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Step-by-Step Solution
Step 1: Identify the Principle (Bernoulli’s Principle)
An airplane wing generates lift due to the pressure difference between its upper and lower surfaces. According to Bernoulli’s principle for airflow:
P + \frac{1}{2}\rho v^2 = \text{constant}
Here, P is the fluid (air) pressure, \rho is the air density, and v is the velocity of the airflow. A higher velocity of airflow over a surface leads to lower pressure on that surface, and a lower velocity leads to higher pressure.
Step 2: Convert Speeds to SI Units (m/s)
The given speeds are in km/h. To use Bernoulli’s equation, we convert them to m/s:
1 \,\text{km/h} = \frac{1000 \,\text{m}}{3600 \,\text{s}} = \frac{1}{3.6}\,\text{m/s}
\\
v_{\text{lower}} = 180 \times \frac{1}{3.6} = 50 \,\text{m/s}
\\
v_{\text{upper}} = 252 \times \frac{1}{3.6} = 70 \,\text{m/s}
Step 3: Calculate the Pressure Difference
Let P_{\text{lower}} be the pressure on the lower wing surface and P_{\text{upper}} that on the upper wing surface. From Bernoulli’s principle:
P_{\text{lower}} - P_{\text{upper}} = \frac{1}{2} \rho \left( v_{\text{upper}}^2 - v_{\text{lower}}^2 \right).
Substituting the values:
\\
\rho = 1 \,\text{kg}/\text{m}^3,
\quad v_{\text{upper}} = 70 \,\text{m/s},
\quad v_{\text{lower}} = 50 \,\text{m/s}.
First compute v_{\text{upper}}^2 - v_{\text{lower}}^2 :
\\
70^2 - 50^2 = 4900 - 2500 = 2400.
Therefore,
\\
P_{\text{lower}} - P_{\text{upper}}
= \frac{1}{2} \times 1 \times 2400
= 1200 \,\text{N/m}^2.
Step 4: Compute the Total Lift Force
Each wing has an area of 40 \,\text{m}^2 , so the total wing area is
\\
A_{\text{total}} = 40 + 40 = 80 \,\text{m}^2.
The lift force F_{\text{lift}} is given by:
\\
F_{\text{lift}}
= ( P_{\text{lower}} - P_{\text{upper}} ) \times A_{\text{total}}
= 1200 \times 80
= 96000 \,\text{N}.
Step 5: Relate Lift Force to Weight to Find the Mass
In level flight, the lift force balances the weight ( mg ), where m is the airplane's mass and g is the gravitational acceleration:
F_{\text{lift}} = mg
\quad\Longrightarrow\quad
96000 = m \times 10.
Therefore, the mass of the plane is:
\\
m = \frac{96000}{10} = 9600 \,\text{kg}.
Final Answer
The mass of the plane is 9600 \,\text{kg}.
