Question

The phase difference between displacement and acceleration of a particle in a simple harmonic motion is:

• A. $\pi$ rad
• B. $\frac {3\pi}{2}$ rad
• C. $\frac {\pi}{2}$ rad
• D. $zero$

Answer 1

The Correct Option is A

In simple harmonic motion (SHM), the displacement of a particle and its acceleration are related through the differential equation:

\(a(t) = -\omega^2 x(t)\),

where \(x(t)\) is the displacement, \(a(t)\) is the acceleration, and \(\omega\) is the angular frequency.

This equation shows that displacement and acceleration are out of phase by \(\pi\) radians because acceleration is proportional to the negative of displacement.

To understand the phase relationship, consider the expressions for displacement and acceleration in terms of phase:

• \(x(t) = A \sin(\omega t + \phi)\),
• \(a(t) = -\omega^2 A \sin(\omega t + \phi)\).

Since \(a(t) = -\omega^2 x(t)\), the acceleration is inverted with respect to the displacement, indicating a phase difference of \(\pi\) radians.

This means that when the particle is at a displacement of \(x(t)\), the acceleration \(a(t)\) is maximally negative, and vice versa, completing the understanding of their phase difference.

Thus, the correct answer is \(\pi\) radians.

1. The phase difference between displacement and acceleration in SHM is \(\pi\) radians because they are in opposite directions.
2. Options like \(\frac{\pi}{2}\) radians and \(\frac{3\pi}{2}\) radians are incorrect because they do not correspond to the nature of the phase relationship defined by \(a(t) = -\omega^2 x(t)\).
3. A zero phase difference is incorrect as acceleration is not in sync with displacement; it is, in fact, the opposite.