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10.27 A plane is in level flight at constant speed and each of its two wings has an area of 25 m². If the speed of the air is 180 km/h over the lower wing and 234 km/h over the upper wing surface, determine the plane’s mass. (Take air density to be 1 kg m^–3).

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Vishal Baghel | Contributor-Level 10

5 months ago

The area of the wing of the plane, A = 2 $\times 25 m^{2}$ = 50 $m^{2}$

Speed of air over the lower wing, $V_{1}$ = 180 km/h = 50 m/s

Speed of air over the upper wing, $V_{2}$ = 234 km/h = 65 m/s

Density of air, $ρ$ = 1 kg/ $m^{3}$

Let us assume, pressure over the lower wing = $P_{1}$ and pressure over the upper wing = $P_{2}$

The upward force on the plane can be obtained using Bernoulli’s equation:

On lower wing= $P_{1}$ + $\frac{1}{2} ρ V_{1}^{2}$ , On upper wing = $P_{2}$ + $\frac{1}{2} ρ V_{2}^{2}$

From equilibrium of momentum

$P_{1}$ + $\frac{1}{2} ρ V_{1}^{2}$ = $P_{2}$ + $\frac{1}{2} ρ V_{2}^{2}$

( $P_{1}$ - $P_{2}) = \frac{1}{2} ρ (V_{2}^{2} - V_{1}^{2}$ )

The net upward force = ( $P_{1}$ - $P_{2}) A$ = $\frac{1}{2} ρ (V_{2}^{2} - V_{1}^{2}$ )

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The area of the wing of the plane, A = 2 $\times 25 m^{2}$ = 50 $m^{2}$

Speed of air over the lower wing, $V_{1}$ = 180 km/h = 50 m/s

Speed of air over the upper wing, $V_{2}$ = 234 km/h = 65 m/s

Density of air, $ρ$ = 1 kg/ $m^{3}$

Let us assume, pressure over the lower wing = $P_{1}$ and pressure over the upper wing = $P_{2}$

The upward force on the plane can be obtained using Bernoulli’s equation:

On lower wing= $P_{1}$ + $\frac{1}{2} ρ V_{1}^{2}$ , On upper wing = $P_{2}$ + $\frac{1}{2} ρ V_{2}^{2}$

From equilibrium of momentum

$P_{1}$ + $\frac{1}{2} ρ V_{1}^{2}$ = $P_{2}$ + $\frac{1}{2} ρ V_{2}^{2}$

( $P_{1}$ - $P_{2}) = \frac{1}{2} ρ (V_{2}^{2} - V_{1}^{2}$ )

The net upward force = ( $P_{1}$ - $P_{2}) A$ = $\frac{1}{2} ρ (V_{2}^{2} - V_{1}^{2}$ ) $\times A$

= $\frac{1}{2} 1 (65^{2} - 50^{2}$ ) $\times 50$ = 43125 N

Using the relation F = mg, we get m = F/g

= (43125/9.8) Kg = 4400.51 kg

Hence the mass of the plane is about 4400 kg.

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10.27 A plane is in level flight at constant speed and each of its two wings has an area of 25 m². If the speed of the air is 180 km/h over the lower wing and 234 km/h over the upper wing surface, determine the plane’s mass. (Take air density to be 1 kg m^–3).

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10.27 A plane is in level flight at constant speed and each of its two wings has an area of 25 m2. If the speed of the air is 180 km/h over the lower wing and 234 km/h over the upper wing surface, determine the plane’s mass. (Take air density to be 1 kg m–3).

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10.27 A plane is in level flight at constant speed and each of its two wings has an area of 25 m². If the speed of the air is 180 km/h over the lower wing and 234 km/h over the upper wing surface, determine the plane’s mass. (Take air density to be 1 kg m^–3).