There are two fixed heavy masses of magnitude M of high density on x-axis at a distance 2d apart….

Problem :  There are two fixed heavy masses of magnitude M of high density on x-axis at a distance 2d apart.  A small mass m moves in a circle of radius R in the y-z plane between the heavy masses.  Find the speed of the small particle.

Solution :

Force of attraction between M and m is

$ \displaystyle F = \frac{G M m}{R^2 + d^2} $

By symmetry Fx components will cancel.

  ∴ The net force, which provides the centripetal force, is given by

$ \displaystyle 2F_y = 2.\frac{G M m}{R^2 + d^2} \frac{R}{\sqrt{R^2 + d^2}} $

$ \displaystyle = \frac{2 G M m R}{(R^2 + d^2)^{3/2}} $

$ \displaystyle \frac{2 G M m R}{(R^2 + d^2)^{3/2}} = \frac{m v^2}{R}$

$ \displaystyle v = \sqrt{\frac{2 G M R^2}{(R^2 + d^2)^{3/2}}} $

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