Problem | Teachers' Notes | Hint | Solution | Printable page |

A simple pendulum released from rest from an angle $x$ of less that $20^\circ$ is well-modelled by the linear differential equation
$$m\frac{d^2 y}{dt^2} +k^2y=0$$
Can you work out the meaning of the various letters in the equation, the units of the constant $k$ ,and find the general solution to this equation?

Look at the solution: it keeps on oscillating forever. Clearly, this is an unreasonble assumption, as real pendulums stop swinging due to friction.

Here friction is modelled by a force term which is directly proportional to the velocity of the object, acting in the opposite direction to the motion. Do you think that this is a good modelling assumption? What are its strengths and weaknesses?

With this assumption, the differential equation becomes
$$m\frac{d^2 y}{dt^2} + \lambda \frac{d y}{dt}+k^2y=0$$
Can you see how this equation satisfies all of requirements of a motion under friction? What are the units of $\lambda$, and what features of the motion would contribute to it in practice? It is assumed that $\lambda$ is constant: is this a reasonable assumption?

Although this equation might look complicated it is still only a second order differential equation with constant coefficients, so can be solved using standard methods (and a lot of algebra) to give
$$y = \exp\Big[-\frac{\lambda t}{2m}\Big]\Big[A\cos(\Lambda t)+\sin(\Lambda t)\Big] \quad \Lambda = \sqrt{\frac{k^2}{m}-\frac{\epsilon^2}{4m^2}}$$

Test your skills of differentiation by showing that this equation indeed satisfies the differential equation. More importantly, analyse the solution: how well does it model friction? Consider in particular the amplitude and the frequency of the damped oscillation. Does this meet with your intuitive expectation?

engineering. chemistry. physics. Displacement, velocity and acceleration. Graph sketching. 1st and 2nd order linear differential equations with constant coefficients. Energy. Simple harmonic motion. Differential equations. biology.