Hooke's Law

• Elongation of a spring (Hooke's law)

« Hooke sowed the seeds of several sciences […]. In the study of insects, friction, the strength and elasticity of materials, meteorology, oceanography, comets, evolution, geology, crystals, light, heat, sound, combustion and respiration, his work was seminal. His insights into light and planetary motion were important in preparing the way ».

In mechanics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress). Materials for which Hooke's law is a useful approximation are known as linear-elastic or "Hookean" materials. For systems that obey Hooke's law, the extension produced is directly proportional to the load.

F is the restoring force exerted by the material (usually in newtons), and k is the force constant (or spring constant). The constant has units of force per unit length (usually in newtons per meter).

If this holds, we say that the behaviour is linear. If shown on a graph, the line should show a direct variation. There is a negative sign on the right hand side of the equation because the restoring force always acts in the opposite direction of the x displacement (when a spring is stretched to the left, it pulls back to the right).

If the measurements have been taken carefully and if the spring has not been subjected to any strain, there will be a unique relationship between the number of masses and length of the spring.

If you make a graph of the length of the spring as a function of the applied weight, the experimental points will lie (approximately) along a straight line, so in this case force and length are the typical application of the mathematical relation describing direct proportionality.

It is useful, for a correct comprehension of direct proportionality, to plot some experimental data above.

We must observe, however, that the measurements required for this experience are taken with an inverse experimental procedure: we are plotting the weight of the loaded masses with respect to the elongation of our spring. Actually, for every known mass we are measuring the correspondent elongation.

This inverse procedure both simplifies the acquisition and limits the errors occurring during the measurements, that is the reason why it is preferred in several other experiments.

The deformation of an elastic body is directly proportional to the deforming force. The value of k depends only on the nature of the body.

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