Question

One-fourth length of a spring of force constant K is cut away. The force constant of the remaining spring will be
A. \[\dfrac{3}{4}K \\ \]
B. \[\dfrac{4}{3}K \\ \]
C. 4K
D. K

Answer 1

Hint: Spring constant gives the stiffness of a spring and is equal to the force needed to stretch the spring divided by the distance that the spring is compressed or stretched.

Formula used:
The relation of spring constant K with length can be given by,
\[K \propto \dfrac{1}{l}\]

Complete step by step solution:
A spring of constant K is cut away by \[\dfrac{1}{4}\] of its length, we have to find the spring constant of the remaining part of the spring. We know that spring constant is inversely proportional to length of the spring and let the original length of spring be l then the relation of spring constant K with length can be given by,
\[K \propto \dfrac{1}{l}\,.......(1)\]

As the \[\dfrac{1}{4}\]of length l of spring is cut away then the remaining length l’ of spring will be,
\[l' = \dfrac{3}{4}l\]
Let the spring constant of remaining length be K’ then according to equation (1) it will be,
\[K' \propto \dfrac{4}{{3l}}\,.......(2)\]
On dividing equation (2) by equation (1) we get,
\[\dfrac{{K'}}{K} = \dfrac{4}{{3l}} \times l \\
\Rightarrow K' = \,\dfrac{4}{3}K\].
Hence, the constant of remaining length will be \[\dfrac{4}{3}K\].

Therefore, option B is the correct answer.

Note: Springs having larger spring constant will have smaller displacement and one having smaller spring constant will have larger displacement, it always has positive magnitude because negative spring constant will mean that when a compressive force is applied to spring it will compress itself further, which is against the nature of a spring.