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An ordinary spring has behavior described by a force law having this form. It possesses a normal length, xe, and if stretched or compressed, it experiences a force of strength k |(x-xe )| pushing or pulling it back toward its normal position. This can be described by the following force law:
We can describe such a force by a potential
energy function V given by which can be rewritten as
This is exactly the differential equation obeyed
by
where c and d are constants that depend on the initial values
of position and velocity of the spring .The general solution
can also be written as the sum of a sine and a cosine term. This system has the interesting property that the parameter
w that appears here, which by the way is a measure of the
frequency of the sinusoidal motion, and its period, depends
only on m and k, and not on the initial conditions. The solution here, that the spring oscillates on for ever is obviously unrealistic; springs stop moving. This is because the model we have just used obeys conservation of energy in the spring motion, while in reality there is friction in the spring motion, and some of the energy in it gets converted into heat. Similarly air resistance changes the motion of a thrown ball, for example in the constant gravity case. We will consider force laws that model friction and air resistance soon. First we consider an important reformulation of Newton’s Laws of motion that can be applied to conservative systems, that is those that are like the two we have considered here, in which energy is conserved. |
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,
and so again have conservation of energy.
(and cos wt as well).
.