How does friction affect the motion of a pendulum? 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? 
 A: 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.)
