Conservative Forces and Conservation of Energy (Proof)
In this short section we will
a) define conservative forces and
b) prove that such forces conserve a quantity we call energy for constant mass systems.
A conservative force is defined as:
Where is a function of the coordinates.Now we multiply both sides by . We get:
Where we used Newton’s second law in the first equal sign. It is left to the reader to prove:
Where is the velocity. Using the equations above we have:
Defining energy to be the sum of the kinetic energy and the potential energy we have:
In other words , the standard conservation of energy.
For the first equality we have:
For the second equality we have:
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