16.07 Dynamics Fall 2004 Ⅴ ersIon1.1 Lecture D8-Conservative Forces and Potential Energy We have seen that the work done by a force F on a particle is given by dw= F. dr. If the work done by F, when the particle moves from any position r to any position T2, can be expressed 2=F=()-V()=1-1, 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. dr(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 r1 and r2 is independent of the path. This follows from(1) since W12 only depends on the initial and final potentials Vi and V2(and not on how we go from r to r2),and ii)the work done by potential forces is recoverable. Consider the work done in going from point ri to W12. If we go, now, from point r2 to ri, we have that W2l ==w12 since the total work W12+ one nsion any force which is only a function of position is conservative. That is, if we have a force, F(a), which is only a function of position, then F(ar)dr is always a perfect differential. This means that we can define a potential function as (x)=-/F()dr here ro 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 sufficient. In multiple dimensions, he condition for a force field to be conservative is that it can be expressed as the gradient of a potential function. That is Note The gradient operator, V The gradient operator, V(called"del"), in cartesian coordinates is defined as a( VO=02+0uj+ azkJ. Peraire 16.07 Dynamics Fall 2004 Version 1.1 Lecture D8 - Conservative Forces and Potential Energy We have seen that the work done by a force F on a particle is given by dW = F · dr. If the work done by F, when the particle moves from any position r1 to any position r2, can be expressed as, W12 = Z r2 r1 F · dr = −(V (r2) − V (r1)) = V1 − V2 , (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 · dr (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 r1 and r2 is independent of the path. This follows from (1) since W12 only depends on the initial and final potentials V1 and V2 (and not on how we go from r1 to r2), and ii) the work done by potential forces is recoverable. Consider the work done in going from point r1 to point r2, W12. If we go, now, from point r2 to r1, we have that W21 = −W12 since the total work W12 + W21 = (V1 − V2) + (V2 − V1) = 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 V (x) = − Z x x0 F(x) dx , where x0 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 sufficient. In multiple dimensions, 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 = −∇V . Note The gradient operator, ∇ The gradient operator, ∇ (called “del”), in cartesian coordinates is defined as ∇( ) ≡ ∂( ) ∂x i + ∂( ) ∂y j + ∂( ) ∂z k . 1