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Mass-Spring Oscillator
The elastic constant of a spring is the force per unit extension.  
Think of it as a measure of how "strong" the spring is; the number of Newtons force needed to stretch the spring by, say 1cm.  
Therefore the elastic constant, k is given by   
   
The diagram below shows a mass hanging on a spring, initially at its equilibrium position, being pulled down to cause an extra extension, x.   
   
  When the mass is displaced downwards (as shown above) the restoring force acting on it, F, is of magnitude kx, upwards.    
If it is displaced upwards the net force is downwards. Therefore, to describe this situation we write   
   
so that the equation describes a force with is always in the opposite sense to the displacement.    
   
If the mass is released after having been given a displacement, x, then its initial acceleration will be   
   
   
So (as we might have guessed!) the motion is s.h.m. because the acceleration is directly proportional to x and always directed in the opposite sense to x.   
By comparison with the basic s.h.m. equation   
   
we see that, in the case of a mass on a spring, the constant relating acceleration to displacement 2) is given by   
   
and, remembering that  
   
we can see that the time period of this mass-spring oscillator is given by   
   
   
We are assuming here that the displacement of the mass is small enough for the spring to behave perfectly elastically.   
   
For another example of s.h.m., see The Simple Pendulum   
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