To find the magnitude of the net force acting on the plane, we need to consider both the centripetal force and the tangential force due to the acceleration.

1. Given Data:

• Mass of the plane $m = 40,000 \, \text{kg}$
• Diameter of the loop = 260 m → Radius $r = 130 \, \text{m}$
• Speed of the plane at the point where it is flying straight down $v = 55 \, \text{m/s}$
• Tangential acceleration $a_{\text{tangential}} = 12 \, \text{m/s}^2$

2. Centripetal Force

The centripetal force is responsible for keeping the plane in circular motion and is given by:

$F_{\text{centripetal}} = \frac{m v^2}{r}$

Substituting the values:

$F_{\text{centripetal}} = \frac{40,000 \times (55)^2}{130} = \frac{40,000 \times 3025}{130}$

$F_{\text{centripetal}} = \frac{121,000,000}{130} \approx 930,769.23 \, \text{N}$

3. Tangential Force

The tangential force is due to the plane’s acceleration in the direction of motion:

$F_{\text{tangential}} = m \cdot a_{\text{tangential}} = 40,000 \times 12 = 480,000 \, \text{N}$

4. Net Force

The net force $F_{\text{net}}$ is the vector sum of the centripetal and tangential forces. Since these two forces are perpendicular to each other (centripetal towards the center of the loop and tangential along the direction of motion), we use the Pythagorean theorem:

$F_{\text{net}} = \sqrt{F_{\text{centripetal}}^2 + F_{\text{tangential}}^2}$

$F_{\text{net}} = \sqrt{(930,769.23)^2 + (480,000)^2}$

$F_{\text{net}} = \sqrt{866,331,027,170.92 + 230,400,000,000}$

$F_{\text{net}} \approx \sqrt{1,096,731,027,170.92} \approx 1,047,009.07 \, \text{N}$

Final Answer:

The magnitude of the net force on the plane is approximately 1,047,009 N.

Would you like a further breakdown or have any questions?

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Here are 5 related questions:

1. How would the force change if the radius of the loop increased?
2. What would happen to the net force if the plane’s speed doubled?
3. How would the net force be affected if the tangential acceleration decreased?
4. How do centripetal and tangential forces differ in circular motion?
5. What would the gravitational force on the plane be at this point in the loop?

Tip: When dealing with circular motion, always separate forces into radial (centripetal) and tangential components for accurate analysis.