Stability
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Stability and Oscillations
[Diagram goes here - download the original to see it.] In a game of shove- halfpenny a coin is resting on the edge of a table partly overhanging the edge but nonetheless in a stable equilibrium. Its stability is characterised by the fact that it is not falling- that is, it is not losing or gaining gravitational potential energy. In fact, it is the case that for any object (in conservative system) that there is no charge of potential energy. An object is a subject to conservative forces if its total energy is conserved. Consider an example, an object such as a planet in orbit, where then energy of the object takes the form of either kinetic energy or potential energy or both. If the system is constant total energy is constant, hence Kinetic Energy + Potential Energy = Total Energy [Equation] Now [Equation] But acceleration is [Equation] So [Equation] Hence [Equation]by Newton's second Law. If an object is in equilibrium the resultant force, F, is zero. Hence [Equation] This tells us that an object where there is a possible exchange only between kinetic and potential energy is in a state of equilibrium if, and only if [Equation] That is, the rate of change of potential energy is zero. In fact, this result can be generalised to any situation where potential energy is a fraction of a single variable. If a body is subject only to conservative forces, but it is free to move so that its potential energy, U, is a functional of a variable, X, such that [Equation], then its equilibrium positions are given as solutions to the equation. [Equation]
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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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