# A pendulum swings through an arc of 36°.  If the displacement is equal on each side of the equilibrium position, what's the amplitude of this vibration?

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A pendulum swinging symetrically about its equilibrium position is an example of simple harmonic motion.  Simple harmonic motion is motion of an object which repeats a symetric, regularly repeating pattern in a specific amount of time called its Period.  If losses due to non-conservative forces are ignored, the period of motion will remain constant and the object will repeat the pattern until it is stopped by an outside force.

Objects which exhibit simple harmonic motion (SHM) have a reference position known as its equilibrium position.  The motion moves away from that point in equal distances on either side.  This is what makes the motion symetric.  The maximum distance away from the equilibrium position the object reaches before it begins to return to the equilibrium is called the amplitude.  To determine the amplitude of a pendulum undergoing SHM we need to do a little geometry and trigonometry.

When the pendulum is at its maximum swing, at its amplitude, it will be a height H above its lowest position L (L being the length of the pendulum rope).  You can then create a right triangle where L is the hypotnuse, L-H being the altitude, and A (amplitude) is the base.  To get A, we use A = Lsin(theta), where theta is the maximum angle of displacement measured from the equlibrium position.

So, to find the amplitude of a pendulum which swings 18 degrees on either side of the equilibrium we would simply solve the equation

A = Lsin(18).  Of course, to do this you must know the length of the pendulum from the pivot point to the center of the mass.

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