Question

Gravitation Exercises Problem 1 Consider the following figure. An illustration shows two particles, each labeled M, one placed at each end of a vertical line. A horizontal line intersects the vertical line right at its center. A point P is at the right end of this horizontal line. The distance between the vertical line and the point P is r. The distance between the horizontal line and each particle M is a (a) Compute the vector gravitational field at a point P on the perpendicular bisector of the line joining two objects of equal mass separated by a distance 2a as shown in the figure above. (Use any variable or symbol stated above along with the following as necessary: G for the gravitational constant.) (b) Explain physically why the field should approach zero as r . (c) Prove mathematically that the answer to part (a) behaves in this way. (d) Explain physically why the magnitude of the field should approach 2GM/r2 as r e) Prove mathematically that the answer to part (a) behaves correctly in this limit. .

Answer 1

Kone g sino Let og and go be due upper and - g = GM the gravitational fields lower mass. g = GM -0 and a Grg - BPZ ca12) 2 = GM 2 so we com see that magnitudes of g & g are equal l.e. - g, = 2 = GM (792) On resolving g and go into vertical and horizontal components we see that Neatical Components a sino & g sino are equal & opposite so cancel each other while horizo- nal component are in same direction (along -Te x-anis 80 met gravitational field at P- g=g coso to coso

= g = 29, coso 2:9, 923 In vector form, g = -29 Mei Ca²+02 3/2 6 when he so then freient "P" lies Just at the mid point of line joining both masses In such situation gravitational fields g and I will be equal a and opposite so cancel each - other that's why not field will be zero as M o 191 © from egun w but r=o- 9= 2GMXO - Ca 402312 = 18=0 @ds et le point P is located at infinity so effect of masses's gravitational field at the point will be negligible so net field becomes zero o From equ. 4 - g=2G me = QGMr = (QGM/02)

As r tends to infinity-

2GM 250
and
और 120

So-

g= (0 +1)3/2 0+1)3/？ < 0