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16.2 Linear Restoring Force

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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 , and so again have conservation of energy.
If the motion is characterized by mass m, the differential equation obeyed by the system is 

which can be rewritten as

This is exactly the differential equation obeyed by (and cos wt as well).
We may therefore conclude that :

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 Potential Energy function for this system is .
You can again use the conservation of energy to deduce the relation between speed and extension of the spring once you know the energy in the system.

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.