A particle at rest starts moving with constant angular acceleration \(4 \mathrm{rad} / \mathrm{s}^2\) in circular path. At what time the magnitudes of its tangential acceleration and centrifugal acceleration will be equal?

In rotational motion,
\(\begin{aligned}
& \omega=\omega_0+\alpha \mathrm{t} \\
& \omega=\alpha \mathrm{t}
\end{aligned}\)
\(\text { ( } \because \omega_0=0 \text {; particle at rest.) }\)
\(\therefore \quad\) Centrifugal acceleration \(\mathrm{a}=\omega^2 \mathrm{r}\)
\(\therefore \quad \mathrm{a}=\alpha^2 \mathrm{t}^2 \mathrm{r}\)
Tangential acceleration \(\mathrm{a}_{\mathrm{t}}=\alpha \times \mathrm{r}\)
Given: \(\mathrm{a}=\mathrm{a}_{\mathrm{t}}\)
\(\Rightarrow \alpha^2 \mathrm{t}^2 \mathrm{r}=\alpha \mathrm{r}\)
\(\mathrm{t}^2=\frac{1}{\alpha}=\frac{1}{4}\)
\(\therefore \quad \mathrm{t}=\frac{1}{2}=0.5 \mathrm{~s}\)