The periodic time of a mass suspended by a spring (force constant k) i

1.	The periodic time of a mass suspended by a spring (force constant k) is T. If the spring is cut in three equal pieces, what will be the force constant of each part? If the same mass be suspended from one piece, what will be the periodic time?
Answer» Consider the spring be made of combination of three springs in series each of spring constant k. The effective spring constant K is given by: \(\frac{1}{K}=\frac{1}{k}+\frac{1}{k}+\frac{1}{k}=\frac{3}{k}\) Or K = \(\frac{k}{3}\) Or k = 3K ∴ Time period of vibration of a body attached to the end of this spring. T = \(2\pi\sqrt{\frac{m}{K}}\) = \(2\pi\sqrt{\frac{m}{(\frac{k}{3})}}\) = \(2\pi\sqrt{\frac{3m}{k}}\) …(i) When the spring is cut into three pieces, the spring constant = k. Time period of vibration of a body attached to the end of this spring. T₁= \(2\pi\sqrt{\frac{m}{k}}\) …(ii) From the equation (i) and (ii), \(\frac{T_1}{T}=\frac{1}{\sqrt3}\) Or T₁ = \(\frac{T}{\sqrt 3}\)

The periodic time of a mass suspended by a spring (force constant k) is T. If the spring is cut in three equal pieces, what will be the force constant of each part? If the same mass be suspended from one piece, what will be the periodic time?

Answer»

Consider the spring be made of combination of three springs in series each of spring constant k. The effective spring constant K is given by:

\(\frac{1}{K}=\frac{1}{k}+\frac{1}{k}+\frac{1}{k}=\frac{3}{k}\)

Or K = \(\frac{k}{3}\)

Or k = 3K

∴ Time period of vibration of a body attached to the end of this spring.

T = \(2\pi\sqrt{\frac{m}{K}}\)

= \(2\pi\sqrt{\frac{m}{(\frac{k}{3})}}\)

= \(2\pi\sqrt{\frac{3m}{k}}\) …(i)

When the spring is cut into three pieces, the spring constant = k. Time period of vibration of a body attached to the end of this spring.

T₁= \(2\pi\sqrt{\frac{m}{k}}\) …(ii)

From the equation (i) and (ii),

\(\frac{T_1}{T}=\frac{1}{\sqrt3}\)

Or T₁ = \(\frac{T}{\sqrt 3}\)