In a Torsional Pendulum an object is suspended from a wire with rigidity coefficient C. If such a wire is twisted by an angle θ, due to its elasticity it exerts a restoring torque τ = Cθ on the twisted object attached to it. A general torsional pendulum is shown in figure. Here a disc D of radius R and mass M is attached to a stiff wire whose other end is suspended from ceiling as shown.
From the equilibrium position of this disc if it is twisted by an angle θ as shown, the wire applies a restoring torque on it, which is given as
τR = -Cθ [–ve sign,for restoring nature]
If during restoring motion the angular acceleration of disc is α, we can write τ = Iα where I is the moment of inertia of disc about its central axis, thus we have
Equation-(1) resembles with the basic differential equation of SHM in angular form thus we can state the angular frequency of this SHM is
ω = √C/I .....(2)
Thus the period of SHM is
In the above cases of some pendulums we’ve discussed, how to find the angular frequency and time period of a body in SHM. Now we take some similar examples of physical situations in which an object is in SHM.
