Sometimes people say that wd8dpr 4s76 asz2fb6vgater is removed from clothes in a spin dryer by centrifugal fsg 2fdprzda 46sv678borce throwing the water outward. What is wrong with this statement?

Will the acceleration of a car be the+ ua4. og7rvlv same when the car travels around a sharp curve at a constant 60 km/h as when it travels around a g+v4org 7lva .uentle curve at the same speed? Explain.

Suppose a car moves at constant speed along a hilly road. Where does th u msftzbytou88s6 4g 3*ovx+:e 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 leveut x6syo:u8 zo3 vsg bm8tf*+4l stretch near the bottom of a hill?

Describe all the forces acting on a child riding a horse on a badbgia p+xm* )(u ,3wxan)xr-x0rt/ merry-go-round. Which of theseba )a3rxpx+)(b/ ardw,ix0mn *-xt u g forces provides the centripetal acceleration of the child?

A bucket of water can be whirled in a vertical tj vr+5(.t nq1 +irphzcircle without the water spilling out, even at the top of the circle whrn + (hzv.5ipt rtq+1jen the bucket is upside down. Explain.

How many “accelerators” do you have in your car? The//oe;2e nztp/8 bmaih 1bizn.re are at least three controls in the car which can be used to cause the car to accelerate. What are they? What acc/a/ o iht/ebz m8. nbp1;n2zieelerations do they produce?

A child on a sled comes flying over the crest of a small hill, as shown dtob4a+cdn2 pk h0ua77q9 z, vin 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 2ubp o nd aq7h+790kvztcd,a4decrease as he goes over the hill. Explain this decrease using Newton’s second law.

Why do bicycle riders ge m6kf v: .ce1wj(favs n8.-jlean inward when rounding a curve at high speed?

Why do airplanes bank when they turn? --j)js9 drt ygd5ev iygk0,)v How would you compute the banking angle given itsd5tkvyig d rjy)-jg,9 0ve- s) speed and radius of the turn?

A girl is whirling a ball on a string around her head in a horizontal plane.;u/ nlzsf u7ze06q(n1e y.xc6 ws1f8gbkwau . She wants to let go at precisely the right time so that the ball will hit a targc yw gz6nbuxk.n 7;fs.1(wl/8qzu1 6sae fe 0uet on the other side of the yard. When should she let go of the string?

Does an apple exert a gravitational force on the Eo la sys,5wrvbe,n. 9j)n4j/farth? If so, how large a force? Consid/ wol),rj ns54. ,sfy v9bjaneer an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass v zd69qpt hx5bl h qc8qk4pr+l:/ d+1owere double what it is, in what ways would the Moon’s orbit be differk1v+dhqcr:pl zxbdlqt6p+ 5o9q/84 h ent?

Which pulls harder gravitationally, the Earth on the Moon, or the Moon on kjicew 8md 3f +8s yfm)8bm,9qthe Earth? 8)wsfm8 i9myj,kdc3em8 bf +qWhich accelerates more?

The Sun's gravitational pull on the Earth 1f iet ij:m./c4laqo wr, ly5:o0vip ;is much larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint:icrif,/voow.0t1 liy4je5a:;mq: l p Consider the difference in gravitational pull from one side of the Earth to the other.]

Will an object weigh more at the equator or at the poles? What two effects are0agfz .tmeq, +t04m3k.hl hd y at worf0mdy.,04k +htz qt.3h mg aelk? Do they oppose each other?

The gravitational force on the Moon due to the Earth is only about half the for/cke)-eah9b .kky( ss ce on t9k. /yhecs k)esb(k-ahe Moon due to the Sun. Why isn’t the Moon pulled away from the Earth?

Is the centripetal acceleration .g2:2bc5yzbpsa0url go m 0a57vx9h h of Mars in its orbit around the Sun larger or smaller aybvugl7 s2b ph0: m 95o2xrg0cz.ah5than the centripetal acceleration of the Earth?

Would it require less speed to launch a satellite (a) g0 9;sd9dhoq2c5 sz*5excycytoward the east or (b) toward the west? Consider the Earth’s rx c5*s9o 0ch5 gydze;qcd2 9ysotation direction.

When will your apparent weight be the grepm6zs m;h ;7iaiib7p)atest, 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 i)6b ;psm 7zpam;ihi7 speed? In which case would your weight be the least? When would it be the same as when you are on the ground?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long periods in outer space could be adv /j62. ful.)iz*gmthhmso l9uersely 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 i to*i9fghz. s)ljulm/h6.um2nside 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 “fj.-wr+7rlgp 4wsss n38 qy; egree fall” or “apparent weightlessness” between steps.;j3 r+we4 plg.sys7 wr-qs8ng

The Earth moves faster in its orbit around the Sun in January than in*. ili4 )nwhtp July. Is the Earth closer to the Sun in January, or in July? Explain. inp*tl4 w)hi. [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 discoverk w xlbey37m3ovh5k1.ed to have a moon. Explain how this discovery enabled an estimate of Pluto7w 3.1mk3xvek5bh lyo’s mass.

  The Sun's gravitational pull onqklnmv je5;7dv +5rz1 the Earth is much larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pullv+1; lj5 zmkdeq n7v5r 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 travelingw6 23ev xrmei( 1980 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ pulls out of a dive by moving in an arc of radius 6.00 km . What is the plane's acceleration in $g^{\prime}$ 's?    g's

  Calculate the centripetal acceleration of the Earth in ibs qjkjf 91bj/)+vn-uj a)ql,ts orbit around the Sun, and the net force exerted on the Earth. What exerts this force on the Earth? Assume that the Earth's orfj) k buj1,q-+slqa 9 bvjjn)/bit 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 isys i7/c;uofunz 9;b)dzfg l93 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 s7 )yzu99 uc; zi3;nlf/ fogdbdiscus.   

  Suppose the space shuttle is in orbit 400r5gh9(l.,t h0kt en wj km from the Earth's surface, and circles the Earth about oncw,htgth .kl5e0 r(9nj e every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of g , the gravitational acceleration at the Earth's surface. $\approx$    g

  What is the magnitude of the acceleration of nukf1 fq),bz/ a speck of clay on the edge of a potter’sfbk z/1 )fn,qu wheel turning at 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 vert4x w(ld6k/2* gcd-dyp-n y nmhical circle of radius 72.0 cm , as shown iy xdy64ndd-g2ph kwl*(mn-c /n 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 spb quzn( ()px;.7vcfc2vo(xlfeed with which a 1050-kg car can round a turn of radius 77 m on a flat road if the coefficient of stalz(cqfpc2vxux(; . ()nf b7 ovtic friction between tires and road is 0.80? Is this result independent of the mass of the car?   

  How large must the coefficient of static friction be between tg8*pfow+zb, mac/;hl vwrp6q +ne5 l3 he tires and the road if a car3z8;5m fb6llw* ehngacv+ w+ ,/rpoqp is to round a level curve of radius 85 m at a speed of 95km/h ?   

  A device for training astronauts an*x cez6uu b37m hcjs8h.1r.un d jet fighter pilots is designed to rotate a trainee in a horizontal1ecm76u8.c zu*jhxrbn sh 3.u circle of radius 12.0 m . If the force felt by the trainee on her back is 7.85 times her own weight, how fast is 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 turnt8*s1guo fc *ljable 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 the coin and the turntab1o *sc*j8 lgufle?   

  At what minimum speed must a roller coas)ixv kk26-mwd2ejhk+ ter be traveling when upside down at the top of a cik2m6 +wk v2e jdk-hix)rcle
(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 mas-0upzv n0i 5xss 950 kg (including the driver) crosses the rounded top of a hill (ravpzx5u0- 0n isdius =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-diall8u*cnn/ ) tdmeter Ferris wheel need to make fol n/8lt)uc nd*r the passengers to feel “weightless” at the topmost point?    rpm

  A bucket of mass 2.00 k4yl 5qzbdcul2+ ( a;nvg is whirled in a vertical circle of radius 1.10 m. At the lowest point of its motion the tension in the rope supporting the bucketd lc uz y(4;+bv5lnqa2 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 partic 4v5zupr3+ bjtle 9.00 cm from the axis of rotation is t+ up vbr53jt4zo experience an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnivxny ) -w8dosv;al, people are rotated in a cylindrically walled "room." (Sevw)syx-;n 8do e 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-hockey tah;vg k7.bl c14-9 homeig fu0rbletop, and is held in1e9m0- uglrk;7hvocfg.b4ih 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 ignoringcxhb9cx8y - c, the weight of the ball whichyh ,-cbxc xc98 revolves 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 8cta-xuaz8j9m8; .hsadp 7lp) 8 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 masses o/ . vjtrow5-/-ntc nyma7 uw;$ 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 vertica7qe* y3jhtg,6l(c b3nh pw3lvt1k)c r lly at 310 $\mathrm{~m} / \mathrm{s}$ . If he can withstand an acceleration of 9.0 g 's without blacking out, at what altitude must he begin to pull out of the dive to avoid crashing into the sea?   

  Determine the tangential and centripetal componenfdy4 y8fhdh5 iiq (ybl52fc2 -ts of the net force exerted on the car (ih f yb dhf 4l822cidy(5-yq5fby 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 pw l(;opg3 x npv3o52pait area, going from rest to 320km/h in owo; ( axnp32g53plvp a semicircular arc with a radius of 220 m. Determine the tangential and radial acceleration of the car when it is halfway through the turn, assuming constant tangential acceleration. If the curve were flat, what would the coefficient of static friction have to be between the tires and the road to provide this acceleration with no slipping or skidding? $\approx$   

  A particle revolves in a horizontal circle of 8 yme6lm0gd.7goo - 1hoza vt/radius 2.90 m . At a particular instant, its accelethaevgdz o8. oom m 0y/-l17g6ration 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 gravit*q7 1iw3l*twcxebbah3g2w0 rr;4 a cey on a spacecraft 12,800 km (2 Earth radii) above the E 3rx rh 7ibt*cg*a0bae2wq w;4le13cwarth’s surface if its mass is 1350 kg.   

  At the surface of a certain planet, the gravi8zz9 3,kp peynvfw(q8tational acceleration g has a magnitq z vez,w(fp8yp k38n9ude of 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 gravity on the Moon. The Moon's radiurz8*5m dlnop d2,sfto4*nv2b s is 1.74 ln8df,2zvmd 2 po4on *5r*sbt$\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has a radius 1..u1+f ia+l4o): 5 vzdefoezwx5 times that of Earth, but has the same mass. What is the acceleration due to gravity near its surface? o:leo1xuvdezfw a) ++if 45.z   

  A hypothetical planet has a mass 1.66 times that of Earth, bpo: .qd. vi4kqj 8/aanut the same radius. What is g near its surfac v.k8 qaja.pd4n/o iq:e?   

  Two objects attract each other gravitationally with a forceh3h1)hg . zup ./c+ws -zspozq 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 , ngg;sc.*mxc ) the acceleration of gravity, at (a) 3200 m nsx gcc) g;.m*   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside the Earth where tg)94exis;ml hy8vz5c hegrly9s)h 4 g58e izmvc;xavitational 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 tklkg8o w/ c.k+4suin )he mass of our Sun packed into a sphere about 10 km in radius. Estimate the surisn)k +lku 8.o wgk/c4face gravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once wxk 1oo4e5xkrgifd o4o7:. s9+jruu *g as an average star like our Sun but is now in the last koo.ud4xgif+ 7u5s xe4 *1 ok:rjgo 9rstage 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:,h yog ;dg9mo feel weightless 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 isn’t so by calculating the accele:m 9ggh;dyo,o ration of gravity 250 km above the Earth’s surface in terms of g.   

  Four 9.5-kg spheres are locate-jmjev+mc/y18g7b b jd at the corners of a square of side 0.60 m. Calculate the magnitude and direction ofmg1 e8mj+ v-jbc/7ybj the total gravitational force exerted on one sphere by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred years c+ 8(khnd,ijiy6g yw-most of the planets line up on the same side of the Sun. Calculate the total hyi iy8(dngkj6cw -,+force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in a line (Fig. 5-38). The masses are $M_{\mathrm{V}}=0.815 M_{\mathrm{E}}, M_{\mathrm{J}}=318 M_{\mathrm{E}}, \quad M_{\mathrm{S}}=95.1 M_{\mathrm{E}}$ , and their mean distances from the Sun are 108,150,778 , and 1430 million km , respectively. What fraction of the Sun's force on the Earth is this? $F_{Earth-planets}$ =   $\times 10^{17}$ ${F_{Earth-plantes}}$ \ ${F_{Earth-sun}}$=    $\times 10^{-5}$

  Given that the acceleration of gravity at the surface of Mars is 0.38 of what1u,owr(;l3 ajqssrm(+w e* ey it is on Earth, and that Mars’ radiusol(ew;a+ yrsr1m w su*ej3q(, is 3400 km, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sa f(0 dldzckd8a )2+doun using the known value for the period of the Earth anda+d02dzd a kfcl(o8) d its distance from the Sun. [Note: Compare your answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite moving in a stable circui 0:xns.0z od1tid6my7 pv*qolar orbit about the Earth at a heightyi:pn1d soqvdi6to xm 0. *7z0 of 3600 km.    $\times 10^3$

  The space shuttle releases a satellite into a circularu 2 rv,wmpp(j4 orbit 650 km above the Earth. How fast must the shuttle be moving (relative to Earth) when the r 2pjp,(uvm4wrelease occurs?    $\times 10^3$

  At what rate must a cylindrical setoykvc(yvt1m,zq d-l :.9fi go;2 w3 paceship rotate if occupants are to experience simulw(f1mzltvy; i. qy:ce3t-vd92 ook g,ated gravity of 0.60 g? Assume the spaceship’s diameter is 32 m, and give your answer as the time needed for one revolution. (See Question 21, Fig 5–32.)   

  Determine the time it takes for a satellite to orb 4sinn o0rya+0it the Earth in a circular “near-Earth” orbit. A “near-Earth” orbit is one at a height above the surface of the Earth which is ve+on0 r0y 4sainry 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 havk8exuxctgl w;r: 4, ;ve to be launched from the top of Mt. Everest to be plwe x,l4gukr;vt:c x8 ;aced in a circular orbit around the Earth?   

  During an Apollo lunar landing mission, the command module continued to orbit tlz-yg qn(m xy2pfy mzp,k*.gm l9p// tq;u5a9he Moon at an altitude of about 100 km. How long did it take to go around tp.9zgzkap ,- /y5my/y;l*(gmf9 lxptuqm2 qnhe Moon once?    s

  The rings of Saturn are composed of chunks of ice that orbit th+ yh2 :el:qut2z6zkcs e planet. The inner radius of the :zslz+e 6 c:qk2t2uyhrings 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 onco u+ezsl4gv d/.hs:h8e every 15.5 s (see Fig. 5–9). What is the ratio of a person’s apparent weight 4sd8v s.:g /huoe+lzhto her real weight (a) at the top, and (b) at the bottom?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut 4200 km from the cel)m1gbqz9 cn3j*( u awnter of the Earth's Moon in a space vea (*g wc13 nmul)bqjz9hicle (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 of two stava30eg kiq.ezr.7 ez :rs of equal mass. They are observed to be separated r:.g a7kv.ezeqi 0ze3by 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 r ;bpa-jq:2l 0zv jj bd2fx, ta v0qvh2g7tmg05ead for the weight of a 55-kg woman in an elevator that moves:j,h7 gx2zt2vmgv;l d 0-tjajfaqp2b 0b 5vq 0
(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 cord4c9rlu*hof ld **kg k5 suspended from the ceiling of an elevator. The cord can withstand a tension of 220 N and breaks as the elevator accelerates. What was the*9o*fr lgk5 h4ld k*uc elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orbits very near the surface of a planet with period kck71y d*,xmtT , kt, dcmy*7k1xthe density (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 the Moon (27.4 3 lxlbm 0*n,czd) to determine the period of an artificial satellite oz *l3n,m0 xcblrbiting very near the Earth’s surface.    s

  The asteroid Icarus, though only a few hundred meters acrofr cn+/cj.jk;ashx ,y(+8 ep oss, orbits the Sun like the planets. Its peri nca+x sh;c k.ye+/8fro,pjj( od is 410 d. What is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average distaelk/g ase,6-edy +6jg nce 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 rougnj tq-tj35 .oi* 5-hhj+dkfu vnhqf, *4b wnj*hly once every 76 years. It comes 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 Keplei hjqtj-54kvwfh nd j5-n+ n3uhq*.j* *,f botr’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 Gah wkj *vhl 725slbl(jmr 8g2 :q)ti(evlaxy $\left(M_{G} \approx 4 \times 10^{41} \mathrm{~kg}\right)$ 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 moons o2th) :vxrfunb b p/wq3mr i,2m42qlb(i .4xvo f Jupitermxb iv43/(4p,b:x 2b 2m2.)ifrqvt uql owhrn (those discovered by Galileo in 1609).
(a) Determine the mass of Jupiter using the data for Io.$M_{\substack{\text { Jupiter- } \\ \text { Io }}}$ =    $\times 10^{27}$
(b) Determine the mass of Jupiter using data for each of the other three moons. Are the results consistent?
$M_{\substack{\text { Jupiter- } \\ \text { Europa }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Ganymede }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Callisto }}}$ =    $\times 10^{27}$

  Determine the mass of the bzu8 r:f2vx; 3znwl6cEarth from the known period and distance of the Moon. bzurnw3 cxlfv6;z:8 2    $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jupiter’s moons, u -bqk 11(st2x ;w6sgwj8teily sing Kepler’s third law 1sleww kt(gbx8sq2j6 1;yi- t. Use the distance of Io and the periods given in Table 5–3. Compare to the values in the Table.
$r_{\text {Farip }}^{r}$ =    $\times 10^3$
$r_{\text {Guarymale }}$ =    $\times 10^3$
$r_{\text {Culimio }}$ =    $\times 10^3$

  The asteroid belt between Mars and Jupiter consists of many fragmeaabz0)jrq )x2-gv r6pnts (which some space scientists think came from a planet that once orbited the Sun but wasa6vp)-g 2rzrbj0xqa) destroyed).
(a) If the center of mass of the asteroid belt (where the planet would have been) is about three times farther from the Sun than the Earth is, how long would it have taken this hypothetical planet to orbit the Sun?$\approx$    y
(b) Can we use these data to deduce the mass of this planet?

  A science-fiction tale describes an artificial "planet" in the fok ,e.:t: aic- f4ifwgqrm 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 gravigi tfc,qaf -:kw4i.:ety of g as on Earth. What will be the period of revolution, this planet's year, in Earth days?   

  Tarzan plans to cross a gorge by swinging in an arc from a h6bor)8x ppz; vanging vine (Fig. 5–41). If his arms6bp ;pxo 8zrv) are capable of exerting a force of 1400 N on the vine, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 80 kg, and the vine is 5.5 m long.   

  How far above the Earth’s surface will the acceleration of gravity bai* 9r 4trtiqa4:lx1mxby:.x e half what it is on the surfacel1xrix:4rttaq a m4yix.b*:9?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spin in a muk v9w fd90uem/tual circle once d9/vumw9 e0kf every 2.5 s. If we assume their arms are each 0.80 m long and their individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates once per day, the appare(hog)3hhcoo8x*) (yi xx qr n7nt acceleration of gravity at the equator is slightly less than it would bx3 hq oo)8x)r 7nc(gyixho(*he if the Earth didn’t rotate. Estimate the magnitude of this effect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from the Earth will a spacecraft traveling directle in ,3p n7h)bkw4t1ja0moxk:y from the Earth to the Moon experience zero net force because the Earth and Moon pull witnx 1ekha ,3 k4j0 i:ob)nwmp7th equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, but wlme5uwp 8g /)nhen you stand on a bathroom scale in an elevator, it says your mass is 82 kg. What is the acceleration of the elevator, and in w8um)gnep /5lw hich direction?     ----待选择----upwartddown 

  A projected space statiosp:4nz 2fj)dwn consists of a circular tube that will rotate about its center (like a tubuldzs4:2n)pjfw ar 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 felt?$\approx$    $\times 10^3$

  A jet pilot takes his aircraft 2umkd;a,j dnl, w5nt7m zztz x9t,7z4 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 radiu* tudh(t sp.7(0st ufgs r, the acceleration due to gravity at its surface an dt(g0(u7t. t ups*fshd the gravitational constant G.

A plumb bob (a mass m hanging on a string) is deflected from thehr,a 4pt3j/xa vertical by a3,4h/trj axpa n 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 coefficqjxl6ovv+ as x3.x:)x ient of static friction is 0.30 (wet pavement), at what range of speeds can a car safely handle the ax v36l.: )jq+v xoxsxcurve?

  How long would a day be if the Earth were rotating so fast that objects at*q k s/q ptm)ql;mp)f5 the equator werpsmm pqkqtl;/)f)*q 5e apparently weightless?    $\times 10^3$s

  Two equal-mass stars maintain a constant distance apart of wn ; cu18 lswidi,t2,xe/o(iv 8.0 $\times 10^{10} \mathrm{~m} $ and rotate about a point midway between them at a rate of one revolution every 12.6 yr . (a) Why don't the two stars crash into one another due to the gravitational force between them?
(b) What must be the mass of each star?    $\times 10^{26}$

  A train traveling at a constant speed rounds a curve of radiu.4u6afvx ryr2cpg lcem 5/ w97s 235 m. A lamp suspended from the ceiling swings out to an angle of 17. pcf4a/ ucemlwx9yrr6 g75.2 v5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times as massive as theuk58 ws2 9fp0(vkofxzj6ep f6 Earth. Thus, it has been claimed that a person would be crushed by the force of gravity on a planet the size of Jupiter since people can't survive more than a few g 's. Calculate the number of g 's a person would experience at the equator of such a planet. Usp9 xfzs 0f(6eu8pk 5kv2wo6jfe the following data for Jupiter: mass =1.9 $\times 10^{27} \mathrm{~kg}$ , equatorial radius =7.1$ \times 10^{4} \mathrm{~km}$ , rotation period =9 $\mathrm{hr} 55 \mathrm{~min}$ . Take the centripetal acceleration into account.    g's

  Astronomers using the Hubble Spacet; cpd rk3:dvqi8mf.trhp7 (c) zvo.9 Telescope deduced the presence of an extremely massive core in the distant galaxy M87, so dense that it could be a black hole (from which no light escapes). They did this by measuring the speed of gas clouds orbit z v3r;tcoh t.qp8(.f i97v)kd mpc:dring 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 h gu(-5h tmq*u6v,mvzu ill 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) Wher, gz*mv 5(u-muqh6ut ve 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 Sy90j/o vt3v j+m fn4wtmstem (GPS) utilizes a group of 24 satellites orbiting thej0otmvvt / mj9wnf4+3 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, 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 m nz xdldg6d6-)eant to orbit the astd nlz-d)d6 gx6eroid Eros at a height of about 15 km . Eros is roughly 40 $\mathrm{~km} \times 6 \mathrm{~km} \times 6 \mathrm{~km}$ . Assume Eros has a density (mass/volume) of about 2.3 $\times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ . (a) What will be the period of NEAR as it orbits Eros? $\approx$    $\times 10^4$sec
(b) If Eros were a sphere with the same mass and density, what would its radius be? $\approx$    $\times 10^3$
(c) What would g be at the surface of this spherical Eros?$\approx$    $\times 10^{-3}$

  You are an astronaut in the space shuttle pursuing a satellite in need of repansru41 t3:xf eir. You are in a circulan1 f:e 4urstx3r 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 periny ln; ,kvf;.qod of 3000 years. (a) What is its mean distance from the Sunny;n,v; .kqfl?   
(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 vapv 5 e.ntwaz+:m7uz,a2ye o6a lue of G would need to be if you could actually "feel" yourself gravitationally attracted to someone near you. Make reasonable ase+av, 76zmaauz. y2e5:t ponwsumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the Milky Way Galaxy (Fig. 5-46) at0(2yhy i9o l2amfll /:ecsgw . a distance of about 30,000 light-years from the cente.0s e(w:colihm2lyy lga /92fr $ \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 sqirv t.v yrg g4c62*0jmuare 0.50 m on each side. Find the magnitude and direc2 6vyrjgig.40rm*v tc tion of the gravitational force 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 kgm;y0)elt* bx ihbxnb ,g ,4;rw 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 expeycfybc;m (74zwy 0hooxj7-8cjkd: 0d-tf en ) rienced by the tip of the 1.5-cm-long sweep second hand on your wrist waw-cy j n70ot:f7xbdy(hj)zfy8-k;4e o0d cmctch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weight around in aezrm 5.2cug(t 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 z2e t.ru(c5mg that the fishing line makes with the vertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in a new hz5( -0pti(wmpsygu(l ighway is designed so that a car traveling at sp0g wp5ztlp-(iym (s( ueed $ 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 tiltmvl1 xr1;kxm6 6art7 t 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 xr6v1 ra m617 mklx;ttm 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.$\approx$    $\times 10^2$