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Name Period Date 11-1 Chapter 11: Gravitational Interactions The equation for the law of universal gravitation is tll' F = G, =d. F it tn" force of attraction between masses rr, and m1 separated by distance d , and, G is the universal gravitational constant (and relates to the masses and u-here diqbnss just as the constant rc relates the circumference of a circle to its diameter). By substifuting changes in any of the variables into this equation, we can predict how the others change. For example, we can see how the force changes if we know how either or both of the masses change, or how the distance between their centers changes. Suppose, for example, that one of the masses somehow is doubled. Then substituting Zh, for wl in the equation gives FN€w= ('*lf' "-d. So we see the force doubles also. Then substituting 2d for d 2(6 rl'Yfrz\ -' tr -- arouD d. l Or suppose, instead, that the distance of separation is doubled. in the equation gives [l, mr @'= Fn,., = = ,. <' And we see the force is ortlv 1/4 as much. Use this method to solvi: the following problems. Write the equation and make the appropriate subs titutions. 1. If both masses are doubled, what happens to the force? Fnu*= G 2m1?'ng r'l d* rb",, da ? FoLF} If the masses are not changed, but the distance of separation is reduced to "I/2 the original distance, what happens to the force? ;l&a+i$ =6ffi= 4(6ff)= 4 Fo.u Conceptual Physics: Concept-Development Exercises @ Addison-Wesley Publishing Company, Inc. All rights reserved. Exercise 11-1 29 3. If the masses are not changed, but the distance of separation is reduc ed to 1 / 4 the original distance, what happens to the force? t-= I NEW If both masses are doubled, and the distance of separation is doubled" show what happens to the force. r -rl^rhns - 4r hrt F".*=Cffi a\raF= Fo,oi No cHANGE If one of the masses is doubled, the other remains unchanged, and the distance of separation is tripled, show what happens to the force. ,n 2m1m7 -FnE*,= u = TE;F -,n1\ = Z C /r (G T ) =" aoro O L^ Consider a pair of binary stars that pull on each other with a certain force. Would the force be larger or smaller if the mass of each-star were three times ad great and if their distance apart were three times as far? Show what the new force will be compared to the first one. 3-t3t. 9 la^r^"\-tr tr (] (3dr = ((]-if-)= %,o; No CHANGE! F,,ew=e . 6 30 Exetcise 11-1 @ Conceptual Physics: Concept-Development Exercises Addison-Wesley Publishing Company, lnc. All rights reserved. Date Period \::-',e Cl"apter 1"1: LL-2 Gr-aritational Force and Weight has a mass of 0.1 kg has the same mass wherever --:.:::,e{es up the apple -r-:. a::-e :lut it is' The amount of matter i-5:*@-@@+*€$q- :epends -:e on) . (does ._..;adon of the appte. ;;tli-*r-h€f€ not dePend on) I tt has the same resistance to acceleration wherever it is-its inertia 15 (the same) ..'i (different). may weigl exactly 1 N in San Francisco and 9lg-\tty -*:e rr-eiqht of the apple is a different story. It weigh 0'17 N, o;ifr. surfa'ce or itre mo-on the apple would does i== not change that qriantity The all. at a:,J far out in orrt.. ipu.e ir *uy nuue ai*oit no weight rrith location is ;';[ii;;1;i"D5ff;;i"T;'^"d' , gt*..:i (weight), and the quantity that may change with location is its ;:rr:{ff:ryitrt\ (mass) (weight).; . **:-*.," That's because (mass) il.yi,q_T],3 distance. So we see that is a measure of gravitational force on a body, and this force varies with in contact with the object small some weight is the force of gravity U.t*...t t*o 6odies, usually eartl. When we refer to the (mass)Ljy=:g_tpj it to the earth' of an object, we are usually speaking of the gravitational force that attracts FilI in the blanks: find that we are pulled toward the earth with a force of : 1"" N. Strictly speaking, we weigh '^:, N relative to the 500 N, then we say we weigh -- and repeat the weighing earth. How much aoes tt,""eurtr, weigh? tf *. iipit"r. scie upside-dowo . force of i' I N, and process, we can say that the earth is being attracted to us with a If we stand on a weighing scale and earth therefore, rerative to us, the whore 6 000 000 000 000 000 000 000 000-kg Weight, unlike mass, is a relative quantity' weigllt ::'ilir 5t DO YOU SEE WHY IT MAKES SENSE TO DISCUSS THE EARTH'S A^ASS, BUT NOT ITS WEI6HT? 44: i', e are pulled to the earth with a iote of 60Q N, so we weigh 500 N. The earth is pulled toward us with force of 500 N, so it weighs 500 N. a tfr Conceptual Phvsics: Concept-DeveloPmcnt Exercises O aa,iit."-fV&1ey Publishing Company, Inc' AII rights reserved' Exercise 11-2 31 \:::e Date Period 12-1 C:ep:er l-:: Satetrlite \{otion ::;*-- -i is r-rf ,,Newton's Mountain," high enough so_ its top is.above the drag ::---:: a:r.L1sphere. The cannonball is fired and hits the ground aS shown' . *, the path the cannonball might take if it were fired a ':-ie -it faster. : ]a.-r t : ? t 5 i<m/s. c d t 3 t Then draw the orbital path of the cannonball if its speed rvere 8 E t Repeat for a speed that is greater still, but still less than s. km/s. lVhat is the shape of the curve for a speed of 8 km/s? * :,-'lr= Tfgure a What would be the shape of the orbital path if the cannonball were fired at a speed of about 9 km/s? e llrose Figure B shows a satellite in circular orbit' ,or a. At each of the four positions draw a vector that represents the gravitational foice exerted on the satellite' F ,* b. Label the force vectors F. c. Then draw at each position a vector to represent the uelocity of the satellite at thit position, and lable itV' d. Are all four F vectors the same length? Why or why not? Yes, n*ear.rs* the foree i3 l,:-:e san-ir: girengih at equa! distane e* Figure B 12-1 33 ir*nr the earlh' 3: i\ vu :-!, rlan r,vYr re,a What is the angle between Your F and V vectors? f.lo, Fand Vare f. Is there any comPonent of F along V ? e. o 6' PerPendiei":lar. satellite? What does this tell you about the work the force of gravity does on the i-ile foree cf gravity dees no 1r,'ork on ihe sateiiite h. Are all four V vectors the same length? Why or why not? : :':;,..ijS* thgre is no er:-;ii];eratloi'i alonq thr': :'ateiiite's path" Conceptual Phvsics: Concept-DcveloPment Exerciscs. aaii.on-fV"tley Publishing Company, Inc' All rights reserved' O Exercise 1. Does the KE of the satellite in Figure B remain constant, or does li remains constant. it vary? k. Does the PE of the satellite remain constant, or does it vary? it rernain* c0flslanr" Figure C shows a satellite in elliptical orbit. 3. a. Repeat the procedure you used for the circular orbit, drawing vectors F and V for each position, including proper labeling. Show equal magnitudes with equal lengths, and greater magnitudes with greater lengths, but don't bother making the scale accurate. b. Are your F vectors all the same length? Why or why not? Na, the fo;'ee decreases wh*r'lthe elistance fro!'n the earth increases c. Is the angle between your F and V vectors the same everywhere, or does it vary? l+ {,^,,.it !{;llrijiJ- d. Are there places where there is a component Yss, q;veri,ivhere excspt of F along V? :t th9 apE.qee ind the nerigee" e. Is work done on the satellite when there is a component of F along and in the same direction asVT If so, does this increase or decrease the KE of the satellite? \r'e*. This inr:;'-;a:e I ihe {[ {f the satellite" f. When there is a component of F along and opposite to the direction of V, does this increase or deirease th-e KE of the satellite? This cieer*ases ii-,s KF of the satellit*. g. Are your V vectors all the same length? Why or why not? f.ia. When ihe KH ,isgsr,;;5gs ias the saieilite moves farther f:.om ti:e earthl, th* speeeJ qjprr*ases. \lJhen sa:eliite nrrves iorryai'd Figure C the KE increases {as the ii:* elrth], the speed increases. h. What can you say about the sum KE + PE along the orbit? ll ir,' g9ns1s6t {in accord wilh eonservatlon r:f energy]. 34 Exercise 12-1 Conceptual Physics: Concept-Development Exercises O Addison-Wesley Publishing Company, Inc. All rights reserved.