An air plane has a mass of (1.7x10 to power 6) kg, and the air flow past the lower surface of the wings at 95m/s. If the both wings have a surface area of 1200cubic metre, how fast must the air flow over the upper surface of the wings if the plane is to stay in the air?
To answer this question, we need to apply Bernoulli's principle, which states that as the speed of a fluid (in this case, air) increases, the pressure it exerts decreases.
In this scenario, the airplane's weight is balanced by the lift force generated by the airflow over the wings. The lift force is a result of the pressure difference between the upper and lower surfaces of the wings.
First, let's calculate the lift force using the formula:
Lift Force = Pressure Difference * Surface Area
Since the pressure on the lower surface of the wings is higher and the pressure on the upper surface is lower, we can calculate the pressure difference by subtracting the pressure on the upper surface from the pressure on the lower surface.
Now we can calculate the pressure difference:
Pressure Difference = ρ * g * Δh
Where:
ρ is the density of air (approximately 1.2 kg/m³)
g is the acceleration due to gravity (approximately 9.8 m/s²)
Δh is the height difference between the upper and lower surfaces of the wings (which we need to find)
From Bernoulli's principle, we know that the pressure difference is also related to the speed difference of the airflow over the wings. The speed on the lower surface is given as 95 m/s, and we need to find the speed on the upper surface (let's call it v).
Pressure Difference = 0.5 * ρ * (95² - v²)
Now we can equate the two expressions for the pressure difference:
0.5 * ρ * (95² - v²) = ρ * g * Δh
Simplifying the equation:
0.5 * (95² - v²) = g * Δh
Now we can solve for Δh:
Δh = 0.5 * (95² - v²) / g
Finally, we have the equation that relates the speed on the upper surface (v) to the height difference (Δh):
v = sqrt(95² - 2 * g * Δh)
Plugging in the known values:
v = sqrt(95² - 2 * 9.8 * Δh)
Substituting the given values for mass (1.7x10⁶ kg) and surface area (1200 m²) into the weight equation:
Weight = mass * g
Weight = 1.7x10⁶ kg * 9.8 m/s²
Now, let's assume the wings are level. This means the lift force must equal the weight for the plane to stay in the air:
Lift Force = Weight
Now we can equate these two equations:
Pressure Difference * Surface Area = mass * g
Let's substitute in the formulas we derived earlier:
0.5 * (95² - v²) * 1200 = 1.7x10⁶ kg * 9.8 m/s²
Simplifying:
(95² - v²) * 1200 = 1.7x10⁶ * 9.8
Dividing both sides by 1200:
95² - v² = (1.7x10⁶ * 9.8) / 1200
Now we can solve for v:
v² = 95² - (1.7x10⁶ * 9.8) / 1200
v = sqrt(95² - (1.7x10⁶ * 9.8) / 1200)
Simplifying further:
v ≈ sqrt(9025 - 13234)
v ≈ sqrt(-4211)
Since the result is a negative square root, it indicates that the speed required for the airplane to stay in the air is not possible under the given conditions.