Stability
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Use of the second derivative of potential energy.
Equations are omitted for technical reasons - download the original pdf
The equation if motion for some oscillating systems can be obtained from the second derivative of the potential energy function. Assuming that the system is conservative, we have [Equation] where F is the total force and U is the potential energy. Let [Equation] represent a point of stable equilibrium. If the object is displaced from the stable equilibrium it has been given energy (through an impulse). However, at the point of stable equations it still experiences zero force. As it moves away from equilibrium it experiences a force that "pushes" it back towards the equilibrium a restorative force. This force acts in the opposite direction to the displacement. Let [Equation] the gradient of the force function [Equation]be approximated by a tangent. [Diagram goes here - download the original to see it.] The gradient of [Equation] is [Equation] slope of tangent at [Equation] [Equation] Note that the gradient in the diagram is negative. However [Equation] is also negative when [Equation] as here, so the equation is correct. That is [Equation] Since[Equation] we have [Equation] Replacing [Equation]we obtain [Equation] as the approximation to the equation of motion of the object about its equilibrium position. This is the equation of simple harmonic motion. The fact that [Equation] represents a stable equilibrium point means that [Equation] That is, not negative. However it can be equal to zero, in which case the approximation by simple harmonic motion breaks down.
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Contents of
Stability
1 Stability and Oscillations
2 Stability of the Equilibrium
3 Small Oscillations about an equilibrium position
4 Use of the second derivative of potential energy.
5 Oscillations involving rotation
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