Let us consider an equation
$\frac{1}{2} m v^{2}=m g h$
where $m$ is the mass of the body. velocity, $g$ is the acceleration do gravity and $h$ is the height. whether this equation is dimensionally correct.
Answer The dimensions of $LHS$ are
$[ M ]\left[ L T ^{-1}\right]^{2}=[ M ]\left[ L ^{2} T ^{-2}\right]$
$=\left[ M L ^{2} T ^{-2}\right]$
The dimensions of $RHS$ are
$[ M ]\left[ L T ^{-2}\right][ L ]=[ M ]\left[ L ^{2} T ^{-2}\right]$
$=\left[ M L ^{2} T ^{-2}\right]$
The dimensions of $LHS$ and $RHS$ are the same and hence the equation is dimensionally correct.
The velocity $v$ (in $cm/\sec $) of a particle is given in terms of time $t$ (in sec) by the relation $v = at + \frac{b}{{t + c}}$ ; the dimensions of $a,\,b$ and $c$ are
If the capacitance of a nanocapacitor is measured in terms of a unit $u$ made by combining the electric charge $e,$ Bohr radius $a_0,$ Planck's constant $h$ and speed of light $c$ then
Einstein’s mass-energy relation emerging out of his famous theory of relativity relates mass $(m)$ to energy $(E)$ as $E = mc^2$, where $c$ is speed of light in vacuum. At the nuclear level, the magnitudes of energy are very small. The energy at nuclear level is usually measured in $MeV$, where $1\,MeV = 1.6\times 10^{-13}\,J$ ; the masses are measured i unified atomicm mass unit (u) where, $1\,u = 1.67 \times 10^{-27}\, kg$
$(a)$ Show that the energy equivalent of $1\,u$ is $ 931.5\, MeV$.
$(b)$ A student writes the relation as $1\,u = 931.5\, MeV$. The teacher points out that the relation is dimensionally incorrect. Write the correct relation.