Theory

What is Torsional Oscillation?

A body suspended by a thread or wire which twists first in one direction and then in the reverse direction, in the horizontal plane is called a torsional pendulum.The first torsion pendulum was developed by Robert Leslie in 1793.

The period of oscillation of torsion pendulum is given as,

$$T=2\pi \sqrt{\frac{I}{C}}................(1)$$

Where I=moment of inertia of the suspended body; C=couple/unit twist

But we have an expression for couple per unit twist C as,

$$C=\frac{1}{2}\frac{\pi nr^4}{l}..............(2)$$

Where l =length of the suspension wire; r=radius of the wire; n=rigidity modulus of the suspension wire

Substituting (2) in (1) and squaring,we get an expression for rigidity modulus for the suspension wire as,

$$n=\frac{8\pi lI}{r^4 T^2}....................(A)$$

We can use the above formula directly if we calculate the moment of inertia of the disc, I, as (1/2)MR².

Now, let I₀ be the moment of inertia of the disc alone, and I₁ & I₂ be the moments of inertia of the disc with identical masses at distances d₁ & d₂ respectively. If I₁ is the moment of inertia of each identical mass about the vertical axis passing through its centre of gravity, then

$$I_1=I_0+2\,I^1+2md_1^2................\text{(3)}$$ $$I_2=I_0+2\,I^1+2md_2^2.................\text{(4)}$$ $$I_2-I_1=2m(d_2^2-d_1^2)................\text{(5)}$$

But from equation (1) ,

$$T_0^2=4\pi^2\frac{I_0}{C}............\text{(6)}$$ $$T_1^2=4\pi^2\frac{I_1}{C}............\text{(7)}$$ $$T_2^2=4\pi^2\frac{I_2}{C}.............\text{(8)}$$ $$T_2^2-T_1^2=\frac{4\pi^2}{C}(I_2-I_1).........\text{(9)}$$

Where T₀, T₁, and T₂ are the periods of torsional oscillation without identical masses, with identical masses at positions d₁ and d₂ respectively.

Dividing equation (6) by (9) and using equation (5),

$$\frac{T_0^2}{(T_2^2-T_1^2)}=\frac{I_0}{[I_2-I_1]}=\frac{I_0}{2m(d_2^2-d_1^2)}\qquad\text{(10)}$$

Therefore,The moment of inertia of the disc,

$$I_0=2m(d_2^2-d_1^2)\left(\frac{T_0^2}{T_2^2-T_1^2}\right)\qquad\text{(11)}$$

Now substituting equation (2) and (5) in (9),we get the expression for rigidity modulus 'n' as,

$$n=\frac{16\pi m(d_2^2-d_1^2)}{r^4}\left(\frac{l}{T_2^2-T_1^2}\right)\qquad\text{(12)}$$

Applications of Torsional Pendulum:

1. The working of "Torsion pendulum clocks" (shortly torsion clocks or pendulum clocks) is based on torsional oscillation.
2. The freely decaying oscillation of a torsion pendulum in a medium (like polymers) helps to determine their characteristic properties.
3. New research promises the determination of frictional forces between solid surfaces and flowing liquid environments using forced torsion pendulums.