Simple harmonic motion

Simple harmonic motion is that described by a mass moving under a restorative force proportional to its displacement. The simple harmonic motion is periodic, but not every periodic motion is harmonic.

The most common example of a harmonic motion is that of a mass attached to a spring within its elastic limit: when the string is stretched or compressed it does not undergo permanent deformations and it goes back to its original state. In this case, the force acting upon the mass is given by Hooke’s law:

Where x is the mass displacement (as well as the spring deformation) and k, the stiffness, is a positive real number characteristic of the spring.

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In the figure below a mass m initially at rest on a frictionless surface is shown. The mass is attached to a spring. 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). 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 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. If the mass is released from this position, it will move to the left, towards x = 0. Hence at time t = 0 the initial position of the mass is x = A.

Then the mass continues to move until it reaches x = 0 (see (c)). From this point on, the force acting on it will be pointed in the positive direction (figure below (d)) until it reaches x = -A. At this point the mass will have zero velocity and then it will start moving towards x = 0 until it reaches again x = A. This motion will repeat itself in a cyclic fashion.

The amplitude of a harmonic simple motion (A) is the maximum displacement of the mass from its equilibrium position. Since the force exerted by the spring is conservative, the mass will go back and forth repeatedly between x = A and x = -A.

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To find the position equation x(t) for a mass undergoing simple harmonic motion we apply Newton’s second law:

We define the angular frequency as:

This equation gives the relationship between the angular frequency, the mass and the stiffness k of the spring.

Substituting into Newton’s second law we get:

From the previous equation it can be seen that x(t) must be a function whose second derivative (without constants) equals the function itself. It can be easily proven that this function is:

The simple harmonic motion is shown graphically in the position-versus-time plot below:

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. It is measured in seconds (s).

The frequency of a simple harmonic motion (f) is the number of oscillations per second. It is measured in hertz (Hz) or s-1:

The period, frequency and angular frequency of a simple harmonic motion are related through the following equations:

There are certain situations where the position of the mass at t = 0 is not x = A. In these case, the equation x(t must be modified to take into account the phase shift δ, expressed in radians):

Graphically:

The velocity and acceleration of a mass undergoing simple harmonic motion are determined by taking the first and second derivatives of the position x(t):

In the following figure the position, velocity and acceleration of a mass undergoing simple harmonic motion are shown. On the vertical axis are the maximum values of the three quantities.

You can also see how the energy in simple harmonic motion is calculated.

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