To determine modulus of rigidity of material of a given wire by using Torsional pendulum.

To determine the modulus of rigidity (shear modulus) of the material of a given wire using a torsional pendulum, both with and without an additional disc.

• Torsional pendulum setup
• Circular disc with a hole at the center
• Additional removable circular disc
• Wire of the material under test
• Vernier caliper
• Micrometer screw gauge
• Digital stopwatch
• Meter scale
• Weight balance
• Support stand with clamp
• Set square

Figure 1: Schematic diagram of the torsional pendulum setup showing the wire, disc, and support.

The torsional pendulum consists of a wire suspended vertically with a disc attached to its lower end. When the disc is rotated and released, the wire experiences a torque that tends to bring the system back to its equilibrium position. This results in an oscillatory motion with a time period that depends on the torsional constant of the wire and the moment of inertia of the disc.

The torque ($\tau$) in a wire is proportional to the angle of twist ($\theta$) and is given by:

where $\kappa$ is the torsional constant of the wire.

For a cylindrical wire, the torsional constant is related to the modulus of rigidity ($\eta$) by:

where:

• $r$ is the radius of the wire
• $L$ is the length of the wire

The time period ($T$) of oscillation of the torsional pendulum is given by:

where $I$ is the moment of inertia of the disc.

For a solid circular disc of mass $M$ and radius $R$, the moment of inertia is:

The modulus of rigidity ($\eta$) can be determined using the following formula:

where:

• $\eta$ is the modulus of rigidity of the wire material
• $L$ is the effective length of the wire
• $I$ is the moment of inertia of the disc
• $T$ is the time period of oscillation
• $r$ is the radius of the wire

For the experiment with two discs:

1. With one disc having moment of inertia $I_1$, the time period is $T_1$.
2. With both discs (total moment of inertia $I_1 + I_2$), the time period is $T_2$.

From these two measurements, we can write:

By solving these equations:

And finally:

A. Preliminary Measurements

B. Experiment with Single Disc

C. Experiment with Two Discs

Table 1: Measurement of Wire Dimensions

Table 2: Measurement of Disc Dimensions

Table 3: Time Period Measurements with Single Disc

Table 4: Time Period Measurements with Two Discs

1. Calculate the mean radius of the wire (r) in meters.
2. Calculate the mean time periods T₁ and T₂ in seconds.
3. Calculate the moments of inertia:
   • $I_1 = \frac{1}{2}M_1R_1^2$ for the first disc
   • $I_2 = \frac{1}{2}M_2R_2^2$ for the second disc
4. Calculate the torsional constant ($\kappa$):
   $$\kappa = \frac{4\pi^2I_2}{T_2^2 - T_1^2}$$
5. Calculate the modulus of rigidity ($\eta$):
   $$\eta = \frac{2L\kappa}{\pi r^4}$$

Sample Calculation:

Let's assume the following values:

• Radius of wire, r = 0.5 mm = 0.0005 m
• Length of wire, L = 1.0 m
• Mass of first disc, M₁ = 200 g = 0.2 kg
• Radius of first disc, R₁ = 8 cm = 0.08 m
• Mass of second disc, M₂ = 300 g = 0.3 kg
• Radius of second disc, R₂ = 8 cm = 0.08 m
• Mean time period with first disc, T₁ = 2.5 s
• Mean time period with both discs, T₂ = 3.2 s

Step 1: Calculate moments of inertia

Step 2: Calculate torsional constant

Step 3: Calculate modulus of rigidity

The modulus of rigidity ($\eta$) of the material of the given wire is __________ N/m² or Pa.

(Fill in the blank with your calculated value.)