Forest UCM Energy TimeDepPE

Time dependent force.

What happens if you have a time dependent force that still manages to satisfy

[math]\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]\Delta T = W = \int \vec F \cdot d \vec r[/math]

or

[math]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]\vec F = - \vec \nabla U(r,t) \cdot d \vec r[/math]

or

[math]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]= - \vec F \cdot d \vec r + \frac{\partial U}{\partial t} dt [/math]

[math]= - dT + \frac{\partial U}{\partial t} dt [/math]

or

[math]dT + dU = \frac{\partial U}{\partial t} dt [/math]

or

[math]d(T + U) = \frac{\partial U}{\partial t} dt \ne [/math]constant

Mechanical Energy is only conserved if the potential is not time dependent.

Example

Charge escaping from a charged conducting sphere into the air.

The force on an external test charge would be decreasing with time.

Even though the coulomb force is conservative it will have a time dependence.

Although the Mechanical Energy is not conserved the TOTAL energy is.

The lost Mechanical energy to the air is gained by the air due to the heating of the air by the discharge.

The potential energy is time dependent for this situation where mechanical energy is being transformed into another form of energy external to the system.

Forest_UCM_Energy#Time_Dependent_PE