Mass spring executes simple harmonic motion

Mass spring system and Simple
Harmonic Motion (SHM)
Mass Spring System:
A mass spring system consists of a mass which is attached to a spring made of certain
material and other end of spring is connected to a firm support.


Motion Of Mass Attached To A Spring
Consider a block of mass ‘m’ attached to one end of elastic spring which can move
freely on a frictionless horizontal surface,


Working
When the block is displaced the elastic restoring force pulls the block towards
equilibrium position ‘O’. For and ideal spring that obeys hooke's law, the elastic
restoring force ‘Fres’ is directly proportional to the displacement ‘x’ from equilibrium
position.

Mathematically:
Fres ∝ -x 一ⅰ
Since F and x always have opposite direction therefore we have negative sign in
relation.

Spring Constant:
The stiffness of the spring or spring constant is represented by the letter ‘k’. The SI Unit
of k is N/m.


Hooke’s Law
The equation for Hooke’s law is,
Fres= -kx
The motion of mass attached to a spring is SHM.

Mathematical Proof:
Restoring force produces acceleration in the body, given by newton’s second law of
motion,
Fres = ma 一ⅱ
Comparing eqⅰand eq ⅱ, we get