Question

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which one of the following equations is only valid if and when the angular measure is expressed in radians?
$\alpha=\frac{\delta \omega}{\delta t}$
$\omega=\frac{\delta \theta}{\delta t}$
$\theta=\frac{1}{2} \alpha t^{2}+\omega_{0} t$
$\omega=\frac{v t}{r}$
$\omega^{2}=\omega_{0}^{2}+2 \alpha \theta$

Explanation:

The equations for angular kinematics (like $\theta=\frac{1}{2}\alpha t^{2}+\omega_{0}t$ and $\omega^{2}=\omega_{0}^{2}+2\alpha\theta$) are derived based on the calculus relationships where the angular measure $\theta$ (and its derivatives $\omega,\alpha$) are in radians. Radians are dimensionless (arc - length/radius) which is crucial for the calculus - based derivations (e.g., $\frac{d\theta}{dt}$ for angular velocity). In the case of $\theta=\frac{1}{2}\alpha t^{2}+\omega_{0}t$, it is analogous to the linear kinematic equation $x = x_{0}+v_{0}t+\frac{1}{2}at^{2}$, and the derivation from $\omega=\frac{d\theta}{dt}$ and $\alpha=\frac{d\omega}{dt}$ (using calculus) requires $\theta$ in radians.

Answer:

$\theta=\frac{1}{2}\alpha t^{2}+\omega_{0}t$