Show that a vibrating mass-spring system performs simple harmonic motion. What is its time period? On what factors does it depend?

Consider a mass attached to a vertical spring with a force constant . When displaced from its equilibrium position, the restoring force acts in the opposite direction, according to Hooke’s Law:

F= -kx
Applying Newton’s second law,

ma= -kx

a= -kx/m

Since acceleration is proportional to displacement and directed toward the equilibrium position, the motion is SHM.

The time period of oscillation is given by:

T=2π√m/k

Factors affecting the time period:

Mass of the body (m): A greater mass increases the time period.

Spring constant (k): A stiffer spring (higher ) results in a shorter time period.

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