Hooke's Law

Table of Contents:

  • Important Mechanical Properties of solids
  • Stress and Strain
  • What is Hooke’s Law?
  • Hooke’s Law Equation
  • Hooke’s Law Experiment
  • Applications of Hooke’s Law
  • Criteria for applying Hooke’s Law

Before we move to learn about Hooke’s Law, we have to understand the subsequent terms which can be used inside the mechanical properties of solids.

Important Mechanical Properties of Solids

Elasticity: Elasticity is the property by virtue of which a material deformed under the influence of load but after the removal of the deforming load the object tends to recover its original dimension. If the body completely regains its original shape and size, then it is called a perfectly elastic body.

Plasticity: It is the belongings of material by which material does not regain its original dimension after the removal of deforming forces. In this material goes in inelastic strain. In this case, permanent deformation occurs.

Ductility: It is the property of material, which permits material to be drawn-out longitudinally to a reduced cross-sectional area, because of the application of tensile force. It also can be defined as the property of material, which permits a material to be drawn-out in the form of wire.

Brittleness: It implies that material can not to be drawn-out in the form of wire. The failure takes place without any significant deformation

Stress and Strain

Stress: It is the resistance offered by the body to any of deformation. Mathematically, it can be expressed as the restoring force per unit area.

Stress = Restoring Force /Area

= F / A

Strain: Deformation per unit length in the direction of deformation is known as strain.

Strain = Change in length / original length

= ∆L / L

What is Hooke’s Law?

When a material behaves elastically and exhibits a linear relationship between stress and strain, it is called linearly elastic material. In this case, stress is directly proportional to strain.

OR

You can say that “for small deformation, stress is directly proportional to strain”

Therefore, in simple terms, Hooke’s law states that the strain in a solid is proportional to the applied stress within the elastic limit of that solid.

Hooke’s Law Equation

Mathematically, Hooke’s Law is expressed as:

Stress α Strain

Stress = Youngs modulus of elasticity* Strain

σ = E ε

Where,

σ is the stress,

E is the modulus of elasticity also known as Young’s modulus of elasticity,

ε is the strain.

In SI units, the spring constant k, and each element of the tensor κ, is measured using units such as newtons per meter (N/m), or kilograms per second squared (kg/s2).

For continuous media, each element is therefore measured in units of pressure, namely pascals (Pa, or N/m2, or kg/(m·s2). The elements of the strain tensor ε are also expressed in units of pressure.

Hooke’s Law Experiment

All materials exhibit some degree of elasticity. Because of this elastic property, an excellent concept of restoring force comes into the picture. We can measure this elasticity property in the form of a restoring force. This restoring force opposes the deformation force and tries to maintain the original dimensions of the material.

Now consider the elasticity in only one dimension. To verify Hook’s Law on this spring-mass system now we find the relation between the restoring force and stretch (elongation) for a spring.

As we know, the restoring force is proportional to the magnitude of the deformation. This restoring force can be written mathematically as

F = – kx.

This expression for this spring-mass system is known as Hooke’s Law.

Here,

F is restoring force.

x represents the magnitude of the distortion or displacement from equilibrium as exhibited in the stretching of a spring or rubber band.

k is the proportionality constant, also known as spring constant.

Note- Here the direction of the force is in the direction opposite that of the displacement, so consider minus sign.

Procedure:

  • Assemble the apparatus, as shown in the figure.
  • Construct a data table. Now take the reading of the weight of a known mass that you hang from the spring and the position of the end of the spring before and after the mass is added.
  • Calculate the force applied to the spring and the resulting stretch of the spring by using W = mg.
  • Take approximately ten readings.
  • For each reading, record the mass, the starting position of the spring (before hanging the mass) and the ending position of the spring (while it is being stretched).

Results:

  • Calculate the value of the restoring force applied to the spring in each trial.
  • Calculate the elongation of the spring in each trial.
  • Draw graph of restoring force versus elongation for the spring.

Conclusion:

The graph of restoring force versus elongation for the spring is the straight line. This straight-line shows that elongation of a spring is directly proportional to the applied force.so we can conclude that this spring-mass system follows Hooke’s Law.

Similarly, If we draw the stress-strain curve for mild steel under tension test for static loading condition, then we see that till the proportional limit Hooks law is valid. After that, plastic deformation starts and validity of Hooke’s Law lost.

Applications of Hooke’s Law

The applications of Hooke’s Law are as given below:

  • Most commonly, in everyday life, Hooke’s Law is applied in springs because of their elasticity.
  • They are used not only in the engineering field but also used in the field of medical science.
  • It is used in breathing (lungs), skin, spring beds, diving boards and cars suspension system.
  • It is used as a fundamental principle behind manometer, spring scale and balance wheel of the clock.
  • It is also used in the foundation for seismology, acoustics and molecular mechanics.

Disadvantages of Applying Hooke’s Law

The disadvantages of Hooke’s Law are as follows:

  • Hooke’s Law is applied only in the elastic region.
  • Hooke’s Law gives accurate result only for solid bodies if the forces and deformations are small.
  • Hooke’s Law is not a universal law.