A force F̅=k(y^2 î+x ĵ), where k is a positive constant,...

A force $\bar{F}=k\left(y^{2} \hat{i}+x \hat{j}\right)$, where $k$ is a positive constant, acts on a particle when it is at position $(x, y)$. First, the particle is taken through path $\mathrm{OAB}$ and the work done by the force is $\mathrm{W}_{\mathrm{OAB}^{*}}$. Then the particle is taken on through path $\mathrm{OCB}$ and the work done by the force is $\mathrm{W}_{\mathrm{ocB}^{\prime}}$. Then (a) $\mathrm{W}_{\text {OAB }}=\mathrm{ka}^{2}$ (b) $\mathrm{W}_{\mathrm{OCB}}=\mathrm{ka}^{2}$ (c) the force is conservative (d) the force is non-conservative

A force $\bar{F}=k\left(y^{2} \hat{i}+x \hat{j}\right)$, where $k$ is a positive constant, acts on a particle when it is at position $(x, y)$. First, the particle is taken through path $\mathrm{OAB}$ and the work done by the force is $\mathrm{W}_{\mathrm{OAB}^{*}}$. Then the particle is taken on through path $\mathrm{OCB}$ and the work done by the force is $\mathrm{W}_{\mathrm{ocB}^{\prime}}$. Then
(a) $\mathrm{W}_{\text {OAB }}=\mathrm{ka}^{2}$
(b) $\mathrm{W}_{\mathrm{OCB}}=\mathrm{ka}^{2}$
(c) the force is conservative
(d) the force is non-conservative

Key Concepts

Path Independence

Path independence is the characteristic that the work done by a conservative force between any two points does not depend on the specific path taken. This property simplifies the computation of work and is key in differentiating conservative forces from non?conservative forces.

Potential Function

A potential function is a scalar field whose gradient (with a negative sign) gives rise to a conservative force field. When a force is conservative, the work done in moving along any path between two points can be computed as the difference in the potential function evaluated at those points.

Curl Test for Conservativeness

The curl test is a mathematical tool used to determine whether a given vector field is conservative. In a simply connected region, if the curl of the force field is zero, the force is conservative, meaning its work is path independent.

Work Done (Line Integral)

Work done by a force is calculated as the line integral of the force along a path. This involves integrating the dot product of the force vector and the differential displacement vector over the path. It represents the energy transferred by the force to the particle as it moves along that path.

Conservative Forces

A conservative force is one in which the work done on a particle moving between two points is independent of the path taken. Such forces can be derived from a potential energy function, and the total work around any closed path is zero.

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