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Question
A particle of mass m is attatched to three springs A, B and C of equal force constants kas shown in figure . If the particle is pushed slightly against the spring C and released, find the time period of oscillation.
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Solution
(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 \[= \frac{2kx}{m}\]
\[\text { Time period }= 2\pi\sqrt{\frac{\text { Displacement }}{\text { Acceleration }}}\]
\[ = 2\pi\sqrt{\frac{x}{2kx/m}}\]
\[ = 2\pi\sqrt{\frac{m}{2k}}\]
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