An airplane is moving through the air at a velocity v = 227 m/s. Streamlines just over the top of the wing are compressed to 80% of their original 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 between the air just over the wing, P, and that under the wing, P'. (The density of air is 1.3 kg/m3.)
(c) Find the net upward force on both wings due to the pressure difference, if the area of the wing is 44 m2 and the density of the air is 1.3 kg/m3.
Do you know how to solve it?
Oh boy, physics questions! Hang on tight, I'll try to make this a bit more fun for you. Ready for some humorous equations?
(a) To determine the velocity of the air just over the wing, we can use the principle of continuity. You see, the principle of continuity states that the mass flow rate is constant in an incompressible fluid. So here's the equation for velocity (v) and area (A):
v1 * A1 = v2 * A2
In this case, the velocity just under the wing is the same as the airplane's velocity (v = 227 m/s) and the area just under the wing is the same as the original area (A1). The velocity just over the wing is what we need to find (v2). But hold on, the area just over the wing is compressed to 80% of its original area (0.8A1). That means we have:
227 m/s * A1 = v2 * (0.8A1)
Now, we can cancel out A1 because math is cool like that:
227 m/s = v2 * 0.8
And solving for v2, we get:
v2 = 227 m/s / 0.8
So, the velocity of the air just over the wing is v2 = 283.75 m/s.
(b) Now, let's calculate the pressure difference between the air just over the wing (P) and that under the wing (P'). We can use Bernoulli's equation here, which tells us that the total pressure is the sum of dynamic pressure and static pressure. But because we love humor, we can also call it "the equation of clown physics":
P + (1/2) * ρ * v^2 = P' + (1/2) * ρ * v2^2
In this equation, ρ represents the density of air. So let's plug in the values we know:
P + (1/2) * 1.3 kg/m^3 * (227 m/s)^2 = P' + (1/2) * 1.3 kg/m^3 * (283.75 m/s)^2
Now, let's simplify:
P + 0.5 * 1.3 kg/m^3 * 51529 m^2/s^2 = P' + 0.5 * 1.3 kg/m^3 * 80531.56 m^2/s^2
Oh boy, that's a lot of numbers! Now we subtract P' from both sides:
P - P' = 0.5 * 1.3 kg/m^3 * 80531.56 m^2/s^2 - 0.5 * 1.3 kg/m^3 * 51529 m^2/s^2
And now, let me have a little fun with the calculations:
P - P' = 0.5 * 1.3 kg/m^3 * (80531.56 m^2/s^2 - 51529 m^2/s^2)
P - P' = 0.5 * 1.3 kg/m^3 * (29002.56 m^2/s^2)
P - P' = 18878.336 kg/m^3 * m^2/s^2
So, the difference in pressure between the air just over the wing (P) and that under the wing (P') is approximately 18878.336 kg/m^3 * m^2/s^2. Wow, that's some serious clown physics right there!
(c) Finally, let's find the net upward force on both wings due to the pressure difference. The net upward force (F) can be calculated using the equation:
F = (P - P') * A
Where A represents the area of the wing. So let's plug in the values we know:
F = 18878.336 kg/m^3 * m^2/s^2 * 44 m^2
Now, it's time for clown magic:
F = 18878.336 kg/m^3 * m^2/s^2 * 44 m^2
F = 830211.584 kg/m/s^2 * m^2
F = 830211.584 N
So, the net upward force on both wings due to the pressure difference is approximately 830,211.584 N. That's quite a bit of clown power pushing those wings up!
I hope these clown physics equations made the journey through aerodynamics a little more entertaining for you! Safe travels with your airplane!
To solve this problem, we can use the principle of continuity equation and Bernoulli's equation. Let's go step by step.
(a) To determine the velocity of the air just over the wing, we can use the principle of continuity equation, which states that the mass flow rate remains constant for an incompressible fluid. The equation is given as:
A1v1 = A2v2
where A1 and A2 are the respective cross-sectional areas of the streamlines just over and under the wing, and v1 and v2 are the respective velocities of the air just over and under the wing.
Let's assume the area of the streamlines under the wing (A2) is equal to their original area, and the area of the streamlines over the wing (A1) is 80% of their original area. We can calculate the velocity just over the wing (v1) as:
A2v2 = A1v1
v1 = (A2v2) / A1
Since the area of streamlines under the wing (A2) is not compressed, its area remains the same. The original area can be calculated as:
A2 = (Original area of streamlines)
The original area of the streamlines over the wing can be calculated as:
A1 = 0.8 * (Original area of streamlines)
Now, substitute the values into the equation and calculate:
v1 = (A2v2) / A1
Note: The original area of the streamlines is not mentioned in the given information, so you'll need to provide that value to proceed with the calculation.
(b) To find the difference in pressure between the air just over the wing (P) and that under the wing (P'), we can use Bernoulli's equation, which states that the sum of pressure energy, kinetic energy, and potential energy is constant along a streamline. The equation is given as:
P + (1/2)ρv^2 + ρgh = constant
where P is the pressure, v is the velocity, ρ is the density of air, g is the acceleration due to gravity, and h is the height.
Substituting the values for the air just over and under the wing into the equation, we can find the pressure difference:
P - P' = (1/2)ρ(v1^2 - v2^2)
Now, substitute the values into the equation and calculate:
P - P' = (1/2)ρ(v1^2 - v2^2)
Remember to use the density of air (ρ) provided in the given information.
(c) To find the net upward force on both wings due to the pressure difference, we can use the formula:
Force = Pressure difference x Area
The pressure difference (P - P') can be obtained from part (b) above. Now, substitute the values into the formula and calculate:
Force = (P - P') x Area
Remember to use the area of the wing provided in the given information.
To answer these questions, we need to apply Bernoulli's principle, which states that the total energy per unit volume (or the sum of kinetic energy, potential energy, and pressure energy) remains constant along a streamline in an ideal fluid flow.
(a) To determine the velocity of the air just over the wing, we can use Bernoulli's principle. Let's assume the velocity of the air just under the wing is V1 and the velocity just over the wing is V2.
According to Bernoulli's principle, we have:
P1 + 1/2 * ρ * V1^2 = P2 + 1/2 * ρ * V2^2
Where P1 and P2 are the pressures just under and over the wing, respectively, ρ is the density of air, and V1 and V2 are the velocities just under and over the wing, respectively.
Since the problem states that streamlines under the wing are not compressed (implying no change in velocity), V1 remains equal to the velocity of the airplane, v = 227 m/s.
Let's assume P2 is the pressure just over the wing. We also know that streamlines just over the wing are compressed to 80% of their original area. The area reduction corresponds to an increase in velocity, so V2 = (1/√0.8) * v.
Substituting these values into Bernoulli's equation, we have:
P1 + 1/2 * ρ * v^2 = P2 + 1/2 * ρ * ((1/√0.8) * v)^2
Simplifying the equation gives:
P2 = P1 - 0.2 * ρ * v^2
Therefore, the velocity of the air just over the wing is (1/√0.8) * v, and the pressure just over the wing is P2 = P1 - 0.2 * ρ * v^2.
(b) To find the difference in pressure between the air just over the wing (P2) and the air just under the wing (P1), we use the equation obtained in part (a):
P - P' = P2 - P1 = -0.2 * ρ * v^2
Substituting the values, we have:
P - P' = -0.2 * 1.3 kg/m^3 * (227 m/s)^2
Solve the equation to find the pressure difference.
(c) To find the net upward force on both wings due to the pressure difference, we need to consider the difference in pressure acting on both the upper and lower surfaces of the wing.
The net upward force, F, is equal to the pressure difference multiplied by the wing area:
F = (P - P') * A
Substitute the values for the pressure difference (P - P') and the wing area A, and then calculate the net upward force, F.