[Physics] How does friction affect the motion of a pendulum

Question

I would like to know what is the difference in the equation of motion of a pendulum in the presence or the absence of frictional forces. And how this translates to the solution of those equations?

Answer 1

Without friction, the equation of motion for a pendulum of length L is,

$$ m\frac{d^2\theta}{dt^2} + \frac{mg \sin(\theta)}{L} = 0. $$

Or for small oscillations, (i.e., $\sin(\theta) \approx \theta$),

$$ m\frac{d^2\theta}{dt^2} + \frac{mg \theta}{L} = 0. $$

Assuming an initial angle $\theta_0$ and a pendulum that starts at rest, the solution to this differential equation is,

$$ \theta(t) = \theta_0 \cos( \sqrt{\frac{g}{L}} t). $$

Frictional force adds an additional damping term into the equation of motion,

$$ m\frac{d^2\theta}{dt^2} + \lambda \frac{d\theta}{dt}+\frac{mg \theta}{L} = 0, $$

where $\lambda$ is a coefficient of kinetic friction.

Assuming an initial angle $\theta_0$ and a pendulum that starts at rest, the solution to the damped differential equation is,

$$ \theta(t) = \theta_0 e^{-\frac12 \frac{\lambda}{m} t} \cos\left(\left( \sqrt{\frac{g}{L} - \frac{\lambda^2}{4m^2}}\right) t\right). $$

(Note: If you would like to consider closed form solutions for large angles, I would recommend consulting the mathematics section. The solutions to that problem are called elliptic integrals of the first kind.)