu2 tle forces provides the centripetal acceleration of the child?

A bucket of water can be whirled in a vertical circle without the water spr*; (dfnv atw03wws- killing out, even at the top of the circle when the bucke w w3t- ns;kvf(0rd*wat is upside down. Explain.

How many “accelerators” do you have in l0wt kj 82oojm3+peg,your car? There are at least three controls in the car which can be used to cause the car to accelerate. W2 8 j kj+3,geptwlom0ohat are they? What accelerations do they produce?

A child on a sled comes flying ov8dk3rc*vnuvl6u *n5;op( ; lwlct7rner 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 hil*k3wnpn;ull*nu;7r 5l cr8dov6v c(t l. Explain this decrease using Newton’s second law.

Why do bicycle riders lean inward when rouof/i di.te)z o*yu ti56/j c-pnding a curve at high speed?

Why do airplanes bank when they tq7zmh:j ef3sx3.rfdi /oz , z x0fs8/rv.qd.u urn? How would you compute the banking angle given its spz7f.s:du3 ,fqv.ex8 jfx0s hmi rzqodr3.//z eed and radius of the turn?

A girl is whirling a ball o 7a,.ru:-7 pqqv(guulxsgoe1 n a string around her head in a horizontal plane. She wants to let ggouaqv u7,xl(1r.-qg7:s upeo at precisely the right time so that the ball will hit a target on the other side of the yard. When should she let go of the string?

Does an apple exert a gravitational force on the Earth? If so,t ac7:,u 4iilc9yxeq* how large a force? Consider an applc*7iltaxq4u y9,i ec:e
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, in wjp(f hs 9-f*izx g*3chhat ways would the Moon’s orbit be dizsp3** -j (c9higfxf hfferent?

Which pulls harder gravitationally, the Earth on 60a dbzmjb21jthe Moon, or the Moon on the Earth? Which accelerates d1bbmz6jj0 2a more?

The Sun's gravitational pull on the Earth is mucb7 bqh-+f a2ruh larger than the Moon's. Yet the Moon'au h+f7b b-r2qs is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]

Will an object weigh more at the equator or at the poles? 5fpu :n3n+:yl6at lfs What two effects are at work? Do theyl:pt lyau3ff+s n5n6: oppose each other?

The gravitational force on the Moon due to the Earth is only about half thz8mwgzr4) pr6 eso+ 12h;vyqz0myc2a ldva32e force on the Moon ddy4 2rlear2+ 8;6) 3zvy1 zzop0awscm2qhvgmue to the Sun. Why isn’t the Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in its orbit around the Sun larger or s-(+s.(lfi.h:cklu khci em8 jmaller than the centrip k8c:. k(c ehl+sfu-ilh.jm(ietal acceleration of the Earth?

Would it require less speed to launch a satellite (a0oz )aon585ky x.blvd ) toward the east or (b) toward the west? Consider the Earth’s rotation directod v0585ynxab ol.)zk ion.

When will your apparcgujs+l7 7qd; fz93 ub ub9yt0ent weight be the greatest, as measured by a scale in a moving elevator: when the elevator (a) accelerate7ju7 + u0b9bq;9sz glyutcd f3s downward, (b) accelerates upward, (c) is in free fall, (d) moves upward at constant speed? In which case would your weight be the least? When would it be the same as when you are on the ground?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long periods in oiy8 me ;n )rn1;wu5vrruter 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 um;r;eyr8i1 )nrwv5n(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 weio8: ygv)5h vmpt 7bcb)ghtlessness” between st7)o yg: 58bhv mcvbp)teps.

The Earth moves faster in its orbit around the Sun in January e-w9w7(d;g t y c*x xgn( g8lf)yvmll-than in July. Is the Earth closer to the(l;ngxm y)w cx* vlg8w gy-9d l7-ft(e 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 ujk 3f2je 7.spzntil it was discovered to have a moon. Explain how thj7 p2kfz e.js3is discovery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth is much larger than thek2th43x,c6 up nujann+tkhrmn96 ;1 d Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]A child sitting 1.10 m from the center of a merry-go-round moves with1;pt ju+ut4 6c,hm6arnnkhn2n3k9d x 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 19 c /rl9;ebfy3iws4f q:80 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 thetsttp e(+rf 2g, /l7la Earth in its orbit around the Sun, and the net fors/afg7p trt+ll e2,(t ce exerted on the 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 exerted on a 2.0-kg discus as it thdn-zw0taxd4ium qd 9b;2fjw94q() m7 kqg 6rotates uniformlya;26t duqm)fdw44m(tjnz 799bgwh qi-d qx0k in a horizontal circle (at arm’s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is in orbit 400 km froeok.u0 zpd7* bhs w8k, 0 +v6mq:txdoj 5q(fsom the Earth's surface, and circles the Earth about once every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in term, d hzxwb05*vdjt+uoom qkpfs e8o0.6s(7 kq:s 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 edg +x7b;oqwrgjv (/j9k:5izp j4b;ujei e of a potter’s wheel turning atzbjp:7b5 ki j; eg(v;94+rqj jxiwuo/ 45 rpm (revolutions per minute) if the wheel’s diameter is 32 cm?   

  A ball on the end of a string is revolved at a uniform rate in a vertical circl gwt+ ai(o/ 4tlr+2m3v3zotl me of radius 72.0 cm , as shown in Fig. 5-33. If its speemz+3 (+/oglto2w4 i vrtlm3at d is 4.00 $\mathrm{~m} / \mathrm{s}$ and its mass is 0.300 kg , calculate the tension in the string when the ball is (a) at the top of its path   
(b) at the bottom of its path.   

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

  What is the maximum speed with which a 1050-kg car can 5vrj, h(2sc7; mwci edr2oa0dround a turn of radius 77 m on a flat road if the coefficient of static friction between tirv hacwds5 0 (i2me;r27rdcjo ,es and road is 0.80? Is this result independent of the mass of the car?   

  How large must the coefficient of static fr(t)ij1tnz9jfl7 uo4 miction be between the tires and the road if a car is to round a level curve of radius 85 m at n 4 uot1mj j9l7(fit)za speed of 95km/h ?   

  A device for training astronauts and jmn6- 1xyvcn c0et fighter pilots is designed to rotate a trainee in a horizontal circle of radius 12.0 m . If the force felt by the trainee on her back is 7.8n 0mx vcc-1n6y5 times her own weight, how fast is she rotating?    rev/s    m/s
Express your answer in both $ \mathrm{m} / \mathrm{s}$ and $\mathrm{rev} / \mathrm{s}$ .

  A coin is placed 11.0 cm from the axis of a rotating turntable of vagzgsl:g q61n4 c (que1riable 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 the coin and thgz1c1n :glq qg6e(4use turntable?   

  At what minimum speed must a roller coasteruwg 5tl1nd) iu-;0bmj be traveling when upside down at the topnuugb )- l15tmdij ;0w of a circle
(Fig. 5-34) so that the passengers will not fall out? Assume a radius of curvature of 7.4 m .   

  (II) A sports car of mass 950 kg (including the driver) crosses the; gt/)q+u xz tohhh1i:4ttbk7 rounded top of a hill (ratq/t o1bzxh;kh :+ig 7ht4)u tdius =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 qqxeqfws:s45e g z)77 a 15-m-diameter Ferris wheel need to make for the passengers to feel “weightless” qqgxf 74s)se7:5ez qwat the topmost point?    rpm

  A bucket of mass 2.00 kg is whirled , 13 vo/x6apf o.eojwtin a vertical circle of radius 1.10 m. At the lfo,6j vat/1 p3oowxe.owest point 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 ro p3h,yfwt.t7otate if a particle 9.00 cm from the axis of rotation is to experience t.7f3wopyht, an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotated i s56hl u-apylhu*gqtw 9*t( ,nbbid-7 n a cylindrically walled "room." (See Figd( as bb*, l9twu nhlh--t5pui*7 6yqg. 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 circ:l:j3 oatgip1,ofyr1 le on a frictionless air-hockey tabletop, and is held in this orbit by a light cord connected to a dangling block f: ,li1tr:gy o1oap j3(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 weightq iqla3 z;o6g) of the ball which revolves on a string 0.600 m long. In particular, find the ; )giloqaq3z 6magnitude 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 odo0qkp.c+rnc5q: 0suwo b0b :f 88 m is perfectly 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 masse4fevvsx7f j91 s $ 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 evasil1o g 1s2je/zj1*fo9y2milqkve maneuver by diving vertically at 310 $\mathrm{~m} / \mathrm{s}$ . If he can withstand an acceleration of 9.0 g 's without blacking out, at what altitude must he begin to pull out of the dive to avoid crashing into the sea?   

  Determine the tangential and centripetjlu 2+5k;gk/ymj2 lk6eflk4f6hb :pu al components of the net force exerted on the car (by the ground) in Example 5 :fu6egl6;uj+j5 m2yh/bk2k4l kplfk-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 Indiana2;i9 msspd qm*polis 500 accelerates uniformly from the pit area, going from rest to 320km/h in a semicircular arc with a radius of 220 m. Determine the tangential and radial acceleration of the car when it is halfway through the turn, asims s;9 2dqmp*suming constant tangential acceleration. If the curve were flat, what would the coefficient of static friction have to be between the tires and the road to provide this acceleration with no slipping or skidding? $\approx$   

  A particle revolves in a horizontal circle of radius 2.90 m . At a particulargrn e5t*m9*6 rj (uruc instant, its accelen *cjg5 (r9treu6mu*rration 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 spach 349i7d2c ,ni kl2teafmrycmka10n8ecraft 12,800 km (2 Earth radii) above the tn2i7ck1adf4c 9mke2ynr 83, a h 0ilmEarth’s surface if its mass is 1350 kg.   

  At the surface of a certain plax8vbx5h ra12s net, the gravitational acceleration g has a magnitude of xbvxh 8rs1a5212 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 w ,qzg 6lw8hf.due to gravity on the Moon. The Moon's radius is 1.74 ., lf8zwhgwq 6$\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has a radius 1.5 ti1v ljru alea-so c+u 5 )9q.g1l3w.gvmmes that of Earth, but has the same mass. What is the acceleration due to gravity g1.eagwu) .qrclo+s39 1lv j-lavm5 u near its surface?   

  A hypothetical planet has a mass 1.66 times that of Earth, but the same radiyn8i sraz+8eo wy yzcu/n/*v + zc6w).kdpf;.us. What is g nyw efi +c.o *rk/ n. ypwu/8)+ay8snzvz6cz; dear its surface?   

  Two objects attract each oxqc67 n lep)ot:;mf :tther gravitationally 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 m-+r/( o7izjmrazi3 lgravity, at (a) 3200 m zm3jl-(m ra/+7riozi    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside the Earth reskx;(ej + s22v3 suqwhere thegravitational acceqk2r(vuej+2e s;xs 3s leration 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 packclpw4bf3 /j5oew ,aca v1n7j;ed into a sphere about 10 km in radius. Estimate the surface gravity on this monster3we cj4vf;7,c 5j/ olapn1wab.    $\times10^12$

  A typical white-dwarf star, whiw monze5*+:r kiy( ;cqch once was an average star like our Sun but is now in the 5;:z n oqyw ker(+c*imlast stage of its evolution, is the size of our Moon but has the mass of our Sun. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronauts feel weigh fuqy)f+0;w emtless while 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 isymu0ew+ ; )qffn’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 sidmnk / 6 i43-joyni6idx5kxd+u e 0.60 m. Calculate the magnitude and direction of the total gravitational force jdimnn56 k6ix dok34xyu+-i/ 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: wqdl)jn m7-e03yisr same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, asq nj 7ewmirds)ly0 3:-suming 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 ry*t gmuvdv+ lo.(01j of Mars is 0.38 of what it is on Earth, and that Mars’ radius i (1* tv m0rlujo.+gvdys 3400 km, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value for the pq/zi/10+lx y)84 t d 9u q1yuykrwmjrperiod of the Earth and its distance from the Sun. [Note: Compxqr4011jd)q t+ r9 z/lywyy/p umki 8uare your answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite moving;ydki3d,rn3 b1fkr zr n/f(b5 in a stable circular orbit about the Eartk(r dd3 ;zy5,n/n1rfbfrbki3h at a height of 3600 km.    $\times 10^3$

  The space shuttle releasy//s8v- wuvt res a satellite into a circular orbit 650 km above the Earth. How fast must the shuttle be moving (relative to Earth) when the release oc8vr v-y/ts wu/curs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if l js(7lba0 z-yoccupants are to experience simulated gravity of 0.60 g? Assume the spaceship’s diameter is 32 m, and give your answer as the time needzbsjly7 (-a l0ed for one revolution. (See Question 21, Fig 5–32.)   

  Determine the time it takes for a satellite to orbit thek;ed* t gpwxm-; wd5t+ Earth in a circular “near-Earth” orbit. A “near-Earth” orbit is one at a height above the5tx pdd;emktw ;+gw *- surface of the Earth which is very small compared to the radius of the Earth. Does your result depend on the mass of the satellite?    s ~    min

  At what horizontal velocity would a satellite have to be launp,t1h5vs2 meuched from the top of Mt. Everehm 5 suet1vp2,st to be placed in a circular orbit around the Earth?   

  During an Apollo lunar landing mission, the command module conti :2d;;sq /jhg/;gxglwyqb0j qnued to orbit the Moon at an altitude of about 100 km. How long did it take to go around the Moojlswy ;0q/qb/dgg;qjxg; h2 : n once?    s

  The rings of Saturn are composed of chudeub(4t,: 5-kwvl btll/wk7znks of ice that orbit the planet. The inner rab uk 5tlbl(k4ze:v -l, w7/dtwdius 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 rotates once eoe :b hfbt h(/lb e8xze:w3u cxe*5,f7very 15.5 s (see Fig. 5–9). What is the ratio of a person’s apparent weight to her real weight (a) atf :l(h:8 b/ef x5ubwxhbto zec73ee,* the top, and (b) at the bottom?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut e. 4dw6x o 4s0:glnlaxoo jneoy(*:/q4200 km from the center of the Earth's Moon in a space vehicle (a) moving at co 4yo:ne(l/ ond0gsj6 l :aexo.wq 4x*onstant 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 syste6j-s4 batr+q z7)f1i bxaov.zm consists of two stars of equal mass. They are observed to be separated bxt- 1a7+ zz v.rbsbo4iq)aj6 fy 360 million km and take 5.7 Earth years to orbit about a point midway between them. What is the mass of each?    $\times 10^{29}$

  What will a spring scale read for the weight of a 55-kg woman in an f(o ua4 x tk r;aby.*724ib3ocrohxh:elevator that move b*2out c.oa7kiy hb34 4( oaxx;:frrhs
(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 them:rgtdn4yan+sbb x9*8 +hps* egg-l - ceiling of an elevator. The cord can withstand a tension of 220 N and breaks as the elevator accelerates. What4+yp8b msl s9h*t:*+eg -nabngxd g-r was the elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orbits very neaevq d *t26f+bar the surface of a planet with period T , the den* 26vfq db+atesity (mass/volume) of the planet is $\rho=m / V=3 \pi / G T^{2}$ .
(b) Estimate the density of the Earth, given that a satellite near the surface orbits with a period of about 85 min .

  Use Kepler’s laws and the period of r6*q:fsnys3t the Moon (27.4 d) to determine the period of an artificial satellite osyf t6rsqn*3 :rbiting very near the Earth’s surface.    s

  The asteroid Icarus, though only a few hundred meters across, orbits thc1 hpjhidj(1s 7 tjt+2e Sun like the 1 dj s2i+hjh p(tj1c7tplanets. Its period is 410 d. What is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average diskjko ). lw7sv5jcq hj/dsbh,;dwy4/+ tance of 4.5 $\times 10^{9} \mathrm{~km}$ from the Sun. Estimate the length of the Neptunian
year given that the Earth is 1.50 $\times 10^{8} \mathrm{~km}$ from the Sun on the average.    $\times 10^2$

  Halley’s comet orbits the Sun roughsp8 :5+a oxr j 0cw46qptep/bxly once every 76 years. It comes very close to the surface of the Sun on i/x pj6t rqo 45xseca80wbp :+pts 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.]    $\times 10^6$

  Our Sun rotates about the center of the di(nhy;1*qex 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, andq:;dw xc. jkhp+bu4( k mean distance for the four largest moons of Jupiter (thos q;(x.kbckhjw4pu +d: e 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 distance of pxfu 29s ev/r8 uz,4muw+h6bdthe Moon. sf9w d xbu/26mhpueu4zr v8,+   $\times 10^{24}$

  Determine the mean distance from Jup +n3tur(f fjx-iter for each of Jupiter’s moons, using Kepler’s third law. Use the distance of Io and the periods given in Table 5–3. Compare to the values in ttur(+x nfj 3f-he 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 9yp ;rhtq1cd6and Jupiter consists of many fragments (which some space scientists think came fromth d69c;y p1rq 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 artificukza(u l* at2(ial "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 oa*l(uta2z( kuf revolution, this planet's year, in Earth days?   

  Tarzan plans to cross a gorge by swinging in an arc from a hanging vine (Fig.u3d9cnc 8hjv );1nrq x 5–41). If his arms are capable of exerting a force of 1400 N on the vine, what is the maximum speed he can tole;rh)nccq x38v1jdnu9rate 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 gravitfrsgkz856)hk5w /7gqflo;ouy be half what it is on the sur fsouk7zg8gkr; oql5/ 5f) hw6face?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spin in a mzvuhj8 w5m+:wutual circle once every 2.5 s. If we assuvm+wh5 j8u:zw me their arms are each 0.80 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 pkgl u p6j-4*hy 4x2i 8jro)t,u; piuzcer day, 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 fracx) zp huriiko;l u yg6-, tujc8p424*jtion of g is this?    $\times 10^{-1}$

  At what distance from the Earth will a spacecraft trav.61yccji onpv:: a h*meling directly from the Earth to the Moon experience zero netj6cyio mn*:c1av:hp. force because the Earth and Moon pull with equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, but when you stand on a bathroom lkz w1m4rbj h/rzv: 40scale in an elevator, it szh b4v m :lzjk4w0/r1rays your mass is 82 kg. What is the acceleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consists of a circular tj nop1f94e4y eube 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 (revolutions per day) if an effect equal to gravity at the surface of the Ea49yfnepo1 e4 jrth (1.0 g) is to be felt?$\approx$    $\times 10^3$

  A jet pilot takes hist 2tvtq*/94pzyww ) soilm6m 3 aircraft in a 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 f*pkj0x54 tglbpi(riakc )-*acceleration du04gi5jxk-ca*t rlf p)kib p *(e to gravity at its surface and the gravitational constant G.

A plumb bob (a mass m hanging on a string) is deflected from the veu; mgi o8k, y-hw1mtqnlo96p5rtical by a,p;ou5w8 -mnqo9hmkt l 6iy 1gn 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 banked for a design speed of If the co*w/u jzna88l :oa wrk(efficient of sta:wza j wu on*(r/k8a8ltic friction is 0.30 (wet pavement), at what range of speeds can a car safely handle the curve?

  How long would a day be m t3 u)55re5yhldo+ rmif the Earth were rotating so fast that objects at the equator were apparently we r3tu)m5le+5m5doyhrightless?    $\times 10^3$s

  Two equal-mass stars maintain a constant dis ,n4 :*d)k vmzufw70lst4rjs ktance 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 rol-pprksk1zh10vr6t ( unds a curve of radius 235 m. A lamp suspended from the ceiling swings out to an angle of 17 0zp1 k-ptkh(rv rsl16.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times as massive as the Earth. Thus, it has beetx,;mu)p3xo pn 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 em ;ptxp) ,o3uxquator 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 Hubbc)jz2*egv 8wnle Space Telescope deduced the presence of an extremely massive core in the distant galaxy M87, so dense that en*28c jzw)g vit could be a black hole (from which no light escapes). They did this by measuring the speed of gas clouds orbiting the core to be 780 $\mathrm{~km} / \mathrm{s}$ at a distance of 60 lightyears $ \left(5.7 \times 10^{17} \mathrm{~m}\right)$ from the core. Deduce the mass of the core, and compare it to the mass of our Sun. v =    $\times 10^{39}$solar masses =    $\times 10^9$

A car maintains a constant speed as it traverses the hs9)s:7dd u 7qmpw 37v myjiii6ill and valley shown in Fig. 5–45. Both the hill and valley have a radius of curvature R.6isd7)yq:jsmmvupwd79 3 7 ii (a) How do the normal forces acting on the car at A, B, and C compare? (Which is largest? Smallest?) Explain. (b) Where would the driver feel heaviest? Lightest? Explain. (c) How fast can the car go without losing contact with the road at A?

  The Navstar Global Positioning System (GPS) utilizes a group of 24 satellitesecc05d2;jxhyu 5b vg m :kh+i:z7 -f-fndv7kx 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.: u;k7 g-kxbx5 hf5y 0-izcnvdjh 2d7+mvc :fe 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, is mhw9; 4seqhuv(eant to orbit the asteroid Eros at a height of about 15 km . Eros is rouwhh ; (9eq4usvghly 40 $\mathrm{~km} \times 6 \mathrm{~km} \times 6 \mathrm{~km}$ . Assume Eros has a density (mass/volume) of about 2.3 $\times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ . (a) What will be the period of NEAR as it orbits Eros? $\approx$    $\times 10^4$sec
(b) If Eros were a sphere with the same mass and density, what would its radius be? $\approx$    $\times 10^3$
(c) What would g be at the surface of this spherical Eros?$\approx$    $\times 10^{-3}$

  You are an astronaut in the space shuttle pursuing a satel 4d lobu:szd5*lite in need of repair. You are in *ud:l4sbd 5z oa circular orbit of the same radius as the satellite (400 km above the Earth), but 25 km behind it.
(a) How long will it take to overtake the satellite if you reduce your orbital radius by 1.0 km?$\approx$    h
(b) By how much must you reduce your orbital radius to catch up in 7.0 hours?$\approx$    $\times 10^3$

  The comet Hale-Bopp has a period of 3000 years. (a) d va 252;ehx4vwp(vzj What is its mean distance from the 5j;vvve w2z pd2 (a4xhSun?   
(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 woulo*9mfnlbp5hx .qxy* -i3 m 1 2gqv2jkbd need to be if you could actually "feel" yourself gravitationally attracted to s bhkgf5jqn 23.l *bi9p 1-xom2q*xymvomeone near you. Make reasonable assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of thf):zb,gj i q)rn0l q1usg* qlw88b e.ue Milky Way Galaxy (Fig. 5-46) at a distance of about 30uq.qq*s b)8 gw, ni0uglrfl ezb8:)1j,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 /m-w o1fescp1 a square 0.50 m on each side. Find the magnitude and direction of the gravitational -pecs f/ wmo11force on a fifth 1.0-kg mass placed at the midpoint of the bottom side of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 k. i4ii;ly9jcti- np3q g orbits the Earth $ \left(\operatorname{mass}=6.0 \times 10^{24} \mathrm{~kg}\right)$ and has a period of 6200 s . Find (a) the magnitude of the Earth's gravitational force on the satellite, $\approx$    $\times 10^4$
(b) the altitude of the satellite.    $\times 10^5$

  What is the acceleration experienced by the tip of the 1.5-cm-long sweep seconnbjt( l14-iypfl1 i8b d hand on your wrist watchy bt11-pn fj8ibll4i(?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinkepbrsm9w y57 c,qln3;nr 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 nwnr sb ,;l 5pm7q39cyis 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 higcvs01 xv6w: q:yi 8d9j ldwmen9ns8j*hway is designed so that a car traveling at spej9dns9dls: wwy68j1nc*: i8qe0m x vv ed $ v_{0}$ can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from $v_{\min }$ to $v_{\max }$ . Derive formulas for $ v_{\min }$ and $ v_{\max } $ as functions of $\mu_{s}, v_{0}$ , and R .

  Amtrak’s high speed train, the Acela, utilizes tilt of the cars when negotiati;c7e d4ih xors r (kr,tw3w di/(yo;vrr, j-h:ng 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 traic ,v:r;dr;ok/w- 4rjiho (7iyh,( ts ewxrr3dn 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 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.$\approx$    $\times 10^2$