In our previous discussion of simple harmonic motion, we demonstrated that the position, velocity, and acceleration of a harmonic oscillator can be described by the three following formulas: Position, x equals A times cosine omega t. Velocity, v, equals negative omega times A times sine omega t. Acceleration, a, equals negative omega-squared times A times cosine omega t. Here A is the amplitude of the motion and omega its angular velocity. Here A is the amplitude of the motion and omega its angular velocity. Comparing the expressions for the position x and the acceleration a, we see that, for simple harmonic motion we can express the acceleration as a function of the position: a equals minus omega squared x Because the mass-spring system exhibits simple harmonic motion, we can use the previous relation to describe the acceleration of the mass. Multiplying both sides of the equation by the mass m, and applying Newton's law to the motion we obtain a relation between the force applied to the mass and its displacement. Comparing this expression with Hooke's law F equals minus k, the spring constant times the displacement x, we obtain omega equals square root of k over m. Since the period is T equals 2 pi over omega and the frequency is the inverse of the period, we can express the period and frequency of the mass-spring system as T equals 2pi times the square root of m over k and f is 1 over 2 pi times the square root of k over m.

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