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 MassSpring
Oscillator 
