Does this massless spring affect the system?

Question

I have to write out the differential equation modelling this system: There's a mass connected to a wall with a spring of spring constant $k_1$, sitting on a frictionless surface, with another spring (with spring constant $k_2$) attached to the other end of the mass but not connected to anything on it's other end. Both springs are massless. A force is then applied to the $k_2$ spring and released to start the system moving.

My question is, other than the fact that I need to relate the initial position of the mass to the spring constants $k_1$ AND $k_2$, does the $k_2$ spring affect the system at all? I can't see what affect a massless spring attached to one end of a block, but not attached to anything else could do. Is this spring going to affect my differential equation?

Answer 1

No, it won't.

Part of the reason we use things like massless springs is to avoid thinking about them in detail; they exist to provide a force and to store energy, but are forbidden from doing anything else.

Proof by contradiction:

Suppose that the block is being accelerated in some direction by spring 1. Now the contradictory supposition: suppose that spring 2 exerted some force on the block. Then by Newton's 3rd law the block would exert some force on the spring. However the mass of the spring is zero, so the acceleration of the spring is infinite, so the spring can travel any amount of distance in zero time. The spring will instantaneously move such that the net force applied to it is zero, so that its acceleration remains zero.

Thus the spring cannot exert a force on the block, completing the contradiction.

As you said, the second spring is just there to make your initial conditions more interesting.