Question

The periodic time of two oscillators are \(\mathrm{T}\) and $(5 \mathrm{~T} / 4)$ respectively. Both oscillators starts their oscillation simultaneously from the midpoint of their path of motion. When the oscillator having periodic time \(\mathrm{T}\) completes one oscillation, the phase difference between the two oscillators will be \(\ldots \ldots \ldots\) (A) \(90^{\circ}\) (B) \(112^{\circ}\) (C) \(72^{\circ}\) (D) \(45^{\circ}\)

Short answer

The phase difference between the two oscillators when the first oscillator completes one oscillation is \(72^{\circ}\).

Step-by-step solution

Modify the given information

We are given periodic times of two oscillators as T and (5T/4). When the first oscillator completes one oscillation, the time elapsed is equal to its period, which is T.

Find the fraction of oscillation completed by the second oscillator

To find the fraction of the oscillation completed by the second oscillator at time T, we will divide the time elapsed (T) by the period of the second oscillator (5T/4): Fraction of oscillation completed = Time elapsed / Period of second oscillator Fraction of oscillation completed = T / (5T/4)

Simplify the fraction

We will now simplify T / (5T/4) to find the fraction of the oscillation completed by the second oscillator: Fraction of oscillation completed = T * (4/5T) Fraction of oscillation completed = (4/5)

Convert the fraction into degrees

Since we know the fraction of the oscillation completed by the second oscillator, we can now convert it into degrees. One complete oscillation corresponds to 360 degrees. Therefore, we multiply the fraction by 360 degrees to find the phase difference: Phase difference = Fraction of oscillation completed * 360 Phase difference = (4/5) * 360

Calculate the final result

Now, we perform the multiplication to find the phase difference between the two oscillators: Phase difference = (4/5) * 360 Phase difference = 288 degrees However, we should consider the phase difference within the range of 0 to 360 degrees. So we subtract 360 degrees from 288 degrees to get the phase difference within the required range: Phase difference = 288 degrees - 360 degrees Phase difference = -72 degrees A negative phase difference means that the second oscillator is behind the first oscillator. So, the phase difference between the two oscillators when the first oscillator completes one oscillation is \(72^{\circ}\). Therefore, the correct option is (C) \(72^{\circ}\).