# Derivation of oscillation period and gravitational acceleration

Assumptions presented in the theory of simple pendulum the differential equation describing the motion of a simple pendulum is presented in first row on figure 1, where Newton's second law of motion is applied to a rotational system (not covered in the theory section of this simulation). Here the mass m cancels, i.e. the motion is independent of the attached mass.

For small displacements (small angle approximation) the equation can be further simplified, as presented in second row on figure 1, which is the differential equation of a harmonic oscillator. With the boundary conditions (maximum displacement at t=0 and zero angular velocity at t=0) the solution for the displacement θ(t) can be determined, as in the third row on figure 1.

From the frequency, the oscillation period can be calculated which can be used to determine the gravitational acceleration g by measuring the oscillation period (and known thread length L), as depicted in the last row on figure 1.

Figure 1: Equations for derivations of oscillation period and gravitational acceleration.