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)

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