simple harmonic motion real world examples

The period of a simple harmonic motion (T) is the time it takes for the mass to complete one full cycle, from its initial position x = A to x = -A and back again to x = A.

The simple harmonic motion is shown graphically in the position-versus-time plot below: It can be easily proven that this function is: Substituting into Newton’s second law we get:įrom the previous equation it can be seen that x(t) must be a function whose second derivative (without constants) equals the function itself. This equation gives the relationship between the angular frequency, the mass and the stiffness k of the spring. To find the position equation x(t) for a mass undergoing simple harmonic motion we apply Newton’s second law: Since the force exerted by the spring is conservative, the mass will go back and forth repeatedly between x = A and x = -A. The amplitude of a harmonic simple motion (A) is the maximum displacement of the mass from its equilibrium position. This motion will repeat itself in a cyclic fashion. At this point the mass will have zero velocity and then it will start moving towards x = 0 until it reaches again x = A.

From this point on, the force acting on it will be pointed in the positive direction (figure below (d)) until it reaches x = -A. Then the mass continues to move until it reaches x = 0 (see (c)).

Hence at time t = 0 the initial position of the mass is x = A. If the mass is released from this position, it will move to the left, towards x = 0. When the spring is stretched by an amount x = A (see (b) in the figure below), the force exerted on the mass by the spring is restorative, and thus is pointed in the negative direction. We will take the coordinate x so that it is negative when the spring is compressed, zero at the natural length (when the spring is not deformed) and positive when the spring is stretched. When the spring is neither stretched nor compressed (a), the force acting on the mass is zero (because x = 0 and therefore the mass does not move). In the figure below a mass m initially at rest on a frictionless surface is shown.

Simple harmonic motion real world examples