There are four forces acting on an aeroplane in flight, as shown. (a) The lift force is perpendicular to the wings. To change direction, the aeroplane ‘banks’ so that one wing is lower than the other wing. The lift force is then no longer vertical. The aeroplane flies in a horizontal circular path at a speed of 52 m/s. The diagram below shows the aeroplane banking at an angle of 20° to the horizontal. (i) Show that the radius of the circular path is about 800 m. mass of aeroplane = 1200 kg speed of aeroplane = 54 ms⁻¹ (ii) Determine the time taken by the plane to move through 90° of the circular path. speed of aeroplane = 54 ms⁻¹ (b) The wings of the aeroplane are now levelled, as shown below. The speed and the magnitude of the lift force remain the same. Explain what will happen to the vertical motion of the aeroplane.

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There are four forces acting on an aeroplane in flight, as shown - Edexcel - A-Level Physics - Question 13 - 2023 - Paper 4

Question 13

There are four forces acting on an aeroplane in flight, as shown. (a) The lift force is perpendicular to the wings. To change direction, the aeroplane ‘banks’ so t... show full transcript

Worked Solution & Example Answer:There are four forces acting on an aeroplane in flight, as shown - Edexcel - A-Level Physics - Question 13 - 2023 - Paper 4

Step 1

Show that the radius of the circular path is about 800 m.

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Answer

To determine the radius of the circular path, we first need to identify the forces acting on the aeroplane. The weight (W) of the aeroplane can be calculated using the formula:

$W = mg$

Where:

$m = 1200 ext{ kg}$ (mass of the aeroplane)
$g = 9.81 ext{ m/s}^2$ (acceleration due to gravity)

Substituting values: $W = 1200 ext{ kg} imes 9.81 ext{ m/s}^2 = 11772 ext{ N}$

The lift force (L) can be calculated using suitable trigonometry considering the angle of bank (20°). The vertical component of the lift must counteract the weight:

$L imes ext{cos}(20°) = W$

Thus, $L = \frac{W}{\cos(20°)} = \frac{11772}{\cos(20°)} \\ \approx 12572 ext{ N}$

Now using the centripetal force formula, we have: $L = \frac{mv^2}{r}$

Rearranging for radius (r): $r = \frac{mv^2}{L}$

Substituting:

$m = 1200 ext{ kg}$
$v = 54 ext{ m/s}$
$L \approx 12572 ext{ N}$

$r = \frac{1200 imes (54)^2}{12572} \approx 813.56 ext{ m}$

Thus, the radius of the circular path is approximately 800 m.

Step 2

Determine the time taken by the plane to move through 90° of the circular path.

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Answer

To calculate the time taken to travel through 90° of the circular path, we known the formula:

$t = \frac{\theta}{\omega}$

Where:

$heta = \frac{\pi}{2} \text{ radians} = 90°$
$ext{The angular velocity,} \omega = \frac{v}{r}$ where $v$ is the linear speed and $r$ is the radius calculated previously.

From the first part:

$r \approx 816 m$
$v = 54 ext{ m/s}$

Now, $\omega = \frac{54}{816} \text{ rad/s} \approx 0.066 ext{ rad/s}$

Substituting into the time formula: $t = \frac{\frac{\pi}{2}}{0.066} \approx 24 ext{ s}$

So, it takes approximately 24 seconds to travel through 90°.

Step 3

Explain what will happen to the vertical motion of the aeroplane.

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Answer

When the wings of the aeroplane are levelled, the vertical component of the lift force is equal to that of the weight. Therefore, with the lift force remaining constant while banking ceases:

The resultant upwards force is now balanced with the weight of the aeroplane.
Thus, there will be no net vertical motion, meaning the aeroplane will maintain its altitude.

However, if at any time the lift force exceeds the weight, the aeroplane will start to ascend. Conversely, if the lift is less than the weight, the aeroplane will descend.

There are four forces acting on an aeroplane in flight, as shown - Edexcel - A-Level Physics - Question 13 - 2023 - Paper 4

Worked Solution & Example Answer:There are four forces acting on an aeroplane in flight, as shown - Edexcel - A-Level Physics - Question 13 - 2023 - Paper 4

Show that the radius of the circular path is about 800 m.

Determine the time taken by the plane to move through 90° of the circular path.

Explain what will happen to the vertical motion of the aeroplane.

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