Convert a pressure 76 cm of mercury (Hg) into N/m2 using the method of dimensions.
Answers
ρ = density of mercury = 13.6 *10^3 Kg/m^3
g = 9.8 m/s^2
h = 0.76 m
P = 13.6 x 10^3 x 9.8 x 0.76
= 1.013 x 10^5 N/m^2
Given: the pressure,p = 76 cm of Hg
To Find: the pressure in N/m², P
Solution:
To calculate P, the concept used:
- Pascal is the SI unit of pressure.
- It can also be expressed in dyne / cm².
- Pascal can also be written as N/m².
Applying the above concept:
76 cm of Hg = 76 x 13.6 x 980 dyne / cm²
In terms of dimensions - [M L⁻¹ T⁻²] ⇒1
p [M₁ᵃ L₁ˣ T₁ⁿ] = P [M₂ᵃ L₂ˣ T₂ⁿ]
p = P = [M₁ / M₂]ᵃ [ L₁/ L₂]ˣ [T₁/T₂]ⁿ ⇒ 2
On comparing equations 1 and2:
p = P = [M₁ / M₂]¹ [ L₁/ L₂]⁻¹ [T₁/T₂]⁻²
P = 76 x 13.6 x 980 x [M₁ / M₂]¹ [ L₁/ L₂]⁻¹ [T₁/T₂]⁻² ⇒ 3
here, M₁ = 1 g M₂ = 1 kg
L₁ = 1 cm L₂ = 1 m
T₁ = 1 s T₂ = 1 s
Putting all these values in equation 3:
P = 76 x 13.6 x 980 x [1g / 1kg]¹ [ 1 cm/ 1 m]⁻¹ [ 1s/ 1s]⁻²
= 76 x 13.6 x 980 x [1 / 1000]¹ [ 1 /100]⁻¹ [ 1 / 1]⁻²
= 76 x 13.6 x 980 x [10⁻³]¹ [ 10⁻²]⁻¹ [ 1 / 1]⁻²
= 76 x 13.6 x 980 x 10⁻³ x 10²
= 74480 x 13.6 x 10⁻¹
= 1012928 x 10⁻¹
= 1.01 x 10⁵
P = 1.01 x 10⁵ N / m²