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# Simple Harmonic Motion

Introduction: The motion of a spring, such as the springs on some mattresses, can be explained by simple harmonic motion. Simple harmonic motion refers to an oscillation (motion that occurs at regular intervals) where the restoring force is directly proportional to displacement and that acts in an opposite direction to displacement itself. Simple harmonic motion can also help to explain that which we cannot see, such as the motion of particles as they vibrate in a chemical compound.

With many simple harmonic oscillators, a weight oscillates back and forth while being attached to a stationary object. The force on the spring is represented by $F_s$, and this force is dependent on the spring constant and the displacement of the spring from rest. The equation for the force on the spring is summarized by Hooke's Law, as shown below:

$F_s = -kx$, where k=spring constant, x=displacement from the equilibrium position, and $F_s$=force on the spring

The elastic potential energy, or potential energy of the spring, can be summarized by the equation shown below:

$PE_s = 1/2 kx^2$, where $PE_s$=elastic potential energy, x=displacement of the spring from the equilibrium position, and k=spring constant

What can be seen by both equations above is that, as the displacement from the equilibrium position increases, the force on the spring and the potential energy will increase. The spring constant, on the other hand, typically depends on the material that makes up the spring, and will typically vary on the basis of the rigidity of the spring itself. This spring constant does not change in the equation, regardless of the displacement from the equilibrium position.

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