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.