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Question: What is spring-mass system in Physics? Show that a mass attached to a spring executes SHM.

Spring-mass system

If one end of a spring is attached to a mass and the other end is fixed against a support, and the whole assembly is placed on a frictionless surface, the system is called spring-mass system.

When the mass is displaced to a small distance, the elastic restoring force in the spring pulls it back to the equilibrium position. The mass does not stop at the equilibrium point due to inertia and moves further reaching point B covering the same displacement. Again at the displaced position, a restoring force pulls the mass back toward O. In this way, the mass starts oscillating back and forth.

Spring-mass system executes SHM

Consider a spring-mass system as shown.

When the block is displaced to a small displacement from its equilibrium position, an elastic restoring force pulls it back to the equilibrium position. From Hook’s law, this restoring force is directly proportional to the displacement of the mass. Since Frest and x have opposite directions, therefore,

Frest ∝ -x

 ⇒ Frest = – kx          …          (1)

This equation shows the restoring force and displacement of the mass are proportional to each other and the direction of force is towards the equilibrium position. This is the condition for SHM and, therefore, the spring mass system executes simple harmonic motion.

Moreover, from the second law of motion, Frest = ma       …           (2)

Comparing (1) and (2), we have

Since k and m are both constants, therefore, k/m is also constant. Thus

a -x

Again this is condition of SHM. Therefore, the spring-mass system executes simple harmonic motion.

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