Question

A particle of mass m is attatched to three springs A, B and C of equal force constants k as shown in figure (12−E6). If the particle is pushed slightly against the spring C and released, find the time period of oscillation.

Figure

Answer 1

(a) Let us push the particle lightly against the spring C through displacement x.

As a result of this movement, the resultant force on the particle is kx​.
The force on the particle due to springs A and B is $\frac{kx}{\sqrt{2}}$.
Total Resultant force$=kx+\sqrt{{\left(\frac{kx}{\sqrt{2}}\right)}^2+{\left(\frac{kx}{\sqrt{2}}\right)}^2}$
= kx + kx = 2kx
Acceleration is given by $=\frac{2kx}{m}$
$$\begin{gathered} \mathrm{Time}\mathrm{period}=2\mathrm{\pi }\sqrt{\frac{\mathrm{Displacement}}{\mathrm{Acceleration}}} \\ =2\mathrm{\pi }\sqrt{\frac{x}{2kx\mathit{/}m}} \\ =2\mathrm{\pi }\sqrt{\frac{m}{2k}} \end{gathered}$$