Which of the following is dimensionally correct

1. Pressure = Energy per unit area

2. Pressure = Energy per unit volume

3. Pressure = Force per unit volume

4. Pressure = Momentum per unit volume per unit time

Planck's constant has the dimensions (unit) of 

1. Energy

2. Linear momentum

3. Work

4. Angular momentum

The equation of state of some gases can be expressed as $\left(P+\frac{{\displaystyle a}}{{\displaystyle V^2}}\right)(V-b)=RT$. Here P is the pressure, V is the volume, T is the absolute temperature and a, b, R are constants. The dimensions of ‘a’ are 

1. $M{L}^5{T}^{-2}$

2. $M{L}^{-1}{T}^{-2}$

3. $M^0{L}^3{T}^0$

4. $M^0{L}^6{T}^0$

If V denotes the potential difference across the plates of a capacitor of capacitance C, the dimensions of CV² are [This question includes concepts from 12th syllabus]

1. Not expressible in MLT

2. $ML{T}^{-2}$

3. $M^{2}L{T}^{-1}$

4. $M{L}^2{T}^{-2}$

If L denotes the inductance of an inductor through which a current i is flowing, the dimensions of Li² are  

1. $M{L}^2{T}^{-2}$

2. Not expressible in MLT

3. $ML{T}^{-2}$

4. LT

Of the following quantities, which one has dimensions different from the remaining three 

1. Energy per unit volume

2. Force per unit area

3. Product of voltage and charge per unit volume

4. Angular momentum per unit mass

A spherical body of mass m and radius r is allowed to fall in a medium of viscosity $\eta$. The time in which the velocity of the body increases from zero to 0.63 times the terminal velocity $\left(v\right)$ is called time constant $\left(\tau \right)$. Dimensionally $\tau$ can be represented by 

1. $\frac{m{r}^2}{6\pi \eta }$

2. $\sqrt{\left(\frac{6\pi mr\eta }{g^2}\right)}$

3. $\frac{m}{6\pi \eta rv}$

4. None of the above

The frequency of vibration f of a mass m suspended from a spring of spring constant K is given by a relation of this type $f=C{m}^x{K}^y$; where C is a dimensionless quantity. The value of x and y are 

1. $x=\frac{1}{2},y=\frac{1}{2}$

2. $x=-\frac{1}{2},y=-\frac{1}{2}$

3. $x=\frac{1}{2},y=-\frac{1}{2}$

4. $x=-\frac{1}{2},y=\frac{1}{2}$

The quantities A and B are related by the relation, m = A/B, where m is the linear density and A is the force. The dimensions of B are of

1. Pressure

2. Work

3. Latent heat

4. None of the above

The velocity of water waves v may depend upon their wavelength $\lambda$, the density of water $\rho$ and the acceleration due to gravity g. The method of dimensions gives the relation between these quantities as: 

1. ${\mathrm{v}}^2\propto {\mathrm{g\lambda }}^{-1}{\mathrm{\rho }}^{-1}$

2. ${\mathrm{v}}^2\propto \mathrm{g\lambda \rho }$

3. ${\mathrm{v}}^2\propto \mathrm{g\lambda }$

4. ${\mathrm{v}}^2\propto {\mathrm{g}}^{-1}{\mathrm{\lambda }}^{-3}$