Question

For a system executing simple harmonic motion,

• A. Displacement leads velocity and acceleration by phase of \( \frac{\pi}{2} \) and \( \pi \), respectively
• B. Displacement lags both velocity and acceleration by a phase of \( \frac{\pi}{2} \)
• C. Displacement lags velocity and acceleration by phase of \( \frac{\pi}{2} \) and \( \pi \), respectively
• D. Displacement, velocity and acceleration are in phase

Hint:

In simple harmonic motion, remember that the displacement is always ahead of the velocity by \( \frac{\pi}{2} \), and the velocity leads the acceleration by \( \frac{\pi}{2} \).

Answer 1

The Correct Option is C

For a system executing simple harmonic motion, the displacement, velocity, and acceleration are related through their phase differences:
- The displacement \( x \) is given by: \[ x = A \sin(\omega t) \] where \( A \) is the amplitude, and \( \omega \) is the angular frequency. - The velocity \( v \) is the time derivative of displacement: \[ v = \frac{dx}{dt} = A \omega \cos(\omega t) \] The velocity leads the displacement by a phase of \( \frac{\pi}{2} \). - The acceleration \( a \) is the time derivative of velocity: \[ a = \frac{dv}{dt} = -A \omega^2 \sin(\omega t) \] The acceleration lags the velocity by a phase of \( \frac{\pi}{2} \), and thus, lags the displacement by a phase of \( \pi \).
Thus, displacement lags both velocity and acceleration by the phases of \( \frac{\pi}{2} \) and \( \pi \), respectively.