Question

The potential energy $U$ for a force field $\overrightarrow F $ is such that $U = - Kxy$, where $K$ is a constant. Then
A. $\overrightarrow F = Ky\hat i + Kx\hat j$
B. $\overrightarrow F = Kx\hat i + Ky\hat j$
C. $\overrightarrow F $ is a conservative force
D. $\overrightarrow F $ is a non-conservative force

Answer 1

Hint-Stored energy or potential is defined only for conservative forces.
A force is said to be conservative if the work done by the force on the body in taking it from one position to another is independent of the path taken by the body.
If potential energy is given then the force can be found using the equation
$\overrightarrow F = - \Delta U$
$x$ component of force can be written as
$\overrightarrow {{F_x}} = - \dfrac{{dU}}{{dx}}$
$y$ component of force can be written as
$\overrightarrow {{F_y}} = - \dfrac{{dU}}{{dy}}$
The total force $\overrightarrow F $ can be written as,
$\overrightarrow F = \overrightarrow {{F_x}} \hat i + \overrightarrow {{F_y}} \hat j$

Step by step solution:
Stored energy or potential is defined only for conservative forces.
A force is said to be conservative if the work done by the force on the body in taking it from one position to another is independent of the path taken by the body. It depends only on the initial and final position of the body. Therefore, conservative forces are path independent
Work done around a closed path by a conservative force will be zero.
If potential energy is given then the force can be found using the equation
$\overrightarrow F = - \Delta U$
Therefore, $x$ component of force can be written as
$\overrightarrow {{F_x}} = - \dfrac{{dU}}{{dx}}$ …… (1)
Given,
$U = - Kxy$
Substitute in equation (1)
$
  \overrightarrow {{F_x}} = - \dfrac{{d\left( { - Kxy} \right)}}{{dx}} \\
   = Ky \\
 $
$y$ component of force can be written as
$\overrightarrow {{F_y}} = - \dfrac{{dU}}{{dy}}$
$
  \overrightarrow {{F_y}} = - \dfrac{{d\left( { - Kxy} \right)}}{{dy}} \\
   = Kx \\
 $
The total force $\overrightarrow F $ can be written as,
$\overrightarrow F = \overrightarrow {{F_x}} \hat i + \overrightarrow {{F_y}} \hat j$ …..(3)
Substitute the value of $\overrightarrow {{F_x}} $ and $\overrightarrow {{F_y}} $ in equation (3)
$\overrightarrow F = Ky\hat i + Kx\hat j$

Therefore, option B and option C are correct.

Note:
Formula to remember:
$\overrightarrow F = - \Delta U$
If potential energy $U$ is given then the force can be found using this above equation. This formula is applicable only for conservative forces. For non-conservative forces or path dependent forces such as frictional force we cannot use this relation.