What is the net upward force on an airplane wing of area \\\\(20 m^2\\\\) if the speed of airflow is 300 m/s across the top of the wing and 280 m/s across the bottom? Density of air is \\\\(1.29 kgm^{-3}\\\\).
what's with all the \\\\\\\\\\\\\\\\\\\\\\\\\\\?
To find the net upward force on the airplane wing, we can use Bernoulli's equation, which relates the pressure difference between the top and bottom surfaces of the wing to the speed of airflow across those surfaces.
Bernoulli's equation states that the total pressure at any point in a fluid can be divided into three components: static pressure, dynamic pressure, and potential energy pressure.
Let's break down the equation step by step:
Step 1: Calculate the dynamic pressure on the top and bottom surfaces of the wing:
Dynamic pressure (Pd) = 0.5 * density of air (ρ) * velocity squared (v^2)
On the top surface:
Pd_top = 0.5 * ρ * v_top^2
On the bottom surface:
Pd_bottom = 0.5 * ρ * v_bottom^2
Step 2: Calculate the net pressure difference between the top and bottom surfaces:
Pressure difference (ΔP) = P_top - P_bottom
Static pressure (P_static) cancels out in this scenario since the wing is in motion and we're interested in the net upward force.
Step 3: Calculate the net upward force:
Net upward force (F_net) = ΔP * wing area (A)
F_net = ΔP * A
Now we can plug in the given values to find the net upward force on the wing:
Given:
Area of the wing (A) = 20 m^2
Velocity of airflow across the top surface (v_top) = 300 m/s
Velocity of airflow across the bottom surface (v_bottom) = 280 m/s
Density of air (ρ) = 1.29 kg/m^3
Step 1:
Pd_top = 0.5 * 1.29 * 300^2
Pd_bottom = 0.5 * 1.29 * 280^2
Step 2:
ΔP = P_top - P_bottom
Step 3:
F_net = ΔP * A
Now you can substitute the calculated values from the above steps into the equation to determine the net upward force on the wing.
To calculate the net upward force on the airplane wing, we need to determine the pressure difference between the top and bottom surfaces of the wing and then multiply it by the wing area.
Step 1: Calculate the pressure difference
The pressure difference is equal to the difference in dynamic pressure between the top and bottom surfaces of the wing.
Dynamic pressure, \(q = \frac{1}{2} \rho v^2\)
Where:
\(q\) is the dynamic pressure,
\(\rho\) is the air density,
\(v\) is the velocity of the airflow.
For the top surface, let's calculate \(q_1\):
\(q_1 = \frac{1}{2} \times 1.29 \, \text{kg/m}^3 \times (300 \, \text{m/s})^2\)
For the bottom surface, let's calculate \(q_2\):
\(q_2 = \frac{1}{2} \times 1.29 \, \text{kg/m}^3 \times (280 \, \text{m/s})^2\)
Step 2: Calculate the pressure difference
The pressure difference (\(\Delta p\)) is the difference in dynamic pressure (\(q_1 - q_2\)).
\(\Delta p = q_1 - q_2\)
Step 3: Calculate the net upward force
Finally, we can calculate the net upward force (\(F\)) using the formula:
\(F = \Delta p \times A\)
Where:
\(F\) is the net upward force,
\(\Delta p\) is the pressure difference,
\(A\) is the wing area.
Substituting the values, \(A = 20 \, \text{m}^2\):
\(F = \Delta p \times 20 \, \text{m}^2\)
Now you can substitute the calculated values of \(\Delta p\) and calculate the net upward force on the airplane wing.