Question

Show that the penetration distance $\eta$has units of $m$.

Short answer

The penetration distance $\eta$has units of $m$is given below in step.

Step-by-step solution

Given Information

We need to find the penetration distance $\eta$has units of $m$.

Explanation

The equation of penetration distance is given as:

$\eta =\frac{h}{\sqrt{2m\left(U_0-E\right)}}$

The unit of role="math" localid="1650172801611" $h$is $J.s$. Its dimension can be $\left[h\right]=M\times L^2\times T^{-1}$.

The unit of $m$is $kg$, so dimension is $\left[m\right]=M$.

The unit of $\left(U_0-E\right)$is $J$, then dimension is $\left(U_0-E\right)=M\times L^2\times T^{-2}$

Therefore, the dimension of $\eta$is:

$\left[h\right]=\frac{M\times L^2\times T^{-1}}{\sqrt{M\times L^2\times T^{-2}}}=\frac{M\times L^2\times T^{-1}}{M\times L\times T^{-1}}=L$

Then, $\eta$has the dimension of length. So its unit is$m$.