Air is streaming past a horizontal airplane wing such that its speed is `90 ms^(-1)` at the lower surface and `120 ms^(-1)` over the upper surface. if the wing is `10 m` long and has an average width of `2 m`, the difference of pressure on the two sides and the gross lift on the wing respectively, are (density of air `=1.3kgm^(-3)`)

Options

A. `5Pa, 900N`
B. `95Pa,900N`
C. `4095Pa,900N`
D. `4095Pa,81900N`

Explanation

To solve the problem, we will use Bernoulli's equation to find the pressure difference between the upper and lower surfaces of the airplane wing, and then calculate the gross lift force.

### Step 1: Identify the given values
- Speed of air over the lower surface, \( V_1 = 90 \, \text{m/s} \)
- Speed of air over the upper surface, \( V_2 = 120 \, \text{m/s} \)
- Density of air, \( \rho = 1.3 \, \text{kg/m}^3 \)
- Length of the wing, \( L = 10 \, \text{m} \)
- Width of the wing, \( W = 2 \, \text{m} \)

### Step 2: Calculate the pressure difference using Bernoulli's equation
According to Bernoulli's equation, the pressure difference between two points can be expressed as:

\[ P_2 - P_1 = \frac{1}{2} \rho (V_1^2 - V_2^2) \]

Where:
- \( P_1 \) is the pressure on the lower surface,
- \( P_2 \) is the pressure on the upper surface.

Substituting the known values:

\[ P_2 - P_1 = \frac{1}{2} \times 1.3 \, \text{kg/m}^3 \times (90^2 - 120^2) \]

Calculating \( 90^2 \) and \( 120^2 \):

\[ 90^2 = 8100 \quad \text{and} \quad 120^2 = 14400 \]

Now, substituting these values into the equation:

\[ P_2 - P_1 = \frac{1}{2} \times 1.3 \times (8100 - 14400) \]
\[ = \frac{1}{2} \times 1.3 \times (-6300) \]
\[ = 0.65 \times (-6300) = -4095 \, \text{Pa} \]

The negative sign indicates that the pressure is lower on the upper surface compared to the lower surface.

### Step 3: Calculate the area of the wing
The area \( A \) of the wing can be calculated as:

\[ A = L \times W = 10 \, \text{m} \times 2 \, \text{m} = 20 \, \text{m}^2 \]

### Step 4: Calculate the gross lift force
The lift force \( F \) can be calculated using the formula:

\[ F = \Delta P \times A \]

Where \( \Delta P = P_2 - P_1 = 4095 \, \text{Pa} \).

Substituting the values:

\[ F = 4095 \, \text{Pa} \times 20 \, \text{m}^2 = 81900 \, \text{N} \]

### Final Answers
- The difference of pressure on the two sides is \( 4095 \, \text{Pa} \).
- The gross lift on the wing is \( 81900 \, \text{N} \).

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