The frequency of vibration of a string is given by $\nu = \frac{p}{2l} \left[ \frac{F}{m} \right]^{1/2}$. Here $p$ is the number of segments in the string,$l$ is the length,and $F$ is the tension. The dimensional formula for $m$ will be:

An artificial satellite is revolving around a planet of mass $M$ and radius $R$ in a circular orbit of radius $r$. From Kepler's third law about the period of a satellite around a common central body,the square of the period of revolution $T$ is proportional to the cube of the radius of the orbit $r$. Show using dimensional analysis that $T = \frac{k}{R}\sqrt{\frac{r^3}{g}}$,where $k$ is a dimensionless constant and $g$ is the acceleration due to gravity.

From the following,the quantity (constructed from the basic constants of nature) that has the dimensions of length,as well as the correct order of magnitude for a typical atomic size,is:

If $E, M, J$ and $G$ respectively denote energy, mass, angular momentum, and universal gravitational constant, the quantity which has the same dimensions as the dimensions of $\frac{E J^2}{M^5 G^2}$ is:

Young-Laplace law states that the excess pressure inside a soap bubble of radius $R$ is given by $\Delta P = 4 \sigma / R$,where $\sigma$ is the coefficient of surface tension of the soap. The $EOTVOS$ number $E_0$ is a dimensionless number that is used to describe the shape of bubbles rising through a surrounding fluid. It is a combination of $g$ (acceleration due to gravity),$\rho$ (density of the surrounding fluid),$\sigma$ (surface tension),and a characteristic length scale $L$ (radius of the bubble). $A$ possible expression for $E_0$ is:

The frequency of vibration $f$ of a mass $m$ suspended from a spring of spring constant $K$ is given by a relation of the type $f = C\,{m^x}{K^y}$,where $C$ is a dimensionless quantity. The values of $x$ and $y$ are: