Question

Displacement vs time curve for a particle SHM is as shown in the figure:

Which of the following statements is correct?
A) Phase of the oscillator is the same at t=0s and t=2s.
B) Phase of the oscillator is the same at t=2s and t=5s.
C) Phase of the oscillator is the same at t=1s and t=7s.
D) Phase of the oscillator is the same at t=1s and t=5s.

Answer 1

The phase of a particle SHM is the state of the body according to its position (displacement) and direction of motion at any given time. It is measured in angular units from 0 to $2\pi$ rad.

Complete step by step solution:
Given:
We can determine the phase at each time point according to displacement and the nature of the curve.
So, phase at t=0s is,
${\theta _0} = \dfrac{\pi }{2}$ rad
Phase at t=1s is,
${\theta _1} = \pi$ rad
Phase at t=2s is,
${\theta _2} = \dfrac{{3\pi }}{2}$ rad
Phase at t=5s is,
${\theta _5} = \pi$ rad
and, phase at t=7s is,
${\theta _7} = 0$ rad
Thus, we see that the phase at t=1s and t=5s is the same.

Final Answer: The correct option is D. Phase of the oscillator is the same at t=1s and t=5s.

Note: (1) Phase can also be seen as the angle made by the arm of the particle revolving in a circle with the diameter, where the diametric projection of motion is the SHM.
(2) The phase at t=1s and t=7s can be confused to be the same as they both have the same position of the particle. But since the direction of motion is different and so the phase is different.