Class 11 Physics Online Notes Periodic and Harmonic Motion

11th Physics

Periodic motion of a simple pendulum

If the motion of an object is periodic, then there is a characteristic time: the time it takes for the motion to repeat itself. This is called the period (of the periodic motion) and is usually given the symbol T:

Period (T): The time for one complete cycle of the periodic motion.

For example, the period of the rotation of the earth about its axis is one day. During the quarter, when classes are in session, the period of our activities is one week. We can also speak of the number of cycles repeated per unit time. This is called the frequency of the periodic motion:

frequency (f): The number of cycles per unit time.

Simple Harmonic Motion

If the motion of an object is a perfect single sinusoidal function, we term the motion ”simple harmonic”. Let’s review the mathematics of simple trig functions. We need just three parameters to describe any single sinusoidal function: the period (T), the amplitude (A), and a parameter related to the position at time t = 0. The amplitude of the motion is the maximum value from equilibrium that the object travels. Suppose the object starts off initially at x = 0, has some initial speed in the +x direction and an amplitude A. In this case, the motion of the object is described by:

x(t) = Asin( 2πt /T )

where T is the period of the motion. One often defines 2π/T to be the variable ω:

ω ≡ 2π / T

So one often sees the equation for this case of simple harmonic motion written as x(t) = Asin(ωt). At time t = 0, the position is at x = 0 since sin(0) = 0. The velocity of the particle at any time t is found by differentiating x(t):

v(t) = dx / dt = Aωcos(ωt)

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