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"Area under the Force-Extension Graph"

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Let a force act on a flexible body (e.g. an elastic spring). Due to the action of the force, the body flexibly deforms: it stretches or contracts by and thereby receives elastic energy.

The characteristic of an elastic body is that it returns to its original shape when the force ceases to act. We say that the process is reversible. Elastic energy is converted back into mechanical work or into some other energy, for example, kinetic energy and (or) potential energy.

Elastic energy exists in, for example, a stretched bow with which we intend to shoot an arrow, a ball bouncing off the ground, an elastic rope during adrenaline-pumping jumps from a bridge, etc.

How do we get elastic energy?

In the chapter, Newton's Third Law of Motion, we learned about the formula that connects the force and the extension when an elastic spring is stretched (Hooke's law) which is given as:

where is a constant representing the hardness/stiffness of the spring.

Let's draw a graph of the spring extension and the force acting on the spring.

The force is not constant, it increases with extension . At the final extension , the force is . When calculating the work of the force, we can take the product of the average force together with the distance covered which is equal to the final extension :

The expression on the right-hand side of the above equation is the elastic energy :

If a body deforms elastically due to the work of a force, it gains elastic energy which is given as:

where is the force constant of the spring and is the extension.

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Let's extend the validity of the work-energy theorem: the work of a force is equal to the change in kinetic energy , potential energy , and elastic energy :

or:

If no external force is acting, the total energy is conserved - energies are just converted from one form to another (law of conservation of energy):

The work of a force is equal to the change in kinetic, potential, and elastic energy.

The law of conservation of energy says that if no external force acts on a body, the total energy is conserved - the energies are only converted from one form to another:

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material editor: Habeeb Adenle