Hooke's law and circular motion.

[ehild]

The force acts in a point. If you speak about an extended body, with forces acting at different points, you can sum up the individual equations m_ia_i=F_i, and arrive at the equation for the acceleration of the centre of mass: a_CM∑m_i=∑Fi . If there are only forces of interaction (no external force) the sum of forces is zero. The system, which has 2 kg mass altogether, has zero acceleration. But the individual masses feel the force of the spring and have centripetal acceleration.

ehild

 

[ehild]

It hit me that since we considered the total change in length of the spring due to the 2 masses, we give consideration to the total mass.

A stretched spring has tension. To keep it stretched there must be forces equal to that tension acting at both ends. You can not exert force on a spring if the other end is free. Hook's law refers to the tension which is equal to kΔL, and it is equal to the force acting on one end of the spring (but force of the same magnitude and opposite direction has to act on the other end).
Again, the net force on the stretched spring is zero, as the forces acting at the ends are opposite and of equal magnitude.

ehild