# PE function

1. Feb 15, 2006

### Reshma

A particle of mass M is free to move in the horizontal plane(xy-planne here). It is subjected to force $\vec F = -k\left(x\hat i + y\hat j\right)$, where 'k' is a positive constant.
There are two questions that have been asked here:
1] Find the potential energy of the particle.

$$\vec \nabla \times \vec F = 0$$
The given force is conservative and hence a potential energy function exists.
Let it be U.
$$F_x = -\frac{\partial U}{\partial x} = -kx$$

$$F_y = -\frac{\partial U}{\partial y} = -ky$$

$$U(x,y) = \frac{k}{2}\left(x^2 + y^2) + C$$

2]If the particle never passes through the origin, what is the nature of the orbit of the particle?

I am not sure what the PE function tells about the trajectory of the particle. Explanation needed...

2. Feb 15, 2006

### George Jones

Staff Emeritus
The potential energy can be used to find the Lagrangian, and then Lagrange's equation can be used to find the motion.

Alternatively, $m \ddot{x} = F_x = -kx$ and $m \ddot{y} = F_y = -ky[/tex] can be solved directly. These equations should look very familiar. Regards, George 3. Feb 21, 2006 ### Reshma Thank you for replying. So this is a 2-dimensional harmonic oscillator. The general solution would be: [itex]x = A\cos(\omega_0 t - \alpha)$ & $y = B\cos(\omega_0 t - \beta)$

So, the resultant path of these two SHMs would be an ellipse, right?

4. Feb 21, 2006

### George Jones

Staff Emeritus
Yes.

Regards,
George