Question

An airplane is moving through the air at a velocity \(v=200, \mathrm{~m} / \mathrm{s}\) Streamlines just over the top of the wing are compressed to \(80.0 \%\) of their original cross-sectional area, and those under the wing are not compressed at all. a) Determine the velocity of the air just over the wing. b) Find the difference in the pressure of the air just over the wing, \(P\), and that of the air under the wing, \(P\). The density of the air is \(1.30 \mathrm{~kg} / \mathrm{m}^{3}\). c) Find the net upward force on both wings due to the pressure difference, if the area of the wings is \(40.0 \mathrm{~m}^{2}\).

Short answer

Answer: The velocity of the air just over the wing is 250 m/s, the pressure difference of the air just over and under the wing is 14625 Pa, and the net upward force on both wings due to the pressure difference is 585000 N.

Step-by-step solution

Apply Bernoulli's Equation

To find the velocity of the air just over the wing, we apply Bernoulli's Principle, which states that the total energy per unit volume is constant along a streamline. We can summarize this as: \(p + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}\) Since there is no elevation difference involved and we are neglecting gravitational effects, we can omit the term \(\rho g h\) from our equation. So, our simplified equation will be: \(p + \frac{1}{2}\rho v^2 = \text{constant}\)

Find the velocity over the wing

For this step, we will use the given information that the cross-sectional area of the streamlines just over the wing is compressed to \(80.0 \%\) of their original area. The mass flow rate must be conserved, so we use the equation: \(\rho A v = \text{constant}\) For the original conditions, we have: \(\rho Av_1 = \rho A_\text{reduced} v_2\) With \(v_1 = 200 \,\text{m/s}\) (velocity of the airplane) and \(A_\text{reduced} = 0.8A\) (compressed - 80% of the original area), we can isolate the velocity just over the wing, \(v_2\): \(v_2 = \frac{Av_1}{0.8A} = \frac{200}{0.8} \,\text{m/s}\)

Calculate the air velocity just over the wing

Now we can calculate the air velocity just over the wing, \(v_2\): \(v_2 = \frac{200}{0.8} = 250 \,\text{m/s}\)

Determine the pressure difference

In order to find the difference in pressure between the air just over the wing (compressed area) and the air under the wing (uncompressed), we apply Bernoulli's Equation for both conditions: 1. The uncompressed case: \(p_1 + \frac{1}{2}\rho v_1^2 = \text{constant}\) 2. The compressed case: \(p_2 + \frac{1}{2}\rho v_2^2 = \text{constant}\) To find the pressure difference, we need to subtract the uncompressed case from the compressed case: \(\Delta p = p_2 - p_1 = \frac{1}{2}\rho(v_2^2 - v_1^2)\) Using the given air density, \(\rho = 1.30\,\text{kg/m}^3\), and the calculated air velocity just over the wing, \(v_2 = 250\,\text{m/s}\), we can find the pressure difference \(\Delta p\): \(\Delta p = \frac{1}{2}(1.30)(250^2 - 200^2) \,\text{Pa}\)

Calculate the pressure difference

Now we can calculate the pressure difference, \(\Delta p\): $\Delta p = \frac{1}{2}(1.30)(62500 - 40000) = 0.65 (22500) = 14625 \,\text{Pa}$

Find the net upward force on both wings

To find the net upward force due to the pressure difference, we need to multiply the pressure difference by the area of both wings. Let \(F_\text{net}\) be the net upward force, \(A_\text{wings}\) be the area of both wings, and \(\Delta p\) the pressure difference: \(F_\text{net} = \Delta p \cdot A_\text{wings}\) Using the given area of the wings, \(A_\text{wings} = 40.0\,\text{m}^2\), and the calculated pressure difference \(\Delta p = 14625\,\text{Pa}\), we can find the net upward force: \(F_\text{net} = 14625 \cdot 40.0 \,\text{N}\)

Calculate the net upward force on both wings

Now we can calculate the net upward force on both wings, \(F_\text{net}\): $F_\text{net} = 14625 \cdot 40.0 = 585000 \,\text{N}$ The results are: a) The velocity of the air just over the wing is \(250\,\text{m/s}\). b) The difference in pressure of the air just over and under the wing is \(14625\,\text{Pa}\). c) The net upward force on both wings due to the pressure difference is \(585000\,\text{N}\).