A spring is fixed at one end and at the free end a mass is attached. The system is placed on a horizontal frictionless table. Show that when the mass is displaced from its equilibrium position, it will execute simple harmonic motion. Also find the time period of oscillation.

Let a spring have its one end fixed to a rigid support and mass m attached at the other end as shown in figure. 
               
Let the mass m be pulled towards right by distance y.
The spring will also extend by distance y.
The restoring force set up in the spring is,
                       F=-ky
where,
k is the spring constant of the spring.
Negative sign implies restoring force is opposite to the direction of extension.
When the mass is released, it will move under the action of restoring force.
The acceleration produced in the mass due to restoring force is, 
                    
Thus, the acceleration is directly proportional to displacement and directed towards the mean position. 
Hence, the motion of the mass is simple harmonic motion.
The time period is given by,