# Does the small angle approximation apply to torsional pendulums?

I learned that the period of a Torsional pendulum is equal to $$T = 2\pi\sqrt{\frac{I}{\kappa}}$$ I know the simple pendulum has a similar expression which assumes that $$\sin\theta=\theta$$ as long as $$\theta$$ is less than $$10^{\circ}$$.

Is this also the case for torsional pendulums? In that case what is the largest value of $$\theta$$ through which the mass can be twisted ?

• There is nothing special about 10 degrees. The smaller the angle, the better the approximation. It’s an approximation for any angle except 0. Feb 26, 2020 at 18:17
• For a torsional pendulum, the question is about the material of the wire and how linear the relationship is between twist angle and restoring torque. It will never be perfectly linear. The “limit” depends on how much accuracy you want. Feb 26, 2020 at 18:22

Although, as G Smith comments, the relationship between twist angle and restoring torque is never perfectly linear, it is usually linear (that is the generalised Hooke's law applies) to a very good approximation up to at least an angle of twist of 2$$\pi$$ per metre, for a wire of less than a millimetre diameter.
For a pendulum a small angle approximation is needed for a quite different reason. The restoring torque is due to the pull of gravity on the pendulum, and is easily shown to be proportional to $$\sin \theta$$ in which $$\theta$$ is the angular displacement of the pendulum from the vertical. We have SHM only when $$\theta$$ is small enough for $$\sin \theta = \theta$$ to be a good approximation. For example, for $$\theta=0.1745\ (=10°)$$ the difference between $$\theta$$ and $$\sin \theta$$ is about 0.5%.
• I did mean per metre length of wire, but I suspect that 2$\pi$ isn't a maximum, far from it for very thin wires. Sorry, I can't provide a source. What I claimed is remembered from some experiments I did many years ago with wires under torsion. Feb 26, 2020 at 19:43
For a simple pendulum, $$sin\theta\approx\theta$$ is just an approximation for any nonzero angle. There is nothing special about 10 degrees. The angular “limit” for swinging the pendulum depends on how accurately you are measuring the period, if you are expecting that period to be given by the small-angle approximation.