Expert: Daniel Mazur - 12/7/2007

Question
Hi I am doing several problems in physics for homework and ran into problems with these 3.  Any help on any of them would be greatly appreciated.  I understand if you don't work the problems out and give the answer I am just looking for any little bit of help I can get.

1)  The Yukawa potential energy function for the interaction of neutrons and protons in a nucleus is U(r)= -(a/r)Uoe^(-r/a).  What is the force associated with this potential energy?

2)For what value of the constant c is the force F=i(cxy)+j(5x^2 + 10y) conservative?

3)A particle with mass 10kg starts from rest at r= 10meters and Theta= 20 radians, following the trajectory given by r= 10-2t, theta= 20+0.2t.  Find the radial and angular components Fr and Ftheta of the force, as functions of time, that will cause this motion.

Answer
Hello Marcus,
I think I can help you with most of it.
1) Yukawa potential produces force F=-grad(U)=-dU/dr (partial derivative)
2) Wiki tells us (http://en.wikipedia.org/wiki/Conservative_force) that there are three equivalent ways to ensure a force is conservative. I think the most useful here is that curl(F)=0. If you solve this equation with respect to "c", there will be your answer.
3) I took some time to analyze the problem, because I do not have an off-hand solution (there should be, though). It seems that this is a projectile motion in a uniform gravitational field, only the system of coordinates has been rotated. I'd try to use this as an ansatz, write the equations in a system of coordinates x' and y' rotated by some general angle and then search for an angle value such that any acceleration is observed only along one of the new axis.
I hope this helped a bit.
Cheers,
Daniel