The speed of ripples $(v)$ on a water surface depends on surface tension $(\sigma)$,density $(\rho)$,and wavelength $(\lambda)$. Then the square of speed $(v^2)$ is proportional to

In the equation,pressure $P = \frac{c - t^{2}}{DS}$,where $S$ and $t$ represent the distance and time respectively. The dimensions of $\left(\frac{D}{c}\right)$ are

$A$ 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:

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:

Given that $v$ is speed,$r$ is the radius,and $g$ is the acceleration due to gravity. Which of the following is dimensionless?

To find the distance $d$ over which a signal can be seen clearly in foggy conditions,a railway engineer uses dimensional analysis and assumes that the distance depends on the mass density $\rho$ of the fog,intensity (power/area) $S$ of the light from the signal,and its frequency $f$. The engineer finds that $d$ is proportional to $S^{1/n}$. The value of $n$ is: