Conservative Internal Forces and Potential Energy

1 S. Widnall, J. Peraire Dynamics Fall 2008 Version 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 dr. If the work done by an Internal Forces F , when the particle moves from any position r1 to any position r2, can be expressed as the difference in a scalar function of r between the two ends of the trajectory, r2 W12 = F dr = (V (r2) V (r1)) = V1 V2 , (1) r1 then we say that the force is Conservative .

2 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 .

3 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 , x0 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, 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 r2 V dr = (V2 V1) r1 is independent of the path between r1 and r2.

4 Therefore the work done by Conservative Forces depends only upon the endpoints r2 and r1 rather than the details of the path taken between them. r2 r2 F C dr = Vdr = (V2 V1) r1 r1 In the general case, we will deal with Internal Forces that are a combination of Conservative and non- Conservative Forces . F = F C + F NC = V + F NC . Note The gradient operator, The gradient operator, (called del ), in cartesian coordinates is defined as ( ) ( ) ( ) ( ) x i + y j + z k . When operating on a scalar function V (x,y, z), the result V is a vector, called the gradient of V . The components of V are the derivatives of V along each of the coordinate directions, V V V V x i + y j + z k . If we consider a particle moving due to Conservative Forces with Potential Energy V (x , y, z), as the particle moves from point r = xi + yj + zk to point r + dr = (x + dx)i + (y + dy)j + (z + dz)k, the Potential Energy changes by dV = V (x + dx,y + dy, z + dz) V (x,y, z).

5 For small increments dx,dy and dz, and dV , can be expressed, using Taylor series expansions, as V V V dV = x dx + y dy + z dz = V dr , where dr = dxi + dyj + dzk. This equation expresses the fundamental property of the gradient. The gradient allows us to find the change in a function induced by a change in its variables. If we write V (x , y, z) = C, for some constant C, this is the implicit equation of a surface, which is called a constant Energy surface. This surface is made up by all the points in the x, y, z space for which the function V (x , y, z) is equal to C. It is clear that if a particle moves on a constant Energy surface, dV = 0, since V is 2 constant on that surface. Therefore, when a particle moves on a constant Energy surface, dr will be tangent to that surface, and since 0 = dV = Vdr , we have that V is perpendicular to any tangent to the surface.

6 This situation is illustrated in the picture below for the two dimensional case. Here, the constant Energy surfaces are contour curves, and we can see that the gradient vector is always normal to the contour curves. Note Gradient operator in cylindrical coordinates The gradient operator can be expressed in cylindrical coordinates by writing x = r cos , y = r sin , and r = x2 + y2, = tan 1(y /x). Thus, applying the chain rule for differentiation, we have ( ) r ( ) r ( ) ( ) sin ( ) x = x r + x = cos r r r ( ) r ( ) r ( ) ( ) cos ( ) = + = sin + . y y r y r r r If we note that i = cos er sin e and j = sin er + cos e , we have that ( ) ( ) er +1 ( ) e + ( ) . r r z An expression for spherical coordinates can be derived in a similar manner.

7 Conservation of Energy When all the Forces doing work are Conservative , the work is given by (1), and the principle of work and Energy derived in the last lecture, T1 + W12 = T2 , reduces to, T1 + V1 = T2 + V2 3 or more generally, since the points r1 and r2 are arbitrary, E = T + V = constant . (2) Whenever applicable, this equation states that the total Energy stays constant, and that during the motion only exchanges between kinetic and Potential Energy occur. In the general case, however, we will have a combination of Conservative , F C , and non- Conservative , F NC , Forces . In this case, the work done by the Conservative Forces will be calculated using the corresponding Potential function, , W C = V1 V2, and the work done by the non- Conservative Forces will be path 12 dependent and will need to be calculated using the work integral.

8 Thus, in the general case, we will have, r2 T1 + V1 + F NC dr = T2 + V2 . r1 The work done by non- Conservative Forces which oppose the motion is negative. Therefore the sum of T2 +V2 will be less than T1 + V1. Examples of Conservative Forces Gravity near the earth s surface On a flat earth , the specific gravity g points down (along the -z axis), so F = mgk. Call V = 0 on the surface z = 0, and then z V (z) = ( mg) dz, V (z) = mgz . 0 For the motion of a projectile, the total Energy is then 1 E = mv 2 + mgz = constant .2 1 Since vx and vy remain constant, we also have mv 2 + mgz = constant. 2 z Gravity In a central gravity field M m M m F = G er = ( G ) , r2 r and so, taking V (r ) = 0, M m V m . = Gr = r where G is the universal gravitational constant and = M G is the strength of the gravitational field from a central body of mass M.

9 4 Spring Force For small displacement, the force supported by a spring is F = kx. The elastic Potential Energy of the spring is the work done on it to deform it an amount x. Thus, we have x 1 V = 0 kx dx =2 kx2 . If the deformation, either tensile or compressive, increases from x1 to x2 during the motion, then the change in Potential Energy of the spring is the difference between its final and initial values, or, V =1 k(x22 x12) .2 Gravity Potential for a Rigid Body In this case, the Potential Vi associated with particle i is simply Vi = migzi, where zi is the height of particle i above some reference height. The force acting on particle i will then be F i = Vi. The work done on the whole body will be n 2 nn r i F i dri = ((Vi)1 (Vi)2) = mig((zi)1 (zi)2 = V1 V2 , 1 i=1 ri i=1 i=1 where the gravity Potential for the rigid body is simply, nV = migzi = mgzG , i=1 where zG is the z coordinate of the center of mass.)

10 It s obvious but worth noting that because the gravitational Potential is taken about the center of mass, the inertia plays no role in determining the gravitational Potential . Example Cylinder on a Ramp We consider a homogeneous cylinder released from rest at the top of a ramp of angle , and use conservation of Energy to derive an expression for the velocity of the cylinder. Conservation of Energy implies that T +V = Tinitial +Vinitial. Initially, the kinetic Energy is zero, Tinitial = 0. Thus, for a later time, the kinetic Energy is given by T = Vinitial V = mgs sin , 5 where s is the distance traveled down the ramp. The kinetic Energy is simply T = 21 IC 2, where IC = IG + mR2 is the moment of inertia about the instantaneous center of rotation C, and is the angular velocity.