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

Solutions to SHM Equations

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