Chapter 12 Universal Force due to Gravity Every object in the Universe exerts an attractive force on all other objects The force is directed along the line separating two objects Because of the 3rd law the force exerted by object 1 on 2 has the same magnitude but opposite direction as the force exerted on 2 by 1 m1 F12 F21 r m2 F12 F21 By 3rd law where Gm1m2 F12 F G 2 r And G Universal Gravitational Constant 6 67259 x 10 11 N m2 kg2 G is a constant everywhere in the Universe therefore it is a fundamental constant g is not a fundamental constant but we can calculate it Compare F mg and Gm1m2 F12 2 r Let m1 ME mass of the Earth m2 m mass of an object which is ME r RE object is at surface of the Earth Set the forces equal to each other m GM E m mg RE2 GM E g RE2 g ME 6 67259 x10 11 Nm 2 kg 2 RE 5 9742 x10 24 kg 6 6 378x10 m 2 9 80 sm2 Weight mass Weight the force exerted on an object by the Earth s gravity FG mg W Mass is intrinsic to an object weight is not From previous page W m GME RE2 your weight would be different on the moon Gravity is a very weak force need massive objects Example Problem difficult Two particles are located on the x axis Particle 1 has a mass of m and is at the origin Particle 2 has a mass of 2m and is at x L A third particle is placed between particles 1 and 2 Where on the x axis should the third particle be located so that the magnitude of the gravitational force on both particles 1 and 2 doubles Express your answer in terms of L Solution Principle universal gravitation no Earth F12 Gm1m2 r2 Strategy compute forces with particles 1 and 2 then compute forces with three particles m2 m1 x m1 L Situation 1 m2 m3 x x x1 L Situation 2 Given m1 m m2 2m r12 L Don t know m3 Find x r13 when force on 1 and 2 equals 2F12 Situation 1 Gm1m2 Gm 2m 2Gm Fx F12 r 2 L2 L2 FBD 2 m1 F12 2Gm 2 Fx F21 F12 L2 FBD F21 m2 Situation 2 2Gm 2 Gmm3 4Gm 2 Fx F12 F13 L2 x 2 L2 Since in situation 2 the total force must be 2F12 FBD Solve for x m1 m3 2m 2m m3 4m 2 2 or 2 2 2 L x L x L F13 F12 2 2 m3 x L or x L m3 2m 2m Now consider m2 FBD F23 m2 F21 G 2m m3 2Gm 2 4Gm 2 Fx F21 F23 L2 x 2 L2 1 2m3 2m 2m 2m3 4m 2 2 or 2 2 2 L x1 L x1 L 2 1 2 m3 x L or x1 L 2m3 2m m x x1 L or m3 m3 x L x1 L L L 1 m m Substitute for m3 m3 m3 x x L 2 2m m L x x L 1 2 L 2 x L x 2x x 1 2 L L x 0 414 L or 2 414 L 1 2 m3 m1 Since m3 0 343m or 11 7 m m2 m3 x x1 L x
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