Oscillations and SHM: Diagram-Based Questions (7)

Question

A displacement-time graph of a particle in SHM shows a sinusoid with amplitude $A = 0.05\text{ m}$ and a period of $0.4\text{ s}$. The particle starts from $x = 0$ moving in the positive direction. Write the equation of motion and find the maximum velocity and acceleration.

Solution — Step by Step

Why This Works

SHM is fully captured by three numbers: amplitude $A$, angular frequency $\omega$, and an initial phase $\phi$. The graph gives us all three directly. Velocity is the time derivative of $x$, so its peak is $A\omega$ (when $x = 0$). Acceleration is the second derivative, so its peak is $A\omega^2$ (when $|x| = A$).

The phase relationship is the heart of SHM: at the equilibrium point, speed is maximum and acceleration is zero. At the turning points, speed is zero and acceleration is maximum.

Alternative Method

If the graph instead showed $x(0) = A$ (starting at maximum displacement), the equation would be $x = A\cos(\omega t)$. The amplitude and frequency are read the same way, but the phase shifts by $\pi/2$. Always check the starting point and direction.

Common Mistake

Reading the period as half a cycle. One full period is from peak to next peak, or zero crossing (going up) to next zero crossing going up — not zero to next zero, which is half a period.