Question

The energy $E$ of a particle oscillating in SHM depends on the mass $m$ of the particle, frequency $n$ and amplitude $a$ of oscillation. Show dimensionally that $E\propto m{n}^2{a}^2$

Answer 1

$\begin{array}{c}Energy\left[E\right]=\left[M^{-1}{L}^2{T}^{-2}\right]\\ Mass\left(m\right)=\left[M^1\right]\\ Frequency\left(n\right)=\left[T^{-1}\right]\\ Amplitude\left(a\right)=\left[L^1\right]\\ \left[E\right]=k{\left[M^1\right]}^x{\left[T^{-1}\right]}^y{\left[L^1\right]}^z\\ \left[M^{-1}{L}^2{T}^{-2}\right]={\left[M^1\right]}^x{\left[T^{-1}\right]}^y{\left[L^1\right]}^z\\ x=1\\ -y=-2\\ y=2\\ z=2\\ So,therequiredrelationis,\\ E=km{n}^2{a}^2\end{array}$