The force, F, acting on the wings of an aircraft moving through the air of...
The force, F, acting on the wings of an aircraft moving through the air of velocity, v, and density, ρ, is given by the equation F = \(kv^xρ^yA^z\), where k is a dimensionless constant and A is the surface area of the wings of the aircraft. Use dimensional analysis to determine the values of x, y, and z.
Explanation
F = \(kv^xρ^yA^z\)
For the left-hand side:
F = mass × acceleration = \(MLT^{-2}\)
For the right-hand side:
v = \(LT^{-1}\), ρ = \(ML^{-3}\) and A = \(L^2\)
So,
\(MLT^{-2} = k(LT^{-1})^x(ML^{-3})^y(L^2)^z\)
Since k is dimensionless,
\(MLT^{-2} = (LT^{-1})^x(ML^{-3})^y(L^2)^z\)
\(MLT^{-2} = L^x × T^{(-1)x} \times M^y × L^{(-3)y} \times L^{(2)z}\)
\(MLT^{-2} = L^x × T^{-x} \times M^y \times L^{-3y} \times L^{2z}\)
\(MLT^{-2} = L^{(x - 3y + 2z)} \times T^{-x} \times M^y\)
\(M^1L^1T^{-2}= L^{(x - 3y + 2z)} \times T^{-x} \times M^{y}\)
Comparing the powers:
For M,
y = 1
For L,
x - 3y + 2z = 1 ---- (i)
For T,
-x = -2
∴ x = 2
Substitute (2) for x and (1) for y in equation (i)
⇒ 2 - 3(1) + 2z = 1
⇒ 2 - 3 + 2z = 1
⇒ -1 + 2z = 1
⇒ 2z = 1 + 1
⇒ 2z = 2
∴ z = 1
Hence, x = 2, y = 1 and z = 1.