Dimensions of physical quantities

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Question

The frequency of vibration ' $f$ ' a body of mass 'm' suspended from a spring of spring constant $k$ is given by $f = c m^{x} k^{y}$ , where c is a dimensionless constant. The value of $(x + y)$ is ____________________

Moderate

Solution

$\begin{array}{l} [T^{- 1}] = {[M^{1}]}^{x} {[M^{1} L^{0} T^{- 2}]}^{y} \\ \Rightarrow x + y = 0 and - 2 y = - 1 \end{array}$
$\Rightarrow x = - \frac{1}{2}, y = \frac{1}{2}$
$∴ x + y = 0$

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