$A$ modified gravitational potential is given by $V = -\frac{GM}{r} + \frac{A}{r^2}$. If the constant $A$ is expressed in terms of gravitational constant $G$, mass $M$, and velocity of light $c$, then from dimensional analysis, $A$ is:

The dimensions of the product $\mu_{0} \varepsilon_{0}$ are related to those of velocity as

In an experiment,four quantities $a, b, c$ and $d$ are measured with percentage errors of $1\%, 2\%, 3\%$ and $4\%$ respectively. Quantity $w$ is calculated as follows: $w = \frac{a^4 b^3}{c^2 \sqrt{d}}$. The percentage error in the measurement of $w$ is .......... $\%$.

$A$ beaker contains a fluid of density $\rho \, kg/m^3$,specific heat $S \, J/kg \, ^\circ C$,and viscosity $\eta$. The beaker is filled up to height $h$. To estimate the rate of heat transfer per unit area $(Q/A)$ by convection when the beaker is placed on a hot plate,a student proposes that it should depend on $\eta$,$\left( \frac{S\Delta \theta}{h} \right)$,and $\left( \frac{1}{\rho g} \right)$,where $\Delta \theta$ (in $^\circ C$) is the temperature difference between the bottom and top of the fluid. In that situation,the correct option for $(Q/A)$ is:

State the principle of homogeneity of dimensions.

$A$ quantity $x$ is given by $(IF v^{2} / WL^{4})$ in terms of moment of inertia $I$,force $F$,velocity $v$,work $W$,and length $L$. The dimensional formula for $x$ is the same as that of: