Graphical Analysis of SHM

In simple harmonic motion, both position and velocity vary sinusoidally with time. For an object starting at maximum displacement, the position is described by

$$x(t) = A\cos\left(\frac{2\pi t}{T}\right)$$

where $A$ is the amplitude and $T$ is the period. The position oscillates between $+A$ and $-A$.

The velocity graph is also sinusoidal with the same period, but shifted by one quarter period ($T/4$). The velocity oscillates between $+v_{\max}$ and $-v_{\max}$, where the maximum speed is

$$v_{\max} = \frac{2\pi A}{T}$$

Key relationship: The velocity equals the slope of the position-time graph.

When position is at a maximum or minimum, the position curve is momentarily flat (slope = 0), so $v = 0$. When position passes through zero (equilibrium), the position curve is steepest, so $|v| = v_{\max}$. A positive slope means the object moves in the positive direction ($v > 0$); a negative slope means motion in the negative direction ($v < 0$).