A Level Further Maths: Simple Harmonic Motion - SHM: Light Damping

The following example shows how to model light damping by using a second order differential equation.

The second-order differential equation modelling light damping is given by:

\[ \ddot{x} + 0.5 \dot{x} + 4 x = 0 \]

The characteristic equation is:

\[ m^2 + 0.5m + 4 = 0 \]

Solving for \(m\), we get:

\[ m = -0.25 \pm i \sqrt{3.9375} \]

Therefore, the general solution to the differential equation is:

\[ x(t) = e^{-0.25t} \left( \cos(\sqrt{3.9375} \, t) + 0.1259 \sin(\sqrt{3.9375} \, t) \right) \]

where \( A = 1 \) and \( B \approx 0.1259 \) are determined by the initial conditions:


Simple Harmonic Motion - light damping

  • Initial displacement \( x(0) = 1 \) m
  • Initial velocity \( \dot{x}(0) = 0 \) m/s

The above equation produces a curve that shows light damping in simple harmoinc motion - SHM.

If you want to play with it interactively, please use the following simulation:

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