Conservation of Mechanical Energy Requirement

[fog37]

Hello Forum,
Conservation of mechanical energy ME= KE+PE happens when the net work done by the non conservative forces is zero. Conservation of total momentum, instead, happens when the external net force is zero (or close to zero).

In the case of mechanical energy, the non-conservative forces are also external forces to the system and can be nonzero. If the other external forces are conservative, the ME is conserved but not momentum.
But what we care about is the work they produce. How can nonconservative forces produce zero work? I guess their work may be finite but negligibly small. Maybe the total sum of the work done by the nonconservative force is zero...

Is that correct?

fog37

 

[vanhees71]

First you should clarify the definitions. A conservative force is given by a time-independnet scalar potential,
$$\vec{F}(\vec{x})=-\vec{\nabla} V(\vec{x}).$$
Then you take Newton's 2nd Law, multiply it by $\dot{\vec{x}}$ and integrate over time to get the energy-conservation law
$$\frac{m}{2} \dot{\vec{x}}^2+V(\vec{x})=\text{const}.$$
Then there are forces that do no work. An example is the motion in an external magnetic field. The Lorentz force is always perpendicular to $\dot{\vec{x}}$ and thus it doesn't contribute to the work.
$$\vec{F} \cdot \dot{\vec{x}}=q (\dot{\vec{x}} \times \vec{B}) \cdot \dot{\vec{x}}=0.$$

 

[fog37]

Sure, so there are two types of forces: conservative and non-conservative. Both types of force can perform work. What matters for the mechanical energy to be conserved is that the net work done by the nonconservative forces is zero. How can that happen? If the nonconservative forces act perpendicularly to the object's path or if the algebraic signs of the works produce a net zero work...

As you mention, conservative forces admit a potential function V(x), which is a function of position. Can a time-dependent force be conservative? I don't think so. But why? What is the rigorous argument to explain that?

Thanks!
fog37

 

[Guneykan Ozgul]

Conservative force means that we can define a scalar potential function for that force. To be able to define a scalar function for a vector field the curl (wrt space coordinates) of the vector field should be 0 because curl of a gradient should be always 0. Since force can be written as gradient when we take curl of it, we should obtain 0.

 

[fog37]

Sure, wherever the curl F = 0 we can express F= Delta V, i.e. we can use a scalar potential V.

but what if the force F was F(t), i.e. a function of time?

 

[Guneykan Ozgul]

I think you cannot talk about curl of a time dependent function (without fixing time) because now your force is four vector but curl is defined on 3D. Also notice that if you define a potential when you follow different paths in space between two points, you may not reach same potential due to time dependence which makes your force path dependent . So I don't think a time dependent force is conservative. However, in a fixed time your force can be considered as a conservative force if the curl of it wrt space coordinates 0.