Circular motion - plane going around a loop

[Bensky]

Homework Statement

A pilot flies an airplane in a vertical circular loop at a constant speed of v = 160 m/s. If the pilot's apparent weight at the top of the loop is one-third of his true weight on the ground, find the radius R of the plane's circular path.

Answer: R= 1960m

Homework Equations

F_net = ma
a_normal = v^2 / R

The Attempt at a Solution

Top of the loop
-----------------------
mg - N = mv^2 / R
=> N = mg - mv^2/R

mg - mv^2/R = (1/3) * mg
g - v^2/R = (1/3)g
-25600 / R = (-2/3)g
=> R = 3918.367347 m

This is wrong. The answer should be 1960 m, which is exactly half of my answer strangely enough. I am not seeing what I did wrong here at all, can someone point out where I screwed up?

Thank you

 

[Doc Al]

What direction is the normal force of the plane on the pilot? (Realize that he's upside down.)

 

[Bensky]

What direction is the normal force of the plane on the pilot? (Realize that he's upside down.)

So I guess the normal force should be in the same direction as the weight? Thank you, that did give me the correct answer but I'm confused as to why this is happening or why it matters that he is upside down...I guess I don't fully understand what exactly the normal force is. I've really only seen it used in incline plane problems where it is the force exerted by the "plane" on the object.

EDIT: I think I might understand now. I'm imagining the floor of the plane (which is "above") pressing downward on his feet. Is this why it is in the same direction as the weight?

 

[Doc Al]

The plane pushes up on the pilot's butt, but since the plane is upside down, that's a downward force. The normal force will be whatever it needs to be to maintain the required acceleration. In this problem, you are told that the pilot's weight is insufficient to keep him moving in that vertical circle at the top of the loop (he's going too fast)--so the seat has to exert additional force to help provide the centripetal acceleration.

EDIT: I think I might understand now. I'm imagining the floor of the plane (which is "above") pressing downward on his feet. Is this why it is in the same direction as the weight?

Yep!

 

[Bensky]

The plane pushes up on the pilot's butt, but since the plane is upside down, that's a downward force. The normal force will be whatever it needs to be to maintain the required acceleration. In this problem, you are told that the pilot's weight is insufficient to keep him moving in that vertical circle at the top of the loop (he's going too fast)--so the seat has to exert additional force to help provide the centripetal acceleration.


Yep!

Ok, thanks.

And I know this is not part of the problem, but on the sides of the circle, what would the directions for normal force and weight be?

 

[Doc Al]

And I know this is not part of the problem, but on the sides of the circle, what would the directions for normal force and weight be?

The true weight, of course, always points down. If the plane is traveling in a vertical circle at constant speed--not so easy to arrange--then the net force on the plane is always towards the center. The same is true for the pilot. For him to not slide around in the plane, the seat must exert a force to balance his true weight and provide the inward force to keep him in a circle. So the seat must exert a diagonally upward force on him.