A particle of mass m is executing oscillation about origin on x-axis. Its potential energy is $U\left(x\right)=K|X{|}^n$ .If the time period T is function of its mass, amplitude (a) and (K) ; the value of n if $T \alpha a^{-1/2}$ (K' is a constant) is _______________

$\left[K\right]=\frac{\left[U\right]}{[X{]}^n}=\frac{\left[M{L}^2{T}^{-2}\right]}{\left[L^n\right]}=\left[M{L}^{2-n}{T}^{-2}\right]$

$T\propto \left(\text{ mass }{)}^X\right(\mathrm{amp}{)}^Y(K{)}^Z$

$\begin{array}{l}\left[M^0{L}^{0}T\right]=\left[M{]}^X\right[L{]}^Y{\left[M{L}^{2-n}{T}^{-2}\right]}^Z=\left[M^{X+Z}{L}^{Y+2Z-nZ}{T}^{-2Z}\right]\\ \text{ So, }-2Z=1\end{array}$

$or Z=-\frac{1}{2} ................\left(i\right)$

and, $X+Z=0$

$\text{ or }X=-Z=\frac{1}{2} ...........................\left(ii\right)$

$\text{ and, }Y+2Z-nZ=0$

$\text{ or }Y=1-\frac{n}{2}[\text{ from (i) and (ii)] }$

$\begin{array}{l}\text{ So, }T\propto \left(\mathrm{mass}{)}^{1/2}\right(\mathrm{amp}{)}^{1-n/2}(K{)}^{-1/2}\\ \text{ Given }\\ T\propto (\mathrm{amp}{)}^{-1/2}\\ \text{ So }1-\frac{n}{2}=-\frac{1}{2}\\ \Rightarrow n=3\end{array}$