The characteristic distance at which quantum gravitational effects are significant,the Planck length,can be determined from a suitable combination of the fundamental physical constants $G, h$ and $c$. Which of the following correctly gives the Planck length?

The physical quantity which has the dimensional formula ${M^1}{T^{ - 3}}$ is

Write and explain the principle of homogeneity. Check the dimensional consistency of the given equation: $x = x_{0} + v_{0}t + \frac{1}{2}at^{2}$.

In the equation $[X+\frac{a}{Y^2}][Y-b]= RT$,$X$ is pressure,$Y$ is volume,$R$ is universal gas constant and $T$ is temperature. The physical quantity equivalent to the ratio $\frac{a}{b}$ is

If $L$,$C$,and $R$ denote the inductance,capacitance,and resistance respectively,the dimensional formula for $C^{2}LR$ is

$A$ wave pulse can travel along a tense string like a violin string. $A$ series of experiments showed that the wave velocity $V$ of a pulse depends on the following quantities: the tension $T$ of the string,the cross-sectional area $A$ of the string,and the density $\rho$ (mass per unit volume) of the string. Obtain an expression for $V$ in terms of $T$,$A$,and $\rho$ using dimensional analysis.