Harmonic MotionIf an object repeats the same vibration or oscillation at regular time intervals, such vibrations or oscillations are called as Harmonic Motion.

When you pluck a guitar string, the resulting sound has a steady tone and lasts a long time. Each successive vibration of the string takes the same time as the previous one. We define periodic motion to be a motion that repeats itself at regular time intervals, such as exhibited by the guitar string or by an object on a spring moving up and down.

The time to complete one oscillation remains constant and is called the period T. Its units are usually seconds, but may be any convenient unit of time. The word period refers to the time for some event whether repetitive or not; but we shall be primarily interested in periodic motion, which is by definition repetitive.

A concept closely related to period is the frequency of an event. For example, if you get a pay check twice a month, the frequency of payment is two per month and the period between checks is half a month. Frequency f is defined to be the number of events per unit time. For periodic motion, frequency is the number of oscillations per unit time. The relationship between frequency and period is,

The SI unit for frequency is the cycle per second, which is defined to be a hertz (Hz):

A cycle is one complete oscillation. Note that a vibration can be a single or multiple events, whereas oscillations are usually repetitive for a significant number of cycles.

Simple Harmonic Motion

a) Motion of a pendulumb) Motion of a mass hanging on the springc) Motion of a hydrometer at fluid tankd) Motion of a mass on clamped wooden stripe) Motion of pistons inside the engine of a vehiclef) Vibration of guitar string

Definition: Simple Harmonic Motion (SHM)

mmFF-x+xO

A form of periodic motion of a particle, etc., in which the acceleration is always directed towards some equilibrium point and is proportional to the displacement from this point, is called SHM.Hence, the force always directed towards the opposite to the displacement,

Where, k, is the corresponding force constant.If the object attached to a spring and it is executed the SHM, then k is called the spring constant.By applying, F = ma, (Newtons 2nd Law of Motion) towards point O (Equilibrium/ mean position of the SHM);

Where; and is the angular velocity of the circular motion which corresponds to SHM.Then,Equation of SHMa = - 2x

NOTE: Any kind of motion of an object which satisfies the equation, a = -2x, then it can be identified as SHM. a / m s-2x / m

Acceleration Vs. Displacement of a SHM

a = - 2 x

y = - m x

Graphical representation of SHM

Time Period (T): is defined as the time taken for one complete oscillation.Frequency (f): is defined as the number of oscillations per unit time,f = 1 / TAngular frequency (): is defined by the equation, = 2 f. It is thus the rate of change of angular displacement (measured in radians per sec)Amplitude: The maximum displacement from the equilibrium position.Phase difference (): A measure of how much one wave is out of step with another wave, or how much a wave particle is out of phase with another wave particle.

{x =separation in the direction of wave motion between the 2 particles}

Graphs of x, v and a versus t for SHM

K. Suresh SenanayakeB.Sc. (Hons) Physics (Sp), Grad.IP (SL)1