Dimensions of mass, density, volume, velocity and acceleration are\([{m}]=[{M}]\)
A. Dimensions of the given options are:
\([{E}]=\left[\frac{3}{16} m c^{2}\right]\)
\(\Rightarrow\left[\frac{3}{16} m c^{2}\right]=[m][c]^{2}\)
\(\Rightarrow\left[\frac{3}{16} m c^{2}\right]=[M]\left[L T^{-1}\right]^{2}\)
\(\Rightarrow\left[\frac{3}{16} m c^{2}\right]=\left[M L^{2} T^{-2}\right]\)
The presence of numbers or constant terms does not affect the dimensions.
Thus, option 'A' can not be ruled out.
B. \([E]=\frac{1}{2} m[a \times t]^{2}\)
\(\left.\Rightarrow] \frac{1}{2} m[a \times t]^{2}\right]=[M]\left\{\left[L T^{-2}\right][T]\right\}^{2}\)
\(\Rightarrow\left[\frac{1}{2} m[a \times t]^{2}\right]=\left[{ML}^{2} T^{-2}\right]\)
Thus, option 'B' also can not be ruled out.
C. \([E]=\frac{1}{2}\left[m c^{2}\right]+[m a]\)
\(\Rightarrow \frac{1}{2}\left[m c^{2}\right]+[m a]=[m][c]^{2}+[m][a]\)
\(\Rightarrow \frac{1}{2}\left[m c^{2}\right]+[m a]=\left[M L^{2} T^{-2}\right]+\left[M L T^{-2}\right]\)
The 2nd term is not matched with the first term. From the principle of homogeneity of dimensions, the given formula is wrong.
Thus, option 'C' can be ruled out.
D. \([E]=\left[\frac{m^{2} u^{2}}{d x v}\right]\)
\(\Rightarrow[E]=\left(\left[M^{2}\right]\left[L T^{-1}\right]^{2}\right) /\left(\left[M L^{-3}\right]\left[L^{3}\right]\right)\)
\(\Rightarrow[E]=\left[M L^{2} T^{-2}\right]\)