blazinfury Posted March 2, 2013 Share Posted March 2, 2013 I am confused about whether work is path independent or not. If one were to drop a pen from a height of 5 m or to throw a pen parabolically from 5 m, the work done would be the same-- thus indicating that work is path independent. However, when one is talking about engines and refrigerators, work is deemed to be path-dependent based on U=Q+W. Would someone please clarify and reconcile this confusion. Thanks. Link to comment Share on other sites More sharing options...
elfmotat Posted March 3, 2013 Share Posted March 3, 2013 Work is path-independent if and only if the force acting on the particle is conservative. A conservative force is one which can be written as minus the gradient of a scalar potential function: [math]\mathbf{F}=-\nabla \phi[/math]. The work is path independent for conservative forces because the work done along an arbitrary curve C with endpoints A and B is given by: [math]W=\int_C \mathbf{F}\cdot \mathrm{d}\mathbf{r}[/math] If [math]\mathbf{F}[/math] is conservative, then: [math]\int_C \mathbf{F}\cdot \mathrm{d}\mathbf{r}=-\int_C \nabla \phi \cdot \mathrm{d}\mathbf{r}=\phi(A)-\phi(B)[/math] Gravity is conservative because it can be written as the gradient of the potential function [math]\phi=-\frac{Gm_1m_2}{r}[/math]. Magnetic forces, frictional forces, etc. are not conservative, so the work done in moving a particle through a magnetic field or when working against friction will depend on which path is taken. 2 Link to comment Share on other sites More sharing options...
blazinfury Posted March 3, 2013 Author Share Posted March 3, 2013 That clarified a lot. Thank you for the awesome explanation. So then the forces involved in heat engines are not conservative and thus work is path-dependent. However, why is heat (q) in U=Q+W, path dependent-- also b/c it is not caused by conservative forces? Link to comment Share on other sites More sharing options...
elfmotat Posted March 3, 2013 Share Posted March 3, 2013 That clarified a lot. Thank you for the awesome explanation. So then the forces involved in heat engines are not conservative and thus work is path-dependent. However, why is heat (q) in U=Q+W, path dependent-- also b/c it is not caused by conservative forces? [math]Q[/math] is the heat added to the system. This means that there's some heat transfer process, for example conduction, which is a complicated process that involves molecules bouncing around and smashing into each other. It depends on a number of factors, like temperature, the material, cross-sectional area, the thickness of the material, etc. Link to comment Share on other sites More sharing options...
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