Semantics question about this angular momentum problem

[FisherDude]

Homework Statement

Heading straight toward the summit of Pikes Peak, an airplane of mass 12,000 kg flies over the plains of Kansas at nearly constant altitude 4.30 km with constant velocity 175 m/s west. a) What is the airplane's vector angular momentum relative to a wheat farmer on the ground directly below the airplane? b) Does this value change as the airplane continues its motion along a straight line?

Homework Equations

mag. of angular momentum = position*mass*velocity*angle between

The Attempt at a Solution

The answer to part a) is (-9.03 x 10^9 kg x m^2/s) j, which I have no problem with. But the answer to part b) is "No, L = |r||p|sin(Θ) = mv(rsinΘ), and r*sinΘ is the altitude of the plane. Therefore, L = constant as the plane moves in level flight with constant velocity."

But the problem asks for the plane's angular momentum relative to the wheat farmer. So if the plane keeps on moving west, wouldn't r (the distance between the farmer and the plane) keep on increasing?

The only way the answer makes sense to me is if they're really asking for the angular momentum of the plane relative to the ground, not the wheat farmer, because then, the distance between the plane and the ground would be constant.

Any help would be great...

 

[Doc Al]

But the problem asks for the plane's angular momentum relative to the wheat farmer. So if the plane keeps on moving west, wouldn't r (the distance between the farmer and the plane) keep on increasing?

Sure, the distance r from farmer to plane increases. But r*sinΘ does not. Angular momentum is not just mv*r, but mvr*sinΘ. (Only when the plane is directly overhead does sinΘ = 1.)

The only way the answer makes sense to me is if they're really asking for the angular momentum of the plane relative to the ground, not the wheat farmer, because then, the distance between the plane and the ground would be constant.

Note that r*sinΘ is the distance between plane and ground.

 

[flatmaster]

You're correct, L remains constant even with the farmer. You realized that r*sin(theta) is simply the altitute, which is constant. You're correct that r will increase because the plane-farmer distance increases. However, sin(theta) will decrease at such a rate to keep L constant. Remember what the angle theta is defined as. It's the angle between the vector r and the vector v. You were allowed to ignore this in your original calculation because theta = 90 Sin(theta) = 1

 

[FisherDude]

Wow, I wish i had caught that.

Thanks!