# Dimensionally homogeneous equations - Fluid Mechanics

Q.  Which of the following equations is not dimensionally homogeneous?
Consider standard symbols for quantities.

- Published on 05 Aug 15

a. (Force) F = m x a
b. (Head Loss due to friction) hf = (f L V2) / (2 g d)
c. (Torque) T = F x Distance
d. None of the above

#### Discussion

• Pankaj   -Posted on 05 Oct 15
The equation in which dimensions on left hand side are equal to the dimensions on right hand side is called as dimensionally homogeneous equation.

Taking given equations one by one,

1. (Force) F = m x a

Dimensions of L.H.S.

F = [M L T –2]

Dimensions of R.H.S.

m x a = [M] [L T – 2]

= [M L T –2]

L.H.S = R.H.S.

The equation 'F = m x a' is dimensionally homogeneous

2. (Head Loss due to friction) hf = (f L V2) / (2 g d)

Dimensions of L.H.S.

hf = L

Dimensions of R.H.S.

(f L V2) / (2 g d) = [1 x L x (L/T) 2] / [2 x (L/T) 2 x L]

= [L3 / T2] [T2 / L2]

= L

L.H.S. = R.H.S.

The equation 'hf = (f L V2) / (2 g d)' is dimensionally homogeneous

3. (Torque) T = F x Distance

Dimensions of L.H.S.

Torque = [M L 2 T – 2]

Dimensions of R.H.S.

F x distance = Mass x Acceleration x Distance

= [M] [L / T 2] [L]

[M L 2 T – 2]

L.H.S. = R.H.S.

The equation '(Torque) T = F x Distance ' is dimensionally homogeneous

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