An aeroplane of mass 5000 kg is flying at an altitude of 3 km. If the area of wings is 50 m^2 and pressure at the lower surface of wings is 60000 PA,the pressure on the upper surface of wings is
Apply P1-P2=½P(V1²-V2²)
The answer is 59×10³.
Sashank 9th Class....
Can you give a bit detailed response
Thanku
To find the pressure on the upper surface of the wings, we need to use Bernoulli's principle. Bernoulli's principle states that in a fluid flow, an increase in velocity results in a decrease in pressure.
Given:
- Mass of the airplane(m) = 5000 kg
- Altitude(h) = 3 km = 3000 m
- Area of wings(A) = 50 m^2
- Pressure at the lower surface of wings(P1) = 60000 Pa
To find the pressure on the upper surface of the wings, we'll use the equation:
P2 + (1/2)ρv^2 + ρgh = P1 + (1/2)ρv1^2 + ρgh1
Where:
- P2 is the pressure on the upper surface of the wings (what we want to find)
- ρ is the density of air
- v is the velocity of air above and below the wings
- g is the acceleration due to gravity
- h is the altitude above ground level
The key point here is that the velocity of the airflow above the wings (v) is higher than the velocity of the airflow below the wings (v1).
Let's assume that the velocities above and below the wings are the same. Hence, v = v1.
Plugging in the given values:
P2 + (1/2)ρv^2 + ρgh = P1 + (1/2)ρv1^2 + ρgh1
Since v = v1:
P2 + (1/2)ρv^2 + ρgh = P1 + (1/2)ρv^2 + ρgh1
Since the velocities are canceled out from both sides of the equation, we are left with:
P2 + ρgh = P1 + ρgh1
Now, let's solve for P2:
P2 = P1 + ρgh - ρgh1
To calculate P2, we need to know the density of air (ρ). Assuming standard atmospheric conditions, we can use the value of 1.225 kg/m^3 for the density of air.
Plugging in the rest of the values:
P2 = 60000 + 1.225 * 9.8 * 3000 - 1.225 * 9.8 * 0
Calculating:
P2 ≈ 60000 + 35805000 - 0
P2 ≈ 35865000 Pa
Therefore, the pressure on the upper surface of the wings is approximately 35,865,000 Pa.
To find the pressure on the upper surface of the wings, we need to understand the concept of Bernoulli's principle.
Bernoulli's principle states that as the speed of a fluid (or air) increases, its pressure decreases. This principle is commonly used to explain the lift generated by the wings of an airplane.
To calculate the pressure on the upper surface of the wings, we can use Bernoulli's principle in conjunction with the equation for the lift force.
The lift force on an airplane wing can be calculated using the following formula:
L = 1/2 * rho * A * V^2 * Cl
Where:
L is the lift force
rho is the density of the air
A is the area of the wing
V is the velocity of the air
Cl is the coefficient of lift
Given:
- Mass of the airplane = 5000 kg
- Altitude = 3 km
- Area of the wings = 50 m^2
- Pressure at the lower surface of the wings = 60000 Pa
First, we need to calculate the density of the air at the given altitude. The density of air decreases with increasing altitude. We can use the following approximation:
rho = rho0 * (1 - (lapse_rate * altitude / temperature0))^((gravity * molar_mass) / (R * lapse_rate))
Where:
rho0 is the density of air at sea level (approximately 1.225 kg/m^3)
lapse_rate is the temperature lapse rate (approximately -0.0065 K/m)
altitude is the given altitude (3 km)
temperature0 is the temperature at sea level (approximately 288.15 K)
gravity is the acceleration due to gravity (approximately 9.8 m/s^2)
molar_mass is the molar mass of air (approximately 0.02896 kg/mol)
R is the Universal gas constant (approximately 8.314 J/(mol·K))
Using these values in the formula, we can calculate the density of the air at 3 km altitude.
Next, we need to find the velocity of the air over the wings. This requires taking into account the lift force and the mass of the airplane.
Given the lift force equation, we can rearrange it to solve for V:
V = sqrt((2 * L) / (rho * A * Cl))
Plug in the values for the lift force, density, area, and any appropriate lift coefficient to obtain the velocity of the air.
Once we have the velocity of the air, we can apply Bernoulli's principle to find the pressure on the upper surface of the wings. The equation is:
P1 + 0.5 * rho * V1^2 = P2 + 0.5 * rho * V2^2
Where:
P1 is the pressure on the lower surface of the wings
P2 is the pressure on the upper surface of the wings
rho is the density of the air
V1 is the velocity of air below the wings
V2 is the velocity of air above the wings
Plugging in the values for the density, velocities, and pressure on the lower surface, we can solve for the pressure on the upper surface of the wings.
Please note that this calculation assumes steady, incompressible, and inviscid flow over the wings. It provides a basic estimation and may not consider all factors affecting the pressure distribution over the wing surface in real-world scenarios.