Answer:
Without friction, the equation of motion for a pendulum of length L is,
md2θdt2+mgsin(θ)L=0.
Or for small oscillations, (i.e., sin(θ)≈θ),
md2θdt2+mgθL=0.
Assuming an initial angle θ0 and a pendulum that starts at rest, the solution to this differential equation is,
θ(t)=θ0cos(gL−−√t).
Frictional force adds an additional damping term into the equation of motion,
md2θdt2+λdθdt+mgθL=0,
where λ is a coefficient of kinetic friction.
Assuming an initial angle θ0 and a pendulum that starts at rest, the solution to the damped differential equation is,
θ(t)=θ0e−12λmtcos((gL−λ24m2−−−−−−−−√)t).
(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.)
