Two Gravity force on a point mass by another point mass and

Two Gravity force on a point mass by another point mass and a uniform dust cloud. A point mass m is attracted gravitationally by another stationary point mass M a distance r away. Assume there is also a cloud of mass of uniform density rho surrounding all space. Derive an expression in terms of these variables for the total gravitational force F(r) on m due to both M and the cloud.

Solution

Here, it needs to be understood that as there is a uniform mass density surrounding all the space, the net force due to this mass of cloud on the given mass would be zero, as any force acting on the given body would get balanced by an equivalent force in opposite direction.

Therefore the mass m will suffer a net force only due to the gravitational acceleration of mass M.

Now, for masses m and M, the gravitational pull is given as GMm/r²

Therefore the required total gravitational force is given as F(r) = GMm/r²