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The Simple Pendulum
A simple pendulum is a mass (considered to be a point mass) on the end of a light inextensible rod or string.  
   
The diagrams below represent a simple pendulum displaced from its equilibrium position, O through a small angle, θ.  
   
The diagram on the right is a magnified view of the mass (often called the "bob" of the pendulum) in order to show the forces acting on it.  
At this magnification the arrow representing the force of gravity should really be even longer if any scale is to be adhered to but it's just to show the general idea.  
   
     
   
The restoring force (the force pulling the pulling the pendulum back towards the equilibrium position) F, is a component of the force of gravity acting on the bob.  
   
Therefore, the magnitude of this force is  
 
This force produces an acceleration, a, given by    
 
However, if the angle, θ is small (by that, we generally mean 3 or less) then sinθ is very nearly equal to θ in radians (see here for justification of this approximation).  
Therefore we can write   
 
Also, it is clear that, if x is positive, F acts in the negative sense and when x is negative, F acts in the positive sense, so to describe this situation, we must write  
 
   
We conclude that, for small angles of oscillation, the motion of a simple pendulum is s.h.m.  
   
By comparison with the basic s.h.m. equation    
 
we can see that, in the case of the pendulum, the constant of proportionality between acceleration and displacement is  
 
and since ω = 2π/T, the time period of a simple pendulum is given by  
 
This gives a rather simple way of measuring the acceleration due to gravity at any given place on the earth's surface.  
Simply measure the time period of a pendulum of known length.  
   
For another example of s.h.m., see Mass-Spring Oscillator  
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