Simple Harmonic Motion (Edexcel A Level Further Maths: Core Pure)

What is simple harmonic motion?

• A particle undergoing simple harmonic motion moves along a straight line subject to the following constraints:
• The acceleration of the particle is always directed towards a fixed point on the line of motion
• The acceleration is proportional to the displacement of the particle from the fixed point
• As a result the particle oscillates back and forth along the line around the fixed point
• Many physical systems can be modelled using simple harmonic motion
• One example is an object attached to a spring oscillating in one dimension, when friction, air resistance and other such resistive forces are disregarded

What is the equation that describes simple harmonic motion?

• The standard form of the simple harmonic motion equation is 
• x is the displacement of the particle from the fixed point
• The fixed point is normally indicated by O and is the origin (i.e. zero point) of the coordinate system
• The fixed point O is known as the centre of oscillation
•  is the constant of proportionality, and represents the strength of the force accelerating the particle back towards point O
• The negative sign means that the acceleration is always directed back towards O
• We use  to assure that the constant is positive, and also to simplify the notation for the solution to the equation
•  is the acceleration of the particle
• With simple harmonic motion, Newton’s ‘dot notation’ is often used for the derivatives
• In this notation, and 

What is the solution to the standard simple harmonic motion equation?

• The SHM equation may be solved using the standard techniques for second order differential equations
  • may be rearranged to give the homogeneous second order equation 
  • That has auxiliary equation  with roots 
  • Therefore the general solution is 
    • Initial or boundary conditions given with a question may allow you to find the precise values of the arbitrary constants A and B
• The linear combination of sine and cosine terms in the general solution may be rewritten in the form 
  • This can sometimes be a more useful form for the general solution
  • This form of the equation shows that the particle oscillates around the fixed point O (i.e., around the centre of oscillation)
  • The amplitude of the motion is R
    • 
    • This is the maximum distance that the particle moves away from the fixed point O
  • The period of the motion is 
    • This is the length of time it takes the particle to complete one oscillation
  • The constant α is a ‘phase constant’
    •  
    • If α = 0, then the particle is at point O (i.e., x = 0) when t = 0
    • If  then the particle is at its maximum displacement (i.e., x = R) when t = 0

How can I solve the simple harmonic motion equation to link displacement x and velocity v?

• Because acceleration is the derivative of velocity, and velocity is the derivative of displacement, we may use the chain rule to write

• Substituting that into the standard SHM equation gives

• That form of the SHM equation may be solved using separation of variables:

• • But the velocity is momentarily zero when the particle reaches its maximum displacement
    • After this the particle’s velocity changes direction and the particle heads back towards the centre of oscillation
  • This gives us a boundary condition (because we know that v = 0 when x = R) and allows us to find the value of the constant of integration:

• • Substituting that value of c into the solution and rearranging gives

• • • where R again is the amplitude of the simple harmonic motion
• Note that this version of the solution connects the velocity v to the displacement x, and is independent of the time t
  • Taking square roots gives 
  • Be careful when using this to answer questions
    • The direction of the velocity (plus or minus) will depend on the displacement (plus or minus) and whether the particle is moving towards or away from the centre of oscillation