A pilot of mass 65 kg in a jet aircraft makes a complete vertical circle in mid-air. The vertical circle has a radius of 1.70 km at a constant speed of 215 m/s. Determine the force of the seat on the pilot at (a) the bottom of the loop and (b) the top of the loop. (Note that at the top of the loop, the aircraft is upside down.)

Okay, i know that first i will have to use 1700m as my radius.
Also, i know the equations i will have to work with (uniform circular motion
Centripetal Force= mass times velocity squared, all over radius.)
But, am i looking for the Fn (Normal Force) ..
and ifso,
Centripetal force will equal force of gravity plus the normal force in both situations, so how will the situations differ? like, is there something i'm missing?
Please explain, thanks :)

It is very good that you have mentioned what you have already found, and what bothers you.

Basically, the centripetal force Fc=mv²/r will exert extra load on the seat. The extra load is radially outwards.

Therefore, when the pilot is at the bottom of the loop, Fc adds to gravity, which is mg. With the sign convention of positive upwards, it would be -Fc-mg.

When he is at the top of the loop, gravity is -mg (downwards), which is added to Fc. So the total force on the seat is Fc-mg (upwards).

Therefore you see that the two net forces on the seat are different when at the top and at the bottom of the loop.

Well, let me just say, that pilot sure is going for some serious acrobatics in that jet aircraft! Now let's get to the question at hand.

You're correct that you need to use the equation for centripetal force, which is given by the formula:

Centripetal Force = (mass × velocity^2) / radius

Now, let's talk about the forces involved. At the bottom of the loop, when the pilot is right-side up, there are two forces acting on the pilot: the force of gravity and the normal force. The normal force, which is the force exerted by the seat on the pilot, provides the required centripetal force to keep the pilot in a circular path.

However, at the top of the loop, when the aircraft is upside down, things get a little wacky! The only two forces acting on the pilot are still gravity and the normal force. But in this case, the normal force is providing both the required centripetal force and a force to counteract gravity, since the pilot is now "hanging" upside down.

So, to sum it up: at the bottom of the loop, the normal force only needs to provide the centripetal force, while at the top of the loop, the normal force needs to provide the centripetal force and counteract gravity.

Now, let's plug in the values. We already have the mass of the pilot (65 kg), the velocity (215 m/s), and the radius of the circle (1.70 km or 1700 m). We can use these values to calculate the centripetal force at both the bottom and the top of the loop.

(a) At the bottom of the loop:
Centripetal Force = (mass × velocity^2) / radius
Centripetal Force = (65 kg × (215 m/s)^2) / 1700 m

(b) At the top of the loop:
Centripetal Force = (mass × velocity^2) / radius
Centripetal Force = (65 kg × (215 m/s)^2) / 1700 m

So, now you have the value for the centripetal force at both points of the loop. To find the force of the seat on the pilot (the normal force), you need to consider the vertical forces.

At the bottom of the loop, the net force in the vertical direction is the difference between the force of gravity and the normal force.

At the top of the loop, the net force in the vertical direction is the sum of the force of gravity and the normal force.

By setting up the appropriate equation with these forces, you can solve for the normal force in both cases. And voila, you'll have your answers!

I hope that clears things up for you, and don't forget to buckle up your seatbelt and tighten your wig before attempting any high-flying stunts!

To determine the force of the seat on the pilot at the bottom and top of the loop, we need to consider the forces acting on the pilot.

At the bottom of the loop:
1. Identify the forces: The forces acting on the pilot at the bottom of the loop are the gravitational force (mg) pointing downward and the normal force (Fn) from the seat acting upward.

2. Determine the net force: Since the pilot is moving in a vertical circle, the net force required is the centripetal force (Fc) pointing towards the center of the circle.

3. Apply Newton's second law: Newton's second law states that the net force is equal to the mass times acceleration (F = ma). In this case, the centripetal force is responsible for the acceleration.

Fc = m * a

4. Relate centripetal force to gravitational force and normal force: The centripetal force required to keep the pilot moving in a circle is the sum of the gravitational force and the normal force.

Fc = mg + Fn

5. Apply the centripetal force equation: We can use the centripetal force equation, which states that the centripetal force is equal to the mass times the velocity squared divided by the radius.

Fc = (m * v^2) / r

6. Substitute the given values: Plug in the known values into the equation, where m = 65 kg, v = 215 m/s, and r = 1700 m.

Fc = (65 kg * (215 m/s)^2) / 1700 m

7. Calculate the centripetal force: Calculate the centripetal force using the values.

Fc = 65 kg * 46225 m^2/s^2 / 1700 m
≈ 1770 N

8. Solve for the normal force: Now, substitute the value of the centripetal force back into the equation from step 4, and rearrange to solve for Fn.

Fc = mg + Fn
Fn = Fc - mg
≈ 1770 N - (65 kg * 9.8 m/s^2)
≈ 1233 N

So, at the bottom of the loop, the seat exerts a force of approximately 1233 N on the pilot.

Now, let's consider the top of the loop (when the aircraft is upside down):
At the top of the loop, the forces acting on the pilot are still the gravitational force (mg) pointing downward and the normal force (Fn) from the seat acting upward. The main difference is the direction of the net force required to keep the pilot moving in a circle.

In this case, the net force is directed towards the center of the circle but now points downward, so the normal force and gravitational force have different signs.

Using the same reasoning as before, we can apply the centripetal force equation:

Fc = mg + Fn

However, the gravitational force (mg) acts downward, while the normal force (Fn) acts upward:

Fc = -mg + Fn

Following the same steps as before, we can substitute the values into the equation and solve for the normal force.

Finally, calculate the centripetal force as before and solve for the normal force at the top of the loop.

Remember to take into account the different signs for the forces at the top of the loop.

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