# 2nd derivative of angular displacement wrt time

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

## Answers and Replies

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
Science Advisor
How about this:
\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.

FactChecker
Science Advisor
Gold Member
Try a couple of sanity checks of the proposed equation:
1) If the rotation rate is not changing, the second derivative is zero. Does that mean that it is not rotating at all?
2) Since the right hand side is always positive, does that mean that the rotation rate can only get more positive?

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