Displacement as a continuous function of time for simple harmonic motion. Fig. . (b) A plot obtained from Eq.
( . ). The curves and are for φ = and - π / respectively. The amplitude A is same for both the plots.
Fig. . (a) A plot of displacement as a function of time as obtained from Eq. ( .
) with φ = . The curves and are for two different amplitudes A and B. x ( t ) : displacement x as a function of time t A : amplitude : angular frequency ω t + φ : phase (time-dependent) : phase constant Fig. .
The meaning of standard symbols in Eq. ( . ) argument ( ω t + φ ) in the cosine function. This time-dependent quantity, ( ω t + φ ) is called the phase of the motion.
The value of plase at t = is φ and is called the phase constant (or phase angle ). If the amplitude is known, φ can be determined from the displacement at t = . Two simple harmonic motions may have the same A and ω but different phase angle φ , as shown in Fig. .
(b). Finally, the quantity ω can be seen to be related to the period of motion T . Taking, for simplicity, φ = in Eq. ( .
), we have x ( t ) = A cos ω t ( . ) Since the motion has a period T , x ( t ) is equal to x ( t + T ). That is, A cos ω t = A cos ω ( t + T ) ( . ) Now the cosine function is periodic with period π , i.e., it first repeats itself when the argument changes by π .
Therefore, ω ( t + T ) = ω t + π that is ω = π / T ( . ) ω is called the angular frequency of SHM. Its S.I. unit is radians per second.
Since the frequency of oscillations is simply /T, ω is