# Simple Harmonic Motion

### The most simple example of an object undergoing Simple Harmonic Motion is a mass suspended from a coiled spring :- ## Force against Displacement

### As seen in the statement above, the unbalanced Force experienced by the Mass undergoing SHM is proportional to the Displacement. If this Force is measured experimentally, then the following graph can be generated :- ### As can been seen above, the graph shows also that the force acts in the opposite direction to the displacement. By introducing a Force constant, we can generate the following :- ## SHM in terms of Acceleration

### By equating the above formula with Newton's Second Law, we can derive a relationship for Acceleration :- ## Example 1 - ### x = 4.0 cm = 0.04 m ### In general terms the constant ( k / m ) is related to the period of the motion by :- ### giving the general formula for acceleration in SHM as :- ## Equations of Motion and SHM

### For ease of explanation, we can compare on object undergoing SHM with an object moving in a circle. ### The Displacement on the vertical axis is equal to :- ### By differentiation, an equation of the Velocity can be obtained :-  ### By further differentiation, the acceleration can also be found :- ### By combining the two equations above and by using the trig identity below, the following can be derived :-   ## Graphical Analysis of SHM

### Displacement - ### Velocity - ### Acceleration - ## Example 1 -

### Find the Time taken for the object to move from an equilibrium position to a Displacement of 0.012 m  ## Simple Pendulum as SHM proof

### The following diagram shows a simple pendulum undergoing Simple Harmonic Motion as well as the Forces acting upon it ### If the angle is small ( less than ~ 10o ) then sinq = q in radians and therefore q = x / L   