Question

For a particle moving along a circular path, the radial acceleration $a_{r}$ is proportional to time $t$ . If $a_{t}$ is the tangential acceleration, then which of the following will be independent of time $t$ ?

• A. $a_{t}$
• B. $a_{r}a_{t}$
• C. $\frac{a_{\text{r}}}{a_{\text{t}}}$
• D. $a_{r}(a_t)^2$

Answer 1

Answer: (D)

$\textit{a}_{\text{r}} \propto \textit{t \, }\text{; }\frac{\textit{v}^{2}}{\textit{r}}=\textit{kt}$
$\textit{v}^{2}=\textit{krt} \, ; \, 2\textit{v}\frac{\text{d} \textit{v}}{\text{d} \textit{t}}=\textit{kr}$
$\textit{a}_{\text{t}}=\frac{\textit{kr}}{2 \textit{v}}=\frac{\textit{kr}}{2 \sqrt{\textit{rkt}}}=\frac{1}{2}\sqrt{\frac{\textit{kr}}{\textit{t}}}$
$\textit{a}_{\text{t}}^{2}\textit{a}_{\text{r}}=\frac{1}{4}\frac{\textit{kr}}{\textit{t}}\times \textit{kt}=\text{independent of }\textit{t}$