# 2nd derivative of angular displacement wrt time

## Main Question or Discussion Point

If ##\theta## is angular displacement, does ##\frac{d^2\theta}{dt^2} = (\frac{d\theta}{dt})^2##? Proof?

Parameterized, I think not. A contradiction:

$$\theta = sin(t)$$
$$\frac{d\theta}{dt} = cos(t)$$
$$\frac{d^2\theta}{dt^2} = -sin(t)$$
$$\left(\frac{d\theta}{dt}\right)^2 = (cos(t))^2$$

olivermsun
\begin{align} \theta(t) &= t \\ \frac{d\theta}{dt} &= 1 \\ \frac{d^2\theta}{dt^2} &= 0 \neq \left( \frac{d\theta}{dt} \right)^2 \end{align}

Yeah. Dimensionally they agree because ##\theta## is dimensionless, but they're not equivalent. Thanks.

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