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Physics / Mechanics / Experiment 03

Determination of Modulus of Rigidity
using Barton's Apparatus (Vertical)

An interactive laboratory workspace for studying the torsion of a cylindrical rod and determining its modulus of rigidity.

Level: Undergraduate Topic: Torsion and Elasticity

Objective

Section 01

Aim

To determine the modulus of rigidity of the material of a cylindrical rod using Barton's apparatus in the vertical arrangement.

Learning Objectives

After completing this experiment, students will be able to:

  1. Understand the relationship between torque, angular twist, and torsional rigidity.
  2. Explain the working principle of Barton's apparatus in the vertical configuration.
  3. Measure the dimensions of the rod and the effective length of the suspended rod.
  4. Measure the angular twist produced by known couples.
  5. Determine the modulus of rigidity using the appropriate torsion formula.
  6. Verify that the applied torque is proportional to the angle of twist within the elastic limit.
  7. Estimate experimental uncertainties and identify possible sources of error.
  8. Identify precautions and sources of error in torsion experiments.

Theory

Section 02

1. Modulus of Rigidity

When a cylindrical rod is twisted within its elastic limit, it develops an internal restoring torque. The modulus of rigidity, denoted by η (or G), is the ratio of shear stress to shear strain:

η = shear stress / shear strain

A larger value of η indicates greater resistance to torsional deformation.

2. External Couple in Barton’s Apparatus

In Barton’s apparatus, a pulley of diameter D is attached to the rod. Equal masses M are suspended by strings wound around opposite sides of the pulley. Each mass produces a tangential force Mg; the two equal and opposite forces form a couple.

τ = force × perpendicular distance = (Mg)D

Thus, the externally applied torque is:

τexternal = MgD

Here, g is the acceleration due to gravity and D is the distance between the lines of action of the two forces.

3. Restoring Couple and the Fixed-End Correction

For a uniform solid cylindrical rod of radius r, the restoring torque for a length L twisted through θ radians is given by the torsion equation:

τinternal = (πηr4θ) / (2L)
  • η = modulus of rigidity
  • r = radius of the rod
  • L = length of the rod segment under torsion
  • θ = angle of twist in radians

The clamp at the fixed end may undergo a small amount of slip or twist. Barton’s apparatus eliminates this fixed-end error by measuring the twist at two points along the rod. If the pointers are at distances l1 and l2 from the fixed end, and their readings are θ1 and θ2, respectively, then the relevant segment has length l2 − l1 and twist θ2 − θ1.

τ = πηr4(θ2 − θ1) / [2(l2 − l1)]

4. Modulus of Rigidity

At equilibrium, the external and internal torques are equal:

MgD = πηr4(θ2 − θ1) / [2(l2 − l1)]

Therefore, when the angular readings are in radians:

η = 2MgD(l2 − l1) / [πr4(θ2 − θ1)]

5. Using Readings in Degrees

The circular scales usually give angles in degrees. Convert the measured difference before using the torsion equation:

(θ2 − θ1)rad = (θ2 − θ1)deg × π/180

Substitution gives the working formula:

η = 360MgD(l2 − l1) / [π2r4(θ2 − θ1)deg]

If the apparatus specifies the pulley radius R instead of its diameter, use D = 2R. The equivalent expression is:

η = 720MgR(l2 − l1) / [π2r4(θ2 − θ1)deg]

All lengths must be in metres, masses in kilograms, and the final value of η is obtained in pascals (N/m2).

Apparatus Required

Section 03

Apparatus

  • Barton’s apparatus arranged vertically
  • Uniform cylindrical rod with torsion head
  • Load carriers and slotted weights
  • Graduated circular scale and pointer
  • Meter scale
  • Screw gauge for measuring the rod diameter
  • Spirit level and leveling arrangement
Apparatus

Procedure

Section 04

Procedure (Real Experiment)

  1. Fix the cylindrical rod vertically in Barton’s apparatus. Check that the rod, torsion head, and pulley are properly aligned along the same axis.
  2. Determine the least count and zero error of the screw gauge. Measure the rod diameter at several positions and in different orientations, then calculate the mean diameter and the mean radius r.
  3. Determine the least count and zero error of the Vernier calipers. Measure the diameter of the cylindrical pulley or drum and record it as D.
  4. With no load applied, select two graduated scales and their corresponding pointers. Fix them at two different positions along the rod, set both pointers to the zero reading, and measure the separation l2 − l1 between the two scales.
  5. Apply equal masses to the two pans in small, known steps, starting from 0 g and increasing by 500 g per pan. Allow the rod to come to rest before taking each reading.
  6. For each applied mass, record the deflections θ1 and θ2 shown by pointers 1 and 2, respectively, on both sides of each circular scale.
  7. Continue the observations for equal masses from 0 to 2 kg per pan in steps of 0.5 kg. Keep the load within the elastic limit of the rod.
  8. Remove the masses in the same steps and record the corresponding readings during unloading.
  9. For each mass, take the mean of the loading and unloading readings separately for θ1 and θ2, using the readings from both sides of the scales as appropriate.
  10. For a selected mass, such as 1.5 kg per pan, calculate the angle of twist between the two pointers as θ = θ2 − θ1. Use the magnitude of this difference if only the deflection magnitude is required.
  11. Calculate the modulus of rigidity using the degree-based working formula η = 360MgD(l2 − l1)/[π2r4(θ2 − θ1)deg], where M is the mass on each pan.
  12. Repeat the calculation for the remaining observations and report the mean value of η in N/m2.

Simulation

Simulation
Loading simulation...
Modulus of Rigidity / Barton's Apparatus (Vertical)
Interactive applet / Adjust the controls to explore the torque–twist response

Observation Table

Section 06

Result and Analysis

Section 07

Calculation on freehandTool

η = 360MgD(l2 − l1) / [π2r4(θ2 − θ1)deg]

The modulus of rigidity of the rod = N/m2

Modulus of Rigidity Using Barton’s Apparatus – MCQs

Select the most appropriate answer.

10 questions1 mark eachTotal: 10 marks
Question 1The modulus of rigidity η of a material is defined as the ratio of:
Question 2In Barton’s apparatus, equal masses M are suspended on opposite sides of a pulley of diameter D. The external torque applied to the rod is:
Question 3Why do the two equal tangential forces in Barton’s apparatus form a couple?
Question 4Two pointers are located at distances l1 and l2 from the fixed end, with angular readings θ1 and θ2. Which quantities represent the effective length and twist of the rod segment?
Question 5The purpose of taking readings at two points along the rod is primarily to:
Question 6If the angular difference between the two pointers is measured as (θ2 − θ1) degrees, its value in radians is:
Question 7According to the working formula using angular readings in degrees, the modulus of rigidity is:
Question 8If the radius r of the cylindrical rod is doubled while all other quantities remain unchanged, the calculated modulus of rigidity becomes:
Question 9A Barton’s apparatus specifies the pulley radius R instead of its diameter. Which substitution should be made in the working formula?
Question 10A student obtains l1 = 0.20 m, l2 = 0.80 m, θ1 = 10°, and θ2 = 25°. What are the effective length and angular twist?