Question

Does cutting a spring change the spring constant?

Answer 1

Hint:The spring constant is defined by the force that is required to increase the unit length of a spring. The applied force on the spring is directly proportional to the length of the spring. The answer can be given by assuming a spring is cut into two or three pieces and applying the mathematical representations. The spring constant can be found in these representations.

Formula used:
The applied force, $F = kx$
$x$ is the increase in length of the spring due to the applied force.
$k$ is the spring constant.

 Complete step by step solution:
When a mass is attached to a spring that is connected to a rigid rod, and then a force is applied to this spring, the spring is increased by some length.
The applied force $F \propto x$ . Where $x$ is the increase in length of the spring due to the applied force.
So, $F = kx$, $k$ is the spring constant.
Hence, by definition, The spring constant is the force that is required to increase the unit length of a spring.
Let the spring be cut into three equal pieces. Hence, due to the equal force applied to these parts, each part is increased by the same length $\dfrac{x}{3}$ .
Therefore, the spring constant for each part $k' = \dfrac{F}{{\dfrac{x}{3}}} = \dfrac{{3F}}{x} = 3k$
Hence, the spring constant for each part of the spring will be three times the spring constant of the whole spring.
So, it is concluded that the spring-constant changes after cutting the spring.

Note:Two types of spring combinations can be made and therefore equivalent spring constants can be evaluated. The springs are connected in series or parallel combinations.
For the series combination of two springs, the equivalent spring constant is given by, $k = \dfrac{{{k_1}{k_2}}}{{{k_1} + {k_2}}}$
${k_1},{k_2}$ are the spring constants of these two springs.
For the parallel combination of two springs, the equivalent spring constant is given by, $k = {k_1} + {k_2}$ .