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# Elastic Potential Energy

Introduction: What is elastic potential energy? You might notice the word "elastic" in the title of this lesson. "Elastic potential energy" is exactly what the word "elastic" implies - potential energy that is stored as a result of deforming an elastic material, like a spring. This is important to consider when thinking of the springs on a mattress, or the springs that cause an attached ball to oscillate back and forth.

The elastic potential energy is equivalent mathematically to the work done on a spring, and this work done on the spring is depending on the spring constant, k, and the distance that the spring is stretched, x. The negative product of the spring constant and the distance that the spring is stretched is equivalent to the force required to stretch the spring, as per Hooke's Law (see Equation 1 below). Based on Hooke's Law, the work done to stretch a spring a specific distance is equivalent to the product of the spring constant, the square of the distance that the spring is stretched, and 1/2 (see Equation 2 below).

$F_s = -kx$ (Equation 1)
$Work=∆PE=1/2 kx^2$ (Equation 2)

Note: k=spring constant, F=force, x=distance spring is stretched, ∆PE=Elastic Potential Energy

Based on the equations shown above, as the distance that the spring is stretched increases, the force will increase and the elastic potential energy will increase. The spring constant, clearly, will remain constant, regardless of the value of the distance that the spring is stretched. In addition, it is important to note that, when the spring is not stretched, the elastic potential energy is zero, whereas the elastic potential energy of the spring is at a maximum when the spring stretches to its maximum possible distance.

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