Based on the third law motion of equal and opposite forces, a mass should never accelerate because the net force on it will always be zero. Resolve the issue.

Question

would F=ma answer this question?

If a force is exerted by object A on object B, then B exerts a force of equal magnitude and in opposite direction on A. So, there is then a net force on A causing A to accelrate. But B accelerates in the opposite direction.

But the center of mass of A and B does not accelerate, because the total force exerted on the combined system A plus B is zero.

Try to solve this problem by making clever use of Newton's third Law:

Given a sphere of uniform density rho and radius R, what is the gravitatonal force that one half of the sphere exerts on the other half?