Dimensional Analysis and Its Applications

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 $\alpha$

Young's modulus of steel is $1.9\times {10}^{11} \mathrm{N} {\mathrm{m}}^{-2}.$ When expressed in $C.G.S$. unit, it will be equal to :-

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.

 Suppose we employ a system in which the unit of mass equals $100 \mathrm{kg}$, the unit of length equals $1 \mathrm{km}$ and the unit of time $100 \mathrm{s}$ and call the unit of energy eluoj ($\mathrm{joule}$ written in reverse order), then what is the relation between eluoj and joule?

Convert $1 \mathrm{MW}$ power on a new system having basic units of mass, length, and time as $10 \mathrm{kg}$, $1 \mathrm{dm}$, and $1\min$, respectively. Write value of $x$ considering answer is of the form $x\times {10}^{12}$

$L,C$ and $R$ represent the physical quantities inductance, capacitance and resistance respectively. The combinations which have the dimensions of frequency are:

The pairs of physical quantities that have the same dimensions is (are):

The dimensions of the quantities in one (or more) of the following pairs are the same. Identify the pair(s).

A new system of units is proposed in which unit of mass is $\alpha \mathrm{kg}$, unit of length is $\beta \mathrm{m}$ and unit of time is $\gamma \mathrm{s}$. What will be value of $5\mathrm{ }\mathrm{J}$ in this new system?

If velocity of light $c$, Planck’s constant $h$ and gravitational constant $G$ are taken as fundamental quantities then the dimensions of length will be

A new unit of length is chosen such that the speed of light in vacuum is unity. What is the distance between the sun and the earth in terms of the new unit if light takes $8$ minutes and $20$ seconds to cover this distance?

Choose the incorrect statement:

For a system of particles of masses ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ , centre of mass has speed v at an instant. If relative speed of the particles is u, kinetic energy of the system is