An aeroplane has a mass of 15 tonnes. At cruising speed, air flows over the top of its wings at
150 m/s and under the wings at 120 m/s. Calculate:
9.1 the difference in pressure between the top and bottom of the wings;
9.2 the surface area of the underside of the wings.
To calculate the difference in pressure between the top and bottom of the wings, we can use Bernoulli's equation, which relates the pressure and velocity of a fluid. The equation in this case is:
P + (1/2)ρv^2 = constant
where P is the pressure, ρ is the density of the fluid (air in this case), and v is the velocity of the fluid.
9.1 To find the difference in pressure, we need to calculate the pressure at the top and bottom of the wings separately and then find the difference.
Using Bernoulli's equation, let's start with the top of the wings:
P1 + (1/2)ρv1^2 = constant
where P1 is the pressure at the top of the wings, and v1 is the velocity of air flow over the top of the wings (150 m/s in this case).
Now, let's move to the bottom of the wings:
P2 + (1/2)ρv2^2 = constant
where P2 is the pressure at the bottom of the wings, and v2 is the velocity of air flow under the wings (120 m/s in this case).
Considering that the constant in both equations is the same, we can set them equal to each other:
P1 + (1/2)ρv1^2 = P2 + (1/2)ρv2^2
Now, let's substitute the given values and solve for the pressure difference (P1 - P2):
(P1 - P2) = (1/2)ρ(v2^2 - v1^2)
To find the density of air, we can use the approximation of 1.225 kg/m^3.
Now, plug in the values and calculate:
(P1 - P2) = (1/2)(1.225 kg/m^3)((120 m/s)^2 - (150 m/s)^2)
(P1 - P2) = (1/2)(1.225 kg/m^3)(14,400 m^2/s^2 - 22,500 m^2/s^2)
(P1 - P2) = (1/2)(1.225 kg/m^3)(-8,100 m^2/s^2)
(P1 - P2) = -4,987.5 N/m^2
Therefore, the difference in pressure between the top and bottom of the wings is approximately -4,987.5 N/m^2. The negative sign indicates that the pressure is lower on the top of the wings compared to the bottom.
9.2 To calculate the surface area of the underside of the wings, we need to know the mass of the airplane (15 tonnes) and its cruising speed (assumed to be constant). Given these factors, we can use the lift equation:
L = (1/2)ρv^2AC
where L is the lift force, ρ is the density of air, v is the velocity of the airplane, A is the surface area of the wing, and C is the lift coefficient.
Since we are interested in finding the surface area of the underside of the wings (A), we can rearrange the equation:
A = (2L) / (ρv^2C)
Now, we need to know the lift force (L) and the lift coefficient (C). These values are specific to the airplane and can be obtained from aerodynamic data or empirical formulas.
Once you have the values of L and C, you can substitute them into the formula along with the known values of ρ (density of air), v (cruising speed), and solve for A, the surface area of the underside of the wings.