How does the pendulum method for measuring the moment of inertia work? I'm a high schooler that's trying to design an experiment regarding angular momentum, and that requires me to find out the moment of inertia of a complex object.
I searched online and saw that you could experimentally determine it by suspending it from its center of mass with a rope and measuring the oscillations but I don't get how the calculations work. Can someone explain?
 A: We can find the approximate M.O.I. of a complex object by measuring the time period of small oscillations of the complex object about the required axis.
This is because the time period is given by:- $$ T=2\pi\sqrt{\frac{I_{AA'}}{MgL}}$$ where $L$ is the distance of the axis from the center of mass (which is the major variable to employ this method) but I can still show you the calculation involved:-
If the body is slightly tilted from its equilibrium position by an angle 0, mg will exert a restoring torque on it in opposite direction to restore the equilibrium position. Thus restoring torque on body in dotted position after tilting is:-$$ \tau_R= -mgl\sin\theta \\ \approx -mgl\theta$$ If it's angular acceleration is $\alpha$ then we have $$-mgl\theta = I\alpha \\ \implies \alpha= \frac{mgl\theta}{I}$$ comparing this equation with the standard equation of SHM, we get, $$\omega = \sqrt{\frac{mgL}{I}}$$ which gives us the time period $$T= \frac{2\pi}{\omega}= 2\pi\sqrt{\frac{I_{AA'}}{MgL}}$$ regarding your original problem you still have to determine the distance of the center of mass from the point of suspension which can be determined using these methods for 2-D for 3-D it is a bit more difficult to do it practically using this method, though.
