Question

A spring having a spring constant K is loaded with a mass m. The spring is cut into two equal parts and one of these is loaded again with the same mass. The new spring constant is
A. \[\dfrac{k}{2}\]
B. k
C. 2k
D. \[{k^2}\]

Answer 1

Hint: Spring constant of a spring is inversely proportional to the length of the spring and will be the same for both halves of the spring.

Formula used:
\[k \propto \dfrac{1}{{Length\,of\,spring(l)}}\]
Here k is the spring constant.

Complete step by step solution:
A spring of constant k is loaded with mass, we have to find the spring constant when the spring is cut into two equal halves and one of them is loaded with the same mass again.
As spring constant of a spring is inversely proportional to the length of the spring.

Let the length of the spring be l then, the relation between constant k and length l can be mathematically expressed as:
\[k \propto \dfrac{1}{{Length\,of\,spring(l)}}\]

When the spring is cut into two equal halves then the length of each halves will be \[\dfrac{l}{2}\] and let the spring constant of one half be k’.

From the proportionality relation when length becomes half spring constant will be doubled i.e. k’ = 2k.

Therefore, option C is the correct option.

Note: Spring constant for both halves will be equal i.e. 2k and it does not depend on the mass of the object suspended to the spring. It can be mathematically proved by equation\[F = - kx = - m{\omega ^2}x\], as on substituting \[{\omega ^2} = \dfrac{k}{m}\]in \[k = m{\omega ^2}\]mass m of object will be canceled out.