Air of density `1.2kg m ^(-3)` is blowing across the horizontal wings of an aeroplane in such a way that its speeds above and below the wings are `150 ms^(-1) and 100 ms ^(-1),` respectively. The pressure difference between the upper and lower sides of the wings, is :

Options

A. `60 Nm ^(-2)`
B. `180 Nm ^(-2)`
C. `7500 Nm ^(-2)`
D. `12500 Nm ^(-2)`

Explanation

To find the pressure difference between the upper and lower sides of the wings of the airplane, we can use Bernoulli's principle, which states that for an incompressible fluid, the total mechanical energy along a streamline is constant. The equation can be expressed as:

\[ P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant} \]

Where:
- \( P \) is the pressure,
- \( \rho \) is the density of the fluid,
- \( v \) is the velocity of the fluid,
- \( g \) is the acceleration due to gravity (which can be ignored for horizontal wings),
- \( h \) is the height (which can also be ignored for horizontal wings).

For the air above the wings (1) and below the wings (2), we can write:

\[ P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \]

Where:
- \( P_1 \) is the pressure above the wings,
- \( P_2 \) is the pressure below the wings,
- \( v_1 = 150 \, \text{m/s} \) (speed above the wings),
- \( v_2 = 100 \, \text{m/s} \) (speed below the wings),
- \( \rho = 1.2 \, \text{kg/m}^3 \) (density of air).

We can rearrange the equation to find the pressure difference \( P_2 - P_1 \):

\[ P_2 - P_1 = \frac{1}{2} \rho v_1^2 - \frac{1}{2} \rho v_2^2 \]

Now, substituting the values:

1. Calculate \( \frac{1}{2} \rho v_1^2 \):
\[ \frac{1}{2} \times 1.2 \, \text{kg/m}^3 \times (150 \, \text{m/s})^2 = \frac{1}{2} \times 1.2 \times 22500 = 13500 \, \text{Pa} \]

2. Calculate \( \frac{1}{2} \rho v_2^2 \):
\[ \frac{1}{2} \times 1.2 \, \text{kg/m}^3 \times (100 \, \text{m/s})^2 = \frac{1}{2} \times 1.2 \times 10000 = 6000 \, \text{Pa} \]

3. Now, calculate the pressure difference:
\[ P_2 - P_1 = 13500 \, \text{Pa} - 6000 \, \text{Pa} = 7500 \, \text{Pa} \]

Thus, the pressure difference between the upper and lower sides of the wings is \( 7500 \, \text{Pa} \).