Given that $v$ is speed, $r$ is the radius and $g$ is the acceleration due to gravity. Which of the following is dimensionless
${v^2}/rg$
${v^2}r/g$
${v^2}g/r$
${v^2}rg$
An expression of energy density is given by $u=\frac{\alpha}{\beta} \sin \left(\frac{\alpha x}{k t}\right)$, where $\alpha, \beta$ are constants, $x$ is displacement, $k$ is Boltzmann constant and $t$ is the temperature. The dimensions of $\beta$ will be.
The dimensions of $\frac{\alpha}{\beta}$ in the equation $F=\frac{\alpha-t^2}{\beta v^2}$, where $F$ is the force, $v$ is velocity and $t$ is time, is ..........
If velocity of light $c$, Planck’s constant $h$ and gravitational constant $G$ are taken as fundamental quantities, then express time in terms of dimensions of these quantities.
The displacement of a progressive wave is represented by $y = A\,sin \,(\omega t - kx)$ where $x$ is distance and t is time. Write the dimensional formula of $(i)$ $\omega $ and $(ii)$ $k$.