$\begin {array} {1 1} (A) A = k \large\frac{v}{r} \\ (B) A = k \large\frac{v}{r^2} \\ (C) A = k \large\frac{v^2}{r} \\ (D) A = kvr \end {array}$

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Given that Acceleration of the particle $A \propto r^n v^m$ where $v$ is the velocity and $r$ is the radius of the circle.

$A = k r^n v^m$, where $k$ is a dimensionless unit of propotionality.

We know that the formula for Acceleration $A = \large\frac{L}{T^2}$

$\Rightarrow$ Knowning the dimensions of $A$, $r$ and $v, \large\frac{L}{T^2} $$ = L^n \large(\frac{L}{T})^{\normalsize m}$

For this to be balanced, $n + m = 1 $ and $m = 2$ which means that $n = -1$.

Therefore, we can write the expression for acceleration as $A = k \large\frac{v^2}{r}$

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