A certain uniform spring has spring constant k. Now the spring is cut in half. What is the relationship between k and the spring constant k' of each resulting smaller spring? (Use the following as necessary: k)
To determine the relationship between the spring constant of the original spring, k, and the spring constant of each resulting smaller spring, k', you need to consider the physical properties of a spring.
The spring constant, denoted as k, represents the stiffness of a spring. It quantifies how much force is required to stretch or compress the spring by a certain distance. Mathematically, the force required to stretch or compress a spring by a distance x can be given by Hooke's Law:
F = -kx
In this equation, F represents the force applied to the spring, k is the spring constant, and x is the displacement of the spring from its equilibrium position.
When the uniform spring is cut in half, the total length of the spring is reduced by half, leading to a change in the spring constant for each resulting smaller spring, k'.
To understand the relationship between k and k', you can consider the concept of spring stiffness. The stiffness of a spring is inversely proportional to its length. A longer spring tends to be less stiff, while a shorter spring tends to be more stiff.
In the case of the uniform spring being cut in half, the resulting smaller springs have half the length of the original spring. Since the length is halved, the smaller springs will be stiffer.
Therefore, the relationship between the spring constant of the original spring, k, and the spring constant of each resulting smaller spring, k', is as follows:
k' = 2k
Each smaller spring will have a spring constant, k', that is twice as large as the spring constant of the original spring, k.