We have $$E=E_K+E_P$$
What happens when $\frac{dE}{dt}=0$? Is the potential force conservative?
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3 Answers
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If $\frac{dE}{dt} = 0 $ it basically means that the total energy of the system is not changing as time passes. Now this doesn't clarify whether kinetic energy is changing or potential energy is changing or neither are changing , but Yes, whatever change is there in any of the energies it get transferred to the other form so as to make the total energy constant and conserved. Now as to your question about the potential force, Yes potential force is conservative. Not only this but it is always conservative. Potential energy is defined and derived itself for conservative forces only. When a system contains a conservative force field the work done against that field only matters in the state of the object (it's displacement) and so for this we define potential energy. Mathematically: $$ E_p = W_{conservative} = -\int {F_{conservative} \cdot {dx}}$$ Hope this helps.
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$\begingroup$ The force is still not always necessarily conservative. Imagine a ball rolling down a hill with friction. First potential energy is high, but it steadily decreases. Kinetic energy increases, but at a slower rate than the potential one decreases. We can still use the concept of potential energy. But in the case of OP's question I agree that it's conservative. $\endgroup$– noahCommented Apr 8, 2019 at 18:17
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$\begingroup$ Noah I think you are missing out something...... The mechanical energy in case of rolling not only consists of potential and kinetic energy but also gets distributed into rotational kinetic energy.... As far as I have learnt the idea of introducing something like potential energy was to use the benefit of the fact that in conservative force the energy only depends on the position of object w.r.t. to a reference line ...... So that's what we are taught $\endgroup$– arnav009Commented Apr 9, 2019 at 15:14
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$\begingroup$ Well then have a cube sliding down a ramp, the friction argument still holds. $\endgroup$– noahCommented Apr 9, 2019 at 15:16
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$\begingroup$ If it slides then it definitely experiences kinetic friction and hence energy is lost in the form of heat as the work done by friction..... The conservation of mechanical energy only hold true if the net work done by external force is zero $\endgroup$– arnav009Commented Apr 9, 2019 at 15:18
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$\begingroup$ I don't think you get the point of my first comment. I'm saying that not every time we have some potential energy in a system, all forces must be conservative, as you suggest in your answer. $\endgroup$– noahCommented Apr 9, 2019 at 15:20
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It simply means that the sum of changes in potential and kinetic energy of an isolated system is zero, that is, the total energy of an isolated system is conserved over time. In other words, if there is a decrease in potential energy there must be an equal increase in kinetic energy, and vice-versa.
Also the force associated with potential energy is called a conservative force, not a potential force. The work done by a conservative force depends only on the end points and not the path between the points.
Hope this helps.
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The equation means that the total energy of system is not changing with time thus constant. Kinetic and potential energy may change at any point on path individually but total energy is constant.
Potential is kind of energy which is stored in body when conservative force acts on it. Where conservative force is for which work done is independent of path followed by system. So for conservative force, potential energy between initial and final point remains same regardless of the nature of path.Also for a conservative force a scalar potential is associated for each point on the path followed. Hope you have got your answer.
