A small steel ball of radius r is allowed to fall under gravity through a column of a viscous liquid of coefficient of viscosity $\eta$. After some time the velocity of the ball attains a constant value known as terminal velocity $v_T$. The terminal velocity depends on (i) the mass of the ball m, (ii) $\eta$, (iii) r and (iv) acceleration due to gravity g. Which of the following relations is dimensionally correct?

1. $v_T\propto \frac{mg}{\eta r}$

2. $v_T\propto \frac{\eta r}{mg}$

3. $v_T\propto \eta rmg$

4. $v_T\propto \frac{mgr}{\eta }$ 

The quantity $X=\frac{{\epsilon }_{0}LV}{t}; {\epsilon }_0$ is the permittivity of free space, L is length, V is the potential difference and t is time. The dimensions of X are the same as that of:

1. Resistance

2. Charge

3. Voltage

4. Current

${\mu }_0$ and ${\epsilon }_0$ denote the permeability and permittivity of free space, the dimensions of ${\mu }_0{\epsilon }_0$ are:

(1) $L{T}^{-1}$

(2) $L^{-2}{T}^2$

(3) $M^{-1}{L}^{-3}{Q}^2{T}^2$

(4) $M^{-1}{L}^{-3}{I}^2{T}^2$

The expression $\left[M{L}^2{T}^{-2}\right]$ represents

(1) Pressure

(2) Kinetic energy

(3) Momentum

(4) Power

The dimensions of physical quantity X in the equation Force $=\frac{X}{\text{Density}}$ is given by 

(1) $M^1{L}^4{T}^{-2}$

(2) $M^2{L}^{-2}{T}^{-1}$

(3) $M^2{L}^{-2}{T}^{-2}$

(4) $M^1{L}^{-2}{T}^{-1}$

The dimensions of CV² matches with the dimensions of 

(1) L²I

(2) L²I ²

(3) LI²

(4) $\frac{1}{LI}$

The Martians use force (F), acceleration (A) and time (T) as their fundamental physical quantities. The dimensions of length on Martians system are 

(1) [FT²]

(2) [${\mathrm{F}}^{-1}{\mathrm{T}}^2$]

(3) [${\mathrm{F}}^{-1}{\mathrm{A}}^2{\mathrm{T}}^{-1}$]

(4) [AT²]

The dimension of $\frac{1}{\sqrt{{\epsilon }_0{\mu }_0}}$ is that of 

(1) Velocity

(2) Time

(3) Capacitance

(4) Distance

An athletic coach told his team that muscle times speed equals power. What dimensions does he view for muscle?

(1) $ML{T}^{-2}$

(2) $M{L}^2{T}^{-2}$

(3) MLT²

(4) L

The dimensional formula of wave number is

(1) $M^0{L}^0{T}^{-1}$

(2) $M^0{L}^{-1}{T}^0$

(3) $M^{-1}{L}^{-1}{T}^0$

(4) $M^0{L}^0{T}^0$