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

1. $v^2/rg$

2. $v^{2}r/g$

3. $v^{2}g/r$

4. $v^{2}rg$

A force F is given by F = at + bt², where t is time. What are the dimensions of a and b:

1. $ML{T}^{-3}$ and $M{L}^2{T}^{-4}$

2. $ML{T}^{-3}$ and $ML{T}^{-4}$

3. $ML{T}^{-1}$ and $ML{T}^0$

4. $ML{T}^{-4}$ and $ML{T}^1$

The fundamental physical quantities that have same dimensions in the dimensional formulae of torque and angular momentum are 

1. Mass, time

2. Time, length

3. Mass, length

4. Time, mole

If pressure P, velocity V and time T are taken as fundamental physical quantities, the dimensional formula of force is  

1. $P{V}^2{T}^2$

2. $P^{-1}{V}^2{T}^{-2}$

3. $PV{T}^2$

4. $P^{-1}V{T}^2$

If the speed of light (c), acceleration due to gravity (g) and pressure (p) are taken as the fundamental quantities, then the dimension of gravitational constant is 

1. $c^2{g}^0{p}^{-2}$

2. $c^0{g}^2{p}^{-1}$

3. $c{g}^3{p}^{-2}$

4. $c^{-1}{g}^0{p}^{-1}$

If the time period (T) of vibration of a liquid drop depends on surface tension (S), radius (r) of the drop and density $\left(\rho \right)$ of the liquid, then the expression of T is 

1. $T=k\sqrt{\rho r^3/S}$

2. $T=k\sqrt{{\rho }^{1/2}{r}^3/S}$

3. $T=k\sqrt{\rho r^3/S^{1/2}}$

4. None of these

The physical quantity which has the dimensional formula $M^1{T}^{-3}$ is 

1. Surface tension

2. Solar constant

3. Density

4. Compressibility

The dimensions of the interatomic force constant are:

1. $M{T}^{-2}$

2. $ML{T}^{-1}$

3. $ML{T}^{-2}$

4. $M{L}^{-1}{T}^{-1}$