Class 11PhysicsOscillationsPeriodic, Oscillatory motion and its characteristics and types of SHM and Equation of SHM
DifficultWhy is the periodic time $T = \frac{2\pi}{\omega}$ taken for the function $f(t) = A \cos \omega t$?
(N/A) The periodic time $T$ is defined as the smallest positive time interval after which the function repeats its value.
For the function $f(t) = A \cos \omega t$,the condition for periodicity is $f(t + T) = f(t)$.
Substituting the function,we get $A \cos \omega(t + T) = A \cos \omega t$.
This implies $\cos(\omega t + \omega T) = \cos \omega t$.
We know that the cosine function repeats its value after an interval of $2\pi$,i.e.,$\cos(\theta + 2\pi) = \cos \theta$.
Comparing the arguments,we have $\omega T = 2\pi$.
Therefore,the periodic time is $T = \frac{2\pi}{\omega}$.
For the function $f(t) = A \cos \omega t$,the condition for periodicity is $f(t + T) = f(t)$.
Substituting the function,we get $A \cos \omega(t + T) = A \cos \omega t$.
This implies $\cos(\omega t + \omega T) = \cos \omega t$.
We know that the cosine function repeats its value after an interval of $2\pi$,i.e.,$\cos(\theta + 2\pi) = \cos \theta$.
Comparing the arguments,we have $\omega T = 2\pi$.
Therefore,the periodic time is $T = \frac{2\pi}{\omega}$.
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