Question
Question: The potential energy function associated with the force \[\vec F = 4xy\hat i + 2{x^2}\hat j\] is: ...
The potential energy function associated with the force F=4xyi^+2x2j^ is:
A. U=−4x2y+ constant
B. U=x2y+ constant
C. U=x3y+ constant
D. Not defined
Solution
In this question, we are given a conservative force F in a two-dimensional plane and we need to find out the potential energy function. Now the change in potential energy due to internal conservative force is given by U=−∫F⋅dr . Using this, we can find out the potential energy function.
Formula used:
Potential energy due to internal conservative force is, U=∫dU=−ri∫rfF⋅dr=−W
Where F is the conservative force
dr is the small displacement
Complete step by step answer:
We have given,
F=4xyi^+2x2j^
Now as we know potential energy for a conservative internal force is given by,
U=∫dU=−ri∫rfF⋅dr=−W
We can also say that the potential energy of a body is negative of the work done by the conservative forces in bringing the body from infinity to the present position
U=−∫F⋅dr
As the motion is in two-dimensional plane
Thus, we can write dr=dxi^+dyj^
Substituting the value of F in the equation we get,
U=−∫(4xyi^+2x2j^)⋅(dxi^+dyj^)
Taking dot product we get,
⇒U=−∫4xydx+2x2dy
⇒U=−∫4xydx−∫2x2dy
On integrating we get,
⇒U=−2x2y−2x2y+c
Where c is a constant
⇒U=−4x2y+c
The potential energy function associated with the force F=4xyi^+2x2j^ is −4x2y+c
Hence, option A. is the correct option.
Additional information:
We know F=−drdU . If the force is conservative and is in equilibrium, then F=−drdU=0. Thus, at these points the body is said to be in an equilibrium.
An object is said to be in stable equilibrium if the body tends to return to the same position after a slight displacement. If dr2d2U is positive body is in stable equilibrium
If dr2d2U is negative, the body is said to be in unstable equilibrium
If dr2d2U is zero, the body is said to be in neutral equilibrium
Note: Potential energy is defined for a conservative field only. For non-conservative forces, it has no meaning. Potential energy should be considered as the property of the entire system rather than assigning it to some specific particle. Potential energy depends on the frame of reference.