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Simple harmonic motion

Since force is equal to mass, m, times acceleration, a, the force equation for a spring obeying Hooke's law looks like:

The mass of the spring is small in comparison to the mass of the attached mass and is ignored. Since acceleration is simply the second derivative of x with respect to time,

This is a second order linear differential equation for the displacement x {\displaystyle x} as a function of time. Rearranging:

the solution of which is the sum of a sine and cosine:

A {\displaystyle A} and B {\displaystyle B} are arbitrary constants that may be found by considering the initial displacement and velocity of the mass. The graph of this function with B = 0 {\displaystyle B=0} (zero initial position with some positive initial velocity) is displayed in the image on the right.