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Suppose we are told that the acceleration $A$ of a particle moving with uniform speed $v$ in a circle of radius $r$ is proportional to some power of r, say $r^n$, and some power of v, say $v^m$. Which of the following is the simplest form of an equation for the acceleration?

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

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1 Answer

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 $A = k \large\frac{v^2}{r}$
answered Jun 17, 2013 by meena.p
edited Mar 23, 2014 by balaji.thirumalai

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