Question

Write the dimensional formulae of following quantities:

Gravitational constant; Young's modulus of elasticity, modulus of rigidity; gas constant; Boltzmann constant; coefficient of thermal conductivity; capacity; electric resistance; electric potential.

Answer 1

Gravitational constant G = \(\frac{Fr^2}{m_1m_2}\)

= \(\frac{M^1L^1T^{-2}L^2}{M^1M^1}\) = [M^-1L³T^-2]

Young's modulus of elasticity:

Modulus of elasticity = \(\frac{\text{Stress}}{\text{Strain}}\)

= \(\frac{M^1L^{-1}T^{-2}}{M^0L^0T^0}\) = [M¹L^-1T^-2]

Thus dimensions of pressure, stress and modulus of elasticity are the same i.e., [M¹L^-1T^-2].

Dimensions of all coefficients of elasticity e.g, Young’s modulus of elasticity, Bulk modulus of elasticity and modulus of Rigidity are the same i.e., [M¹L^-1T^-2].

modulus of rigidity:

By definition

Modulus of rigidity = \(\frac{\text {Shear stress}}{\text {Shear strain}}\)

Dimensional formula of Modulus of rigidity are \(\frac{MLT^{-2}}{L^2}\)= ML^-1T^-2

gas constant (R)

∵ Gas equation for one mole PV = RT ⇒ R = \(\frac{PV}{T}\)

∴ R = \(\frac{M^1L^{-1}T^{-2}L^3}{\theta^1}\) = [M¹L²T^-2θ^-1]

electric potential

Electric potential is defined as the work done to move per unit positive charge from one point to another point.
The SI unit of electric potential is volt (V).

V = \(\frac{W}{Q}\)

Hence dimension is \(\frac{ML^2T^{-2}}{AT}\)

= [ML²T^-3A^-1]