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

Pressure depends on distance as $P=\alpha \beta \exp \left(-\alpha zk\theta \right)$, where $\alpha ,\beta$ are constants $z$ is distance, $k$ is Boltzman's constants and $\theta$ is temperature. The dimension of $\beta$ are

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

In the kinematic equation s= ut + (½)at² , the dimensions of all the three terms is-
 

Identify a dimensionally correct expression from the following options.

Choose the physical quantity from the given options that has the same dimensional formula as an impulse.

A particle of mass $m$ is located in a region where its potential energy $U\left(x\right)$ depends on the position $x$ as, $U\left(x\right)=\frac{a}{x^2}-\frac{b}{x}$. Here, $a$ and $b$ are positive constants. If the time period of oscillation, which is calculated from above formula, is stated by a student as $T=4\pi a\sqrt{\frac{ma}{b^2}}$, check whether his answer is dimensionally correct.