# SHM w/ complicated restoring force

1. Sep 21, 2010

### NeedPhysHelp8

The problem statement, all variables and given/known data
An atom of mass m is bonded to surface immobile body by electromagnetic forces. The force binding the atom to the surface has the expression:

$$F= exp\ (a\cos z + b\sin z) + d\tan(z)$$

where a,b, and d are constants and z is upwards. The equilibrium point is defined to be z=0. The system is subject to Earth's gravity

For small oscillations, give an approximate expression for the binding force on the atom.

The attempt at a solution
Ok so in class we were taught to do the Taylor Series expansion on F(x) and keep the linear term so you can get something that looks like F=-kx . So I got this after doing Taylor Series expansion:

where $$F' = (-a\sin z + b\cos z) exp\ (a\cos z + b\sin z) + d\sec^2 (z)$$ then evaluate at z=0 then final result is :

$$F(z)= (b exp(a) + d )z$$

Since $$F(0)$$ is always 0 at the equilibrium I took that out of Taylor series and ignored higher order terms.

Now I'm not sure if I'm going about this the right way? Can someone please tell me if I did this right or not? Much appreciated I love this forum!

Last edited: Sep 21, 2010
2. Sep 21, 2010

### diazona

You've got the right idea, but you seem to have computed F' incorrectly. Take another look at that calculation.

3. Sep 21, 2010

### NeedPhysHelp8

How is F' wrong?? :S please explain I just used the chain rule

4. Sep 21, 2010

### diazona

It's wrong for the F that was in your post when I wrote mine. But after your edit, everything looks OK.

5. Sep 21, 2010

### NeedPhysHelp8

Thanks

Last edited: Sep 22, 2010