Step 1: Understanding the Question:
The question asks for the phase relationship between displacement (\( x \)) and acceleration (\( a \)) for a particle in Simple Harmonic Motion (S.H.M.). Step 2: Key Formula or Approach:
The standard equation for displacement starting from the extreme position is:
\[ x = A \cos(\omega t) \]
The acceleration in S.H.M. is given by the relation:
\[ a = -\omega^2 x \]
Step 3: Detailed Explanation:
Substituting the displacement equation into the acceleration formula:
\[ a = -\omega^2 A \cos(\omega t) \]
Using trigonometric identities, \( -\cos(\theta) = \cos(\theta + \pi) \):
\[ a = \omega^2 A \cos(\omega t + \pi) \]
Comparing the equations for \( x \) and \( a \):
Phase of \( x \) is \( \omega t \).
Phase of \( a \) is \( \omega t + \pi \).
The phase difference \( \Delta \phi = (\omega t + \pi) - \omega t = \pi \text{ rad} \). Step 4: Final Answer:
The phase difference between displacement and acceleration in S.H.M. is \( \pi \text{ rad} \).