To derive the pendulum’s period of oscillation, it is best to analyze it as a rotational motion.
We will still be applying N2L, but Instead of force F, inertia m and acceleration a, we have to work with torque , moment of inertia I (a body’s resistance to rotate, which for the pendulum is mL2) and angular acceleration .
If θ is small, we canmake the approximation. (Yes. This means that our formula is valid only for small amplitude oscillations)
Comparing this with the SHM equation (), we can deduce that
Hence, the natural frequency is and the natural period is .
From , we can see that the natural frequency depends on the (angular) acceleration per unit (angular) displacement ratio. A higher g increases the restoring force (torque), whereas a shorter L decreases the (moment of) inertia, both resulting in a higher (angular) acceleration per unit (angular) displacement.
The mass of the pendulum, surprisingly, does not affect the period. This is because the restoring torque and moment of inertia (mL2) are both proportional to m. So m cancels itself out when it comes to angular acceleration. (This is similar to how all object free fall at g regardless of mass).