Torsional Pendulum Hi from one of my lab classes I had to calculate the moment of inertia
of an object by turning it into a torsional pendulum, and measuring
its period with a known constant A, using the following equation:
$$I=\frac{A}{4\pi^{2}}T^{2}$$
However, this expression comes from a small angle approximation, and
in the experiment I am suposed to rotate the whole system (resembling
a torsional pendulum) at 90º, which clearly isn't a small angle. Can
someone help me to get an expression of the motion of an undamped
torsional pendulum without approximation, so I can run some simulations
to prove the validity of the small angle approximation? That is, discover
at what angles I get an error below 1%.
 A: The small angle refers to the distortion angle of a small part of the wire. Refer to the figure below for an illustration of the distortion angle $\theta$ as a function of the twist angle $\psi$

By equating the tangential movement at one end due to twisting to the movement due to distortion, one arrives as
$$  r \psi = \ell \theta  \; \\ \boxed{\theta = \frac{r}{\ell} \psi} $$
So even if $\psi = 90^\circ$, the distrotion angle $\theta \ll \psi$ since $r \ll \ell$ for any wire.
A: For a torsional pendulum the wire from which the object is suspended exerts a torque $\tau =- \kappa \, \theta$ where $\kappa$ is the torsional constant and $\theta$ is the angle of twist.
This expression holds for large angles of twist and so in the derivation of the period of the torsional pendulum no small angle approximation has to be applied.  

In derivations where the approximation $\sin \theta \approx \theta$ is made where $\theta$ is the angle in radians if the value of $\theta$ is less than about $0.39$ radian or $22^\circ$ the the approximation is good to about $1\%$. 
