Two particles of masses $m$ and $\frac{m}{3}$ are moving along circular paths of radii $r$ given in and x, respectively, with equal speed. If the ratio of their accelerations is $\frac{5}{3}$ then the value of $x$ is __________.

$\frac{3 r}{5}$

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$\frac{5 r}{3}$

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$\frac{7 r}{3}$

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$\frac{3 r}{7}$

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Solution

The correct option is B
$\frac{5 r}{3}$

Step 1: Given data and formula used
The particle of mass $m$ moving on a circular of radius $r$
The particle of mass $\frac{m}{3}$ moving on a circular of radius $x$
Formula of acceleration $a = \frac{v^{2}}{r}$
Step 2: Calculation for the value of $x$
$\frac{a_{1}}{a_{2}} = \frac{\frac{v^{2}}{r_{1}}}{\frac{v^{2}}{r_{2}}} = \frac{r_{2}}{r_{1}} (where v is constant) \Rightarrow \frac{5}{3} = \frac{x}{r} \Rightarrow x = \frac{5}{3} r$
Hence, the correct option is (b).

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