I was wondering if potential energy for spring is always positive. P E = 1 2 k x 2 where k is spring constant and x is the change in length.

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Yair Valentine

Answered question

2022-08-12

I was wondering if potential energy for spring is always positive.

P E = \frac{1}{2} k x^{2}

where k is spring constant and x is the change in length.

Answer & Explanation

Bryant Liu

Beginner2022-08-13Added 15 answers

Actually, the formula, you mentioned, should be for change in potential energy.

C h a n g e i n P . E . = \frac{1}{2} k x^{2} ⧸ \Rightarrow P . E . = \frac{1}{2} k x^{2}

You can always add some negative constant to your P.E. and make it negative.

P . E . = V_{0} + \frac{1}{2} k x^{2}

where

V_{0}

is just a real constant.
This was just the mathematical part.
For a proper physical interpretation, observe that the force corresponding to this potential is given by

\vec{F} = - \vec{\nabla} V = - k x \hat{x}

which indicates a simple harmonic motion (oscillatory motion like that of a pendulum, spring etc.)
So, for any motion of the spring from its equilibrium position (elongation or compression), external energy needs to be supplied. This energy compresses or elongates the spring, which results in storing of potential energy inside the spring. So, the motion involves net increase of potential energy whether it is being compressed or elongated.

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