Simple Harmonic Motion

Oscillation A repeating fluctuation in a physical object or quantity is called an oscillation. In an oscillation, the motion is repetitive, ie. Periodic, and the body moves back and forth around an equilibrium position. A characteristic of oscillatory motion is the time taken to complete one full oscillation. This is called the period.

Oscillations without a fixed period Many objects undergo oscillatory motion with a non-constant period. Example: A leaf on a tree that is blowing in the wind oscillates, but its oscillations do not have a fixed period, and the amount the leaf moves away from its equilibrium position is not a regular function of time.

Other examples of oscillation The motion of a mass at the end of a horizontal or vertical spring after the mass has been displaced away from the equilibrium position. The motion of a ball inside a bowl after it has been displaced away from its equilibrium position at the bottom of the bowl.

More examples The motion of a body floating in a liquid after it has been pushed downwards and then released. A tight guitar string that is set in motion by plucking the string. The motion of a mass attached to a vertical string that is displaced slightly and then released. (A pendulum)

Simple Harmonic Motion Oscillators that are perfectly isochronous (constant period) and whose amplitude does not change with time are called simple harmonic oscillators and their motion is referred to as simple harmonic motion. Although SHM does not exist in the real world, many oscillatory systems approximate to this motion.

Terms related to SHM Displacement –This refers to the distance that an oscillating system is from its equilibrium position at any particular instant. Amplitude – This is the maximum displacement of an oscillating system from its equilibrium position.

Period – this is the time it takes an oscillating system to make one complete oscillation. Frequency – this is the number of complete oscillations made by the system in one second.

Phase Difference Suppose we have two identical pendulums oscillating next to each other. If the displacements of the pendulums are the same at all instances of time, then they are oscillating in phase. If the maximum displacement of one is Θ° when the maximum displacement of the other is -Θ°, then we say they are oscillating in anti-phase or that the phase difference is 180°.

In general, the phase difference between two identical systems oscillating with the same frequency can have any value between 0 and 360° (or 0 to 2π radians).

Angular frequency (ω) A useful quantity associated with oscillatory motion is angular frequency. This is defined in terms of the linear frequency as: ω = 2πf In terms of period, angular frequency is: ω = 2π/T

Defining equation for SHM If it were possible to remove all frictional forces acting on an oscillating pendulum, then the displacement versus time graph for the motion would look sinusoidal. The amplitude does not decay with time, therefore this is SHM. See diagram of displacement versus time on the chalk board.

Acceleration of the System It turns out that if the acceleration a of a system is directly proportional to its displacement x from its equilibrium position and is directed towards the equilibrium position, then the system will execute SHM. This is the formal definition of SHM.

Acceleration expressed mathematically a = -(constant)x The negative sign indicates that the acceleration is directed towards equilibrium. Mathematical analysis shows that the constant is equal to ω2 where ω is the angular frequency. Therefore: a = -ω2x

Force in a SHM system If a system is performing SHM, a force must be acting in the direction of the acceleration. F = -kx Where k is a constant. Note: do not confuse this k with the spring constant of a spring since there are many non-spring examples of SHM.