Question

A particle of mass m is attached 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.

Answer 1

(a) Suppose the particle is pushed lightly against the spring 'C' through displacement 'x'.

Total resultant force on the particle is kx.

Due to spring C and $\frac{kx}{2}$ due to spring A and B.

$\therefore$ Total Rsultant force

$=kx+\sqrt{{\left(\frac{kx}{\sqrt{2}}\right)}^2+{\left(\frac{kx}{\sqrt{2}}\right)}^2}$

$=kx+kx=2kx$

Acceleration $=\frac{2kx}{m}$

Time period $=2\pi \sqrt{\frac{\text{Displacement}}{\text{Acceleration}}}$

$=2\pi \sqrt{\frac{x}{2kx/m}}$

$=2\pi \sqrt{\frac{m}{2k}}$

[Cause : When the body pushed against 'C' the spring C, tries to pull the block XL. At that moment the spring A and B tries to pull the block with force $\frac{kx}{\sqrt{2}}$ and $\frac{kx}{\sqrt{2}}$ respectively towards xy and xz respectively. So the total force on the block is due to the spring force 'C' as well as the component of two spring force A and B ].