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.

A dimensionless quantity is constructed in terms of electronic charge $e$, permittivity of free space $\varepsilon_0$, Planck's constant $h$, and speed of light $c$. If the dimensionless quantity is written as $e^\alpha \varepsilon_0^\beta h^7 c^5$ and $n$ is a non-zero integer, then $(\alpha, \beta, \gamma, \delta)$ is given by

The amount of heat energy $Q$, used to heat up a substance depends on its mass $m$, its specific heat capacity $(s)$ and the change in temperature $\Delta T$ of the substance. Using dimensional method, find the expression for $s$ is (Given that $\left.[s]=\left[ L ^2 T ^{-2} K ^{-1}\right]\right)$ is

A physical quantity of the dimensions of length that can be formed out of $c, G$ and $\frac{e^2}{4\pi \varepsilon _0}$  is $[c$ is velocity of light, $G$ is the universal constant of gravitation and $e$ is charge $] $

In the relation $y = a\cos (\omega t - kx)$, the dimensional formula for $k$ is

A spherical body of mass $m$ and radius $r$ is allowed to fall in a medium of viscosity $\eta $. The time in which the velocity of the body increases from zero to $0.63$ times the terminal velocity $(v)$ is called time constant $(\tau )$. Dimensionally $\tau $ can be represented by