CALC An object moving in the xy-plane is subjected to the force F(arrow on top) =(2xy î+x² ĵ) N, where x and y are in m. c. Is this a conservative force?

Verified step by step guidance

1. Identify the force components: The force given is $\overrightarrow{F}=2xy\hat{i}+x^2\hat{j}$. Here, $F_x=2xy$ and $F_y=x^2$.
2. Recall the condition for a force to be conservative: A force $\overrightarrow{F}$ is conservative if the curl of $\overrightarrow{F}$ is zero, i.e., $\mathrm{\nabla }\times \overrightarrow{F}=0$.
3. Compute the partial derivatives needed for the curl: Calculate $\frac{\partial F_y}{\partial x}$ and $\frac{\partial F_x}{\partial y}$.
4. Compare the partial derivatives: For the force to be conservative, $\frac{\partial F_y}{\partial x}$ should equal $\frac{\partial F_x}{\partial y}$.
5. Conclude whether the force is conservative based on the comparison of the derivatives.

Key Concepts

Conservative Forces

A conservative force is one where the work done by the force on an object moving between two points is independent of the path taken. This means that the work done in moving an object around a closed loop is zero. Examples include gravitational and electrostatic forces. To determine if a force is conservative, one can check if it can be expressed as the gradient of a potential energy function.

Curl of a Vector Field

The curl of a vector field is a measure of the rotation of the field at a point. For a force field to be conservative, its curl must be zero everywhere in the region of interest. Mathematically, the curl is calculated using the determinant of a matrix formed by the unit vectors and the partial derivatives of the vector components. If the curl is non-zero, the force is non-conservative.

Path Independence

Path independence refers to the property of a force where the work done does not depend on the specific trajectory taken between two points. In conservative forces, this implies that the work done is solely a function of the initial and final positions. This concept is crucial for understanding energy conservation in mechanical systems, as it allows for the definition of potential energy associated with the force.