Hooke’s Law Equation – 3 different formulas

Last updated on May 21st, 2023 at 01:51 pm

Often we find more than one expression or formula of Hooke’s law. Here we will find out those forms of Hooke’s Law Equation at the same place and discuss their uses and relevance. We have a very useful post on different types of stress, strain, and elasticity modulus, that you can go through along with this post.

Types of Hooke’s Law

Here, we will study different types of Hooke’s Law – we will explain those, and write the equations related to each type of Hooke’s Law.

Types of Hooke’s Law discussed here are as follows:
(1) Hooke’s Law equation For linear Springs,
(2) Hooke’s law equation for restoring force exerted by a spring, &
(3) Hooke’s law for stress-strain in continuous media.

Hooke’s Law equation For linear Springs

Here we will consider a simple helical spring that has one end attached to some fixed object, while the free end is being pulled by a force whose magnitude is F_s.

Suppose that the spring has reached a state of equilibrium, where its length is not changing anymore.

Let x be the amount by which the free end of the spring was displaced from its initial.

Hooke’s law states that: F_s ∝ x

=> F_s = kx ……………………….. [ Hooke’s Law Equation ] (1)

where k is a positive real number, characteristic of the spring.

Moreover, the same formula holds when the spring is compressed, with F_s and x both negative in that case.

According to this formula, the graph of the applied force F_s as a function of the displacement x will be a straight line passing through the origin, whose slope is k.

Hooke’s law equation for restoring force

Hooke’s law for spring is sometimes stated under the convention that F_s is the restoring force exerted by the spring on whatever is pulling its free end. In that case, the equation gets modified as the following equation since the direction of the restoring force is opposite to that of the displacement.

F_s = – kx …………………………… [ Hooke’s Law Equation ] (2)

Hooke’s Law stress-strain formula | Hooke’s law for continuous media

The stresses and strains of the material inside a continuous elastic material (such as a block of rubber, the wall of a boiler, or a steel bar) are connected by a linear relationship that is mathematically similar to Hooke’s spring law and is often referred to by that name.

Hooke’s Law of Elasticity states that: For small deformations, stress is directly proportional to strain.

 Stress ∝ strain

So, Stress = K. Strain   ………………….  [ Hooke’s Law Equation – Hooke’s Law stress-strain formula ] (3)
  Here K is a constant.

Summary

Here, we have seen 3 different forms of Hooke’s law equation. While solving physics numerical from this Elasticity chapter, use the right formula as per the numerical data and query.