The viscous force $F$ acting on a sphere of radius $r$ moving with velocity $v$ through a fluid of viscosity $\eta$ is given by $F = 6\pi \eta rv$. If a new physical quantity $Q$ is defined as $Q = \frac{Fv^2}{\eta^2 r^3}$, the dimensions of $Q$ are:

The terminal velocity of a small sphere falling through a viscous liquid is given by $v_t = \eta^a \rho^b r^c g^d$. If $\eta$ represents viscosity, $\rho$ density, $r$ radius, and $g$ acceleration due to gravity, find the values of $a$, $b$, $c$, and $d$.

Which of the following expressions, using Planck's constant ($h$), speed of light ($c$), and Newton's gravitational constant ($G$), is dimensionally consistent with length?

The dimensional formula of energy in terms of force (F), momentum (P), and time (T) is:

The dimensions of gravitational constant G are:

If the velocity of light c, gravitational constant G and Planck’s constant h are chosen as fundamental units, the dimensions of length L in the new system is

Find the dimensions of electric permittivity

The equation $\left( {P + \frac{a}{{{V^2}}}} \right).\left( {V - b} \right)$= constant. The unit of a is

The frequency (f) of a vibrating string depends on its length (L), tension (T), and mass per unit length ($\mu$). Using dimensional analysis, find the relationship between f, L, T, and $\mu$.

The Planck length, a fundamental unit in physics, is a combination of Planck’s constant ($h$), the speed of light ($c$), and Newton’s gravitational constant ($G$). Which expression correctly represents the Planck length?

Given $X = {\left( {Gh/{c^3}} \right)^{1/2}}$ , where G,h and c are gravitational constant, Planck’s constant and the velocity of light respectively. Dimensions of X are the same as those of