Forest UCM Energy TimeDepPE
Time dependent force.
What happens if you have a time dependent force that still manages to satisfy
- [math]\displaystyle{ \vec \nabla \times \vec {F}(\vec r, t) = 0 }[/math]?
Because of the above, and Stoke's Theorem , you would be able to find a close loop where zero work is done at some given time.
If we consider the work energy theorem
- [math]\displaystyle{ \Delta T = W = \int \vec F \cdot d \vec r }[/math]
or
- [math]\displaystyle{ d T = \frac{dT}{dt} dt = (m \vec \dot v \cdot v) dt = \vec F \cdot d \vec r }[/math]
If a potential U for the force exists such that
- [math]\displaystyle{ \vec F = - \vec \nabla U(r,t) \cdot d \vec r }[/math]
or
- [math]\displaystyle{ dU(r,t) = \frac{\partial U}{\partial x} dx +\frac{\partial U}{\partial y} dy +\frac{\partial U}{\partial z} dz +\frac{\partial U}{\partial t} dt }[/math]
- [math]\displaystyle{ = - \vec F \cdot d \vec r + \frac{\partial U}{\partial t} dt }[/math]
- [math]\displaystyle{ = - dT + \frac{\partial U}{\partial t} dt }[/math]
or
- [math]\displaystyle{ dT + dU = \frac{\partial U}{\partial t} dt }[/math]
or
- [math]\displaystyle{ d(T + U) = \frac{\partial U}{\partial t} dt \ne }[/math]constant
Mechanical Energy is only conserved if the potential is not time dependent.