rigidity modulus formula in torsional pendulum experiment

The photograph to the right shows the apparatus for the experiment.

The is mounted at the top, and hangs down into the chamber. We show a brass disc hanging from the wire. Its orientation is the same as in the .

You will measure different objects' period of oscillation with the supplied stopwatch.

Barely visible in the photograph is a plexiglas door, which is open in the picture. You will close it before taking data so that air currents do not disturb the motion of the object.

You will measure the period of oscillation for 4 different moments of inertia. The above photograph shows the brass disc oriented for one of the moments of inertia you will measure. Here are the other 3 moments you will use:

You will measure the period of oscillation for each moment of inertia with the stopwatch. A preparatory question below explores the best procedure to use for this measurement.

Although the data collection is simple and straightforward, the analysis of the your results and especially the calculation of the uncertainties in your determinations of the moments of inertia may seem at first to be a bit daunting. A preparatory question below, however, is intended to show you how to make these calculations much simpler than they seem.

We hope that the following questions will guide you in your preparation for the experiment you are about to perform. They are not meant to be particularly testing, nor do they contain any "tricks". Once you have answered them, you should be in a good position to begin the experiment.

• Convert all the above values to SI units.
• What is the percentage error in L ?
• What is the value and error in p = L 2 /12?
• What is the percentage error in p ? How does it compare to the percentage error in L ?
• What is the value, error, and percentage error in a ?
• What is the value and error in a 2 ?
• How does the percentage error in a 2 compare to the percentage error in a ?
• What is the value and error in b 2 ?
• What is the value and error in q = ( a 2 + b 2 )/4?
• For the previous calculation, is one of the errors negligible? If yes, which?
• What is the percentage error in q ? Have you calculated any other percentage errors with this value?
• What is the value and error in r = p + q ?
• What is the value and error in the moment of inertia I = M r ?
• What is the percentage error in I ? Have you calculated any other percentage errors with this value?

As promised, we have prepared a summary of the information presented above. It does not attempt to provide a full discussion, but just reviews the "high points" from above.

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Torsion pendulum test on metal wire.

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🕑 Reading time: 1 minute

Modulus of Rigidity is the coefficient of elasticity of wire for a shearing force. In simple words, rigidity modulus of a metal wire is a measurement of the capability of a material to resist deformation when external tangential (parallel to the surface) force is applied to the metal wire.

Aim of the Test

To determine the rigidity modulus of the suspension wire using torsion pendulum.

Apparatus Required

• Torsion Pendulum
• Cylindrical wire
• Vernier Caliper
• Screw gauge
• Meter scale
• Test Specimens - Steel and Brass Wire.

Torsion Pendulum Principle

For small oscillations of the disc, it is in simple harmonic motion and the formula for simple pendulum holds good.

Where, T = period of oscillation in sec. I = Mass moment of inertia of the rotating system about the longitudinal axis of wire. L = Length of the wire between its grips. N = Modulus of rigidity (Shear modulus).

d = diameter of the given wire in the test

Equations 1 and 2 refer to conditions when no cylindrical weight is added on to the disc and when known cylindrical weights are added, We have,

From which it follows-

(I 2 -I 1 ) is the mass moment of inertia of the cylindrical weights about the axis of rotation of the disc. It is given by,

Where, W = Total weight of cylinders added to the disc. g = Acceleration due to gravity. r = Radius of cylindrical weights. R = distance from center of the cylinder of the cylindrical weight to center of the wire.

Thus from equations (5) and (6)

Test Procedure

Part 1: Determination of Rigidity modulus using Torsion pendulum alone

• The radius of the suspension wire is measured using a screw gauge.
• The length of the suspension wire is adjusted to suitable values.
• The wire for the test is tightened at its bottom to the disc and its top to the bracket.
• The disc is turned and released without the cylindrical weights on it.
• Time for a number oscillation (Say 20) is measured with a stopwatch and the mean period of oscillation ' T 0 ' is determined.

Part 2:  Determination of rigidity modulus and moment of inertia using torsion pendulum with identical masses

• The two identical masses are placed symmetrically on either side of the suspension wire as close as possible to the center of the disc and and the distance d1 is measured which is the distance between the centers of the disc and one of the identical masses.
• Find the time for 20 oscillations twice and determine the mean period of oscillation T 1 .
• The two identical masses are placed symmetrically on either side of the suspension wire as far as possible to the center of the disc and measure d 2  which is the distance between the centers of the disc and one of the identical masses.
• Find the time for 20 oscillations twice and determine the mean period of oscillation T 2 .
• Find the moment of inertia of the disc and rigidity modulus of the suspension wire using the given formulae.

Observation and Calculation

For Part 1 -

Rigidity modulus of the suspension wire =

For Part 2 -

Applications of Torsional Pendulum

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

Gopal Mishra

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