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Unformatted text preview: S. Widnall, J. Peraire 16.07 Dynamics Fall 2008 Version 2.0 Lecture L13- Conservative Internal Forces and Potential Energy The forces internal to a system are of two types. Conservative forces, such as gravity; and dissipative forces such as friction. Internal forces arise from the natural dynamics of the system in contract to external forces which are imposed from an external source. We have seen that the work done by a force F on a particle is given by dW = F d r . If the work done by an internal forces F , when the particle moves from any position r 1 to any position r 2 , can be expressed as the difference in a scalar function of r between the two ends of the trajectory, r 2 W 12 = F d r = ( V ( r 2 ) V ( r 1 )) = V 1 V 2 , (1) r 1 then we say that the force is conservative . In the above expression, the scalar function V ( r ) is called the potential . It is clear that the potential satisfies dV = F d r (the minus sign is included for convenience). There are two main consequences that follow from the existence of a potential: i) the work done by a conservative force between points r 1 and r 2 is independent of the path . This follows from (1) since W 12 only depends on the initial and final potentials V 1 and V 2 (and not on how we go from r 1 to r 2 ), and ii) the work done by potential forces is recoverable . Consider the work done in going from point r 1 to point r 2 , W 12 . If we go, now, from point r 2 to r 1 , we have that W 21 = W 12 since the total work W 12 + W 21 = ( V 1 V 2 ) + ( V 2 V 1 ) = 0. In one dimension any force which is only a function of position is conservative. That is, if we have a force, F ( x ), which is only a function of position, then F ( x ) dx is always a perfect differential. This means that we can define a potential function as x V ( x ) = F ( x ) dx , x where x is arbitrary. In two and three dimensions, we would, in principle, expect that any force which depends only on position, F ( r ), to be conservative. However, it turns out that, in general, this is not sucient. In multiple dimensions, 1 the condition for a force field to be conservative is that it can be expressed as the gradient of a potential function. That is, F C = V . This result follows from the gradient theorem, which is often called the fundamental theorem of calculus, which states that the integral r 2 V d r = ( V 2 V 1 ) r 1 is independent of the path between r 1 and r 2 . Therefore the work done by conservative forces depends only upon the endpoints r 2 and r 1 rather than the details of the path taken between them. r 2 r 2 F C d r = V d r = ( V 2 V 1 ) r 1 r 1 In the general case, we will deal with internal forces that are a combination of conservative and non- conservative forces....
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