When force is applied to a material, we know that it either stretches or compresses in response to the applied force. In mechanics, the force applied per unit area is known as stress and is denoted by the symbol σ. The extent to which the material compresses or stretches is known as strain. Different materials respond differently to applied stress. This information is necessary for engineers while selecting materials for their structures.
In 19th-century, while studying springs and elasticity, English scientist Robert Hooke noticed that many materials exhibited a similar property when the stress-strain relationship was studied. There was a linear region where the force required to stretch the material was proportional to the extension of the material. This is known as Hooke’s Law. In this article, let us discuss Hooke’s law in detail.
What is Hooke’s Law?
Hooke’s law states that the strain of the material is proportional to the applied stress within the elastic limit of that material.
When the elastic materials are stretched, the atoms and molecules deform until stress is been applied and when the stress is removed they return to their initial state.
Mathematically, Hooke’s law is commonly expressed as:
F = –k.x
In the equation,
Interested to learn more about spring? Below are the links:
Solved Example
A spring is displaced by 5 cm and held in place with a force of 500 N. What is the spring constant of the spring?
Solution:
We know that the spring is displaced by 5 cm, but the unit of the spring constant is Newtons per meter. This means that we have to convert the distance to meters.
Converting the distance to meters, we get
5 cm = 0.05 m
Now substituting the values in the equation, we get
F = –k.x
Now, we need to rework the equation so that we are calculating for the missing metric which is the spring constant, or k. Looking only at the magnitudes and therefore omitting the negative sign, we get
500 N/0.05 m = k
k = 1000 N/m
Therefore, the spring constant of the spring is 1000 N/m.
Hooke’s Law Experiment
Consider a spring with load application as shown in the figure.
Different springs, depending on the material, will have different spring constants. This can be calculated. The constant calculated empirically can be used further. The figure shows us three instances, the stable condition of the spring, the spring elongated to an amount x under a load of 1 N, and the spring elongated to 2x under a load of 2 N. If we plug in these values into the equation above we get the spring constant for the material in consideration.
σ = Eε |
Where,
- σ is the stress
- E is the modulus of elasticity also known as Young’s modulus
- ε is the strain
When the stress is removed from the material, there are two types of deformation that can take place: plastic deformation and elastic deformation
Hooke’s Law Graph
The figure below shows the stress-strain curve for low carbon steel.
From the origin till the proportional limit nearing yield strength, the straight line implies that the material follows Hooke’s law. Beyond the elastic limit between proportional limit and yield strength, the material loses its elastic nature and starts exhibiting plasticity. The area under the curve from origin to the proportional limit falls under the elastic range. The area under the curve from a proportional limit to the rupture/fracture point falls under the plastic range.
The ultimate strength of a material is defined based on the maximum ordinate value given by the stress-strain curve (from origin to rupture). The rupture strength is given by the value at a point of rupture.
This video explains to you the general stress-strain graph of an elastic material experiencing tensile load and what are various stages in it. Within the elastic limit it follows Hooke’s law. As we keep on increasing the load, other stages and points like proportionality limit, yield point, fracture point, ultimate tensile strength are achieved, and they have been well explained in this video.
Hooke’s Law FAQs
- List the applications of Hooke’s Law
- Hooke’s Law is used all branches of science and engineering
- It is used as a fundamental principle behind manometer, spring scale, balance wheel of the clock.
- Foundation for seismology, acoustics and molecular mechanics.
- List the disadvantages of Hooke’s Law
- The law ceases to apply past the elastic limit of a material.
- The law is accurate only for solid bodies if the forces and deformations are small.
- The law isn’t a universal principle and only applies to the materials as long as they aren’t stretched way past their capacity.
- How many independent elastic constants does an isotropic, homogeneous and elastic material obeying Hooke’s law have?
- Explain the factor of safety.
- Give an example of non-hoookean material.
The applications of Hooke’s Law is as follows:
The disadvantages of Hooke’s Law is as follows:
The isotropic, homogeneous and elastic material obeying Hooke’s law has 3 constants Young’s modulus, Shear modulus and bulk modulus.
The factor of safety is the ratio of ultimate stress to the permissible stress.
Rubber is generally regarded as a “non-hookean” material because its elasticity is stress dependent and sensitive to temperature and loading rate.
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