In space there exists points in which the combined gravitational effects of larg

Question

In space there exists points in which the combined gravitational effects of large bodies, such as the Earth and the Sun cancel out. These are known as Lagrange points. Ideally if you were to place an object of mass 'm' at one of these points, the object would not have any gravitational attraction to these large bodies. In a simplistic case assume the Sun and the Earth are the only objects in space. Determine the point in between the Earth and the Sun where the gravitational attraction is zero. Express this as either the distance from the Earth to the object or from the Sun to the object. Distance from Sun to Earth (d_SE = 149.6 x 10^9m) Mass of the Sun (1.989 times 10^30kg) Mass of the Earth (5.972 times 10^24kg)

Explanation / Answer

let r1 be the distance from earth where attraction is zero.

we know force of gravity, Fg = (G*m1*m2) / r1^2

Fge = Fgs

Let m2 = 1 kg (the mass of an object between the earth and the sun )

For Earth: Fge = [(6.67x10^-11)(5.972x10^24)(1)] / r1 ^2

For the Sun : Fgs = [(6.67x10^-11)(1.989x10^30)(1)] / (149.6x10^9 - r1)^2

Fge = Fgm

[(6.67x10^-11)(5.972x10^24)(1)] / r1 ^2 = [(6.67x10^-11)(1.989x10^30)(1)] / (149.6x10^9 - r1)^2

or, [(5.972x10^24)(1)] / r1 ^2= [(1.989x10^30)(1)] / (149.6x10^9 - r1)^2

or, 2.4437x10^12/r1 = 1.4103x10^15/(149.6x10^9 - r1)

or, 2.4437/r1 = 1.4103x10^3/(149.6x10^9 - r1)

or, 1410.3r1 = 3.6557x10^11 - 2.4437r1

or, 1412.7437r1 = 3.6557x10^11

or, r1 = 258.76x10^6 m .......................................ans

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