# . A mass m is attached to the free end of a massless spring of spring constant k with its other end fixed to a rigid support as shown in figure

Find out the time period of the mass, if its displaced slightly by an amount x downward.

Answer: The following steps are usually followed in this method:

Step-1: Find the stable equilibrium position which is usually known as the mean position. Net force or torque on the particle ate this position is zero. Potential energy is minimum. In our example initial position is the mean position.

Step-2 Write down the mean position force relation. In above figure at mean position kx0 = mg.........(i)

Step-3: Now displace the particle from its mean position by a small displacement x (in linear SHM) or angle θ (in case of an angular SHM) as shown in figure. From the above figure

Fnet = mg – k (x + x0) ...(2)

Step-5: Now try to reduce this net force equation in the form of F = – kx (in linear S.H.M.) or τ = – kθ (in angular SHM) using mean position force relation in step 2 or binomial theorem. from eq. (2) Fnet = mg – kx – kx0

Using eq (1) in above equation

Fnet = – kx ...(3)

Equation (3) shows that the net force acting towards mean position and is proportional to x, but in this S.H.M. constant KS.H.M. is replaced by spring constant k. So