Stability
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Stability of the Equilibrium
Equations are omitted for technical reasons - download the original pdf
If the potential energy function is a minimum, then the equilibrium point is stable. If the potential energy function is a maximum then the equilibrium point is unstable. [Diagram goes here - download the original to see it.] To explain why - when an object moves away from equilibrium its potential energy is converted to kinetic energy. If the object is in static equilibrium at a minimum of the potential energy function then its total energy is equal to its potential energy. Unless it is given energy, say in the form of a push- then it cannot acquire any further energy to get its motion underway. [Diagram goes here - download the original to see it.] Even when it is given energy, if the introduction of energy is small ( a small push ) then it only has energy to "climb" part of the potential energy "hill" and once it has done s it must return to the minimum since the highest point it reaches on the potential energy "slope" is nonetheless, an unstable equilibrium. The object subsequently performs small oscillations about the equilibrium position- that is, assuming hat the system is conservative (after the initial "push" of course ) and no further energy can leak in or out. [Diagram goes here - download the original to see it.] In order to get the object to leave the stability of the potential energy "well" it must be sufficient energy to overcome a neighbouring maximum. [Diagram goes here - download the original to see it.] In fact potential energy can also be a function of more than one variable. The case considered so far have been one dimensional cases-that is, cases exhibiting one degree of freedom
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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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