Question

The frequency $f$ of vibration of mass $m$ suspended from a spring of spring contact $k$ is given by $f=c{m}^x{k}^y$. Where $c=$ dimensionless constant, then find the values of $x$ and $y$.

Answer 1

In this situation dimension of RHS=LHS

i.e, $\left[f\right]=\left[c{m}^x{k}^y\right]$

$\left[T^{-1}\right]=\left[M^x\right]\left[M^y{T}^{-2y}\right]=\left[M^{x+y}{T}^{-2y}\right]$, since $\left[k\right]=\left[M{T}^{-2}\right]$

comparing both sides we get,

$x+y=0,and-1=-2y\Rightarrow y=1/2$

and $x=-y=-1/2$