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. $M^0{L}^0{T}^0$