Prove that the time period of oscillations of a pendulum is independent of the mass or material of the bob.

First we will treat the pendulum as a point mass (m) a distance L from the axis of rotation. Use newtons second law with the moment of inertia I about the axis of rotation, and alpha is the angular acceleration.
tau=I*alpha
F_g*L*sin(theta)=I*(d^2(theta))/dt^2
-m*g*L*sin(theta)=(m*L^2)(d^2(theta))/(dt^2)
0=L/g*sin(theta)+(d^2(theta))/dt^2
If we assume the angular displacements are small, we can use the small angle approximation.
0=L/g*theta+(d^2(theta))/dt^2
This is a differential equation that has a solution of the form
theta(t)=theta_0*cos(omega*t+phi)
Where the angular frequency is
omega^2=g/L
The period of oscillation is
T=2*pi*omega=2*pi*(g/L)^(1/2)
The period of oscillation is indeed independent of the mass of the pendulum.
https://courses.lumenlearning.com/boundless-physics/chapter/periodic-motion/

https://www.acs.psu.edu/drussell/Demos/Pendulum/Pendula.html