One of the proposed mechanisms for the lift of an aircraft is the pressure difference that develops due to the aerodynamic shape of the wings. Higher pressure develops bellow the airplane which lifts it up. Suppose you have a semi-circular wings with radius 1 m. If air molecules need 1 s to go from one edge of the wing to another and meet again, what is the pressure difference in N/m^2 that pushes the wing up?
Density of air is 1.23 kg/m^3
3.60980
hey whoever is doing this please stop this don't cheat and no one please give the answers.and by the way the answer is wrong
1.23
To calculate the pressure difference that pushes the wing up, you need to determine the velocity at which the air molecules travel from one edge of the wing to another. The distance between the two edges of the semi-circular wing is the diameter, which is twice the radius, so it is 2 meters.
Given that the air molecules take 1 second to travel from one edge of the wing to another and meet again, you can calculate the velocity using the formula:
velocity = distance / time
velocity = 2 meters / 1 second = 2 m/s
Now, you can use Bernoulli's principle, which states that the total energy of a fluid flowing along a streamline is conserved. In this case, you can compare the pressure at the top of the wing (lower velocity) with the pressure at the bottom of the wing (higher velocity).
The equation for Bernoulli's principle is:
P + (1/2) * ρ * v^2 = constant
where P is the pressure, ρ is the density of the air, and v is the velocity.
Let's consider the top of the wing first. Since the velocity is lower, we can rearrange the equation:
P_top = constant - (1/2) * ρ * v_top^2
Now, let's consider the bottom of the wing. The velocity is higher, so the equation becomes:
P_bottom = constant - (1/2) * ρ * v_bottom^2
Since the distance the air molecule travels from the top to the bottom of the wing is 2 meters, we can equate the time of 1 second to the distance divided by the velocity:
1 second = 2 meters / v_bottom
Rearranging this equation, we can solve for v_bottom:
v_bottom = 2 meters / 1 second = 2 m/s
Now, substitute the values into the equation for P_bottom:
P_bottom = constant - (1/2) * ρ * (2 m/s)^2
P_bottom = constant - (1/2) * 1.23 kg/m^3 * 4 m^2/s^2
Now, plug in the values for the known variables:
P_top = constant - 4.92 N/m^2
Now, since the pressure difference that pushes the wing up is the pressure at the bottom of the wing minus the pressure at the top of the wing, we can calculate it:
Pressure difference = P_bottom - P_top
Pressure difference = (constant - 4.92 N/m^2) - constant
The constant terms will cancel out, and we are left with:
Pressure difference = -4.92 N/m^2
Therefore, the pressure difference that pushes the wing up is -4.92 N/m^2. The negative sign indicates that the pressure at the bottom is lower than the pressure at the top, creating an upward force on the wing.