Dimensional Analysis and Its Applications

The ratio of the dimensions of Planck’s constant and that of the moment of inertia has the dimensions of

Of the following quantities, which one has dimension different from the remaining three?

If $\text{C}$ and $\text{R}$ denote capacitance and resistance respectively, then the dimensional formula of $\text{CR}$ is

Let $\left[{\epsilon }_0\right]$ denote the dimensional formula of the permittivity of the vacuum, and $\left[{\mu }_0\right]$ that of the permeability of the vacuum. If $M=$ mass, $L=$ Length, $T=$ time and $I=$ electric current. Choose the correct option.

A new temperature scale uses $X$ as a unit of temperature, where the numerical value of the temperature $t_{\mathrm{x}}$ in this scale is related to the absolute temperature $T$ by $t_{\mathrm{x}}=3T+300$. If the specific heat of material using this unit is $1400 \mathrm{J} {\mathrm{kg}}^{-1} {\mathrm{X}}^{-1}$ its specific heat in the S.I. system of units is :

Dimensional analysis of the equation, $(\text{velocity}{)}^{\mathrm{x}}={\left(\text{pressure difference}\right)}^{\frac{3}{2}}\times (\text{density}{)}^{\frac{-3}{2}}$ gives the value of $x$ as,

A length-scale (l) depends on the permittivity (ε) of a dielectric material, Boltzmann constant (kB), the absolute temperature (T), the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles. Which of the following expression for l is dimensionally correct?

In CGS system the magnitude of the force is $100 \mathrm{dyne}.$ In another system, where the fundamental physical quantities are kilogram, metre and minute, the magnitude of the force is

Expression for the time in terms of $G$ (universal gravitational constant), $h$ (Plank constant) and $c$ (speed of light) is proportional to

The time dependence of a physical quantity $P$ is given by $P=P_{\mathrm{o}}{e}^{-\alpha t^2}$, where $\alpha$ is a constant and $t$ is time.
The constant $\alpha$