Question

How do you derive the equation τ = k θ for a torsion spring? τ is torque, k is torsion spring’s constant, and θ is displacement in radians. Thanks!  

Answer 1

Torsional pendulum

Simple harmonic motion equation written as

$I\frac{d^2\theta}{dt^2}=c\theta$

We know that

Torque

$\tau=I\frac{d^2\theta}{dt^2}$

Both equation equal we can written as

$\tau=c\theta$

Now c=k

$\tau=k\theta$

Where $\theta$ =displacement (rad)