Force F⃗-y î |x ĵ acts upon a particle of mass m. (a)...

Force $\vec{F}-y \hat{i} \mid x \hat{j}$ acts upon a particle of mass $m$. (a) Potential energy is given by $U--\frac{x y}{2}+c$, where $C$ is constant (b) Force is conscrvative in nature and potential energy $U=-x y+C$, where $C$ is constant (c) Force is non conservative in nature (d) Origin is the unstable cquilibrium position

Force $\vec{F}-y \hat{i} \mid x \hat{j}$ acts upon a particle of mass $m$.
(a) Potential energy is given by $U--\frac{x y}{2}+c$, where $C$ is constant
(b) Force is conscrvative in nature and potential energy $U=-x y+C$, where $C$ is constant
(c) Force is non conservative in nature
(d) Origin is the unstable cquilibrium position

Key Concepts

Conservative Forces

A conservative force is one that can be expressed as the negative gradient of a potential energy function. In such fields, the work done by the force along any path depends only on the end points and not on the specific path taken. This property implies that the circulation of the force around any closed loop is zero, which is a key criterion for a force field to be considered conservative.

Curl of a Vector Field

The curl of a vector field is a measure of the rotation or the swirling strength of the field at a point. For a force field, if the curl is non-zero throughout the region, it indicates that the field has an inherent rotational character and cannot be derived from a scalar potential function. This is a standard test used to determine whether a force is conservative.

Potential Energy Functions

In the context of conservative forces, a potential energy function is defined such that the force is the negative gradient of this function. This function represents the stored energy in the system due to its position in the force field. If the force field is conservative, the potential energy difference between any two points is independent of the path taken between them.

Equilibrium and Stability Analysis

An equilibrium position is a point where the net force acting on a particle is zero. The stability of an equilibrium is determined by analyzing the behavior of the system in the vicinity of this point, often through the curvature of the potential energy function in conservative systems. If a small perturbation leads the system to return to equilibrium, it is stable; if it moves away, the equilibrium is unstable. In non-conservative fields, stability analysis typically requires examining the dynamics directly since a potential energy function may not exist.

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