A particle slides on frictionless x - y plane. It is acted on by a conservative force described by the potential - energy function `U(x, y) =(1)/(2)k(x^(2)+y^(2))`. Derive an expression for the force acting on the particle.

Explanation

To derive the expression for the force acting on a particle in a conservative force field described by the potential energy function \( U(x, y) = \frac{1}{2} k (x^2 + y^2) \), we will follow these steps:

### Step 1: Understand the relationship between force and potential energy
The force \( \vec{F} \) acting on a particle in a conservative field can be derived from the potential energy function \( U \) using the formula:
\[ \vec{F} = -\nabla U \]
where \( \nabla U \) is the gradient of the potential energy function.
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