Experiment using Torsion Pendulum

EXPERIMENT USING TORSION PENDULUM

Determination of the moment of inertia of the disc and rigidity modulus of the wire using torsion pendulum with mass.

To determine the moment of inertia of the disc and the rigidity modulus of the wire, the disc is set into torsional oscillations without any mass over it and the time period of oscillations(T) is measured.

(i.e.) Time period of oscillation without mas

 T = 2π √(I/C)

(or)

 T² = 4π²I / C           …………(1)

Where, I is the moment of inertia of the disc about the axis of rotation and C is the restoring couple.

Now, two equal cylindrical masses are placed over the disc at equal distances say d₁, from the centre of the disc as shown in Fig.4.9 and is set into torsional oscillations. Time period of oscillation (T₁) is measured.

Time period of oscillation when the masses is at a distance

(d₁) = T₁ = 2π √(I₁/C)

(or)

 T₁² = 4π²I₁ / C           …………(2)

Where, I₁ is the moment of inertia of the disc along with the cylindrical masses placed over the disc at a distance d₁.

Now, the cylindrical masses are placed at the edges of the disc at equal distance say d₂ from the centre of the disc as shown in Fig.4.10 and the time period of oscillation (T₂) is measured.

Time period of oscillation when the mass is at a distance

(d₂) = T₂ = 2π √(I₂/C)

(or)

 T₂² = 4π²I₂ / C           …………(3)

Where I₂ is the moment of inertia of the disc along with two equal masses placed at a distance d₂.

a) Moment of Intertia of the disc

From equations (1), (2) and (3), we can write

From the parallel axis theorem we can write the moment of inertia I₁ as

 I₁ = I + 2I_m + 2md₁²               .....(5)

Where,

I_m → Moment of inertia of each cylindrical mass passing through its centre.

M → Mass of the cylindrical weights placed over the disc.

Similarly from the parallel axis theorem, we can write

I₂ = I + 21_m + 2md₂²              .....(6)

From equation (5) and (6), we can write,

I₂ ‒ I₁ = 2m (d₂² ‒ d₁²)            .....(7)

Substituting equation (7) in equation (4)

Moment of inertia of the disc about the axis of rotation

      ………..(8)

b) Rigidity modulus of the wire

From the theory of torsion pendulum, we know

The Rigidity modulus of the wire η = 8πIL / T²r⁴           ………(9)

Substituting equation (8) in (9) we have

Equation (10) represents the rigidity modulus of the wire.