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- The Torsional Pendulum

The Torsional Pendulum Experiments were performed to measure shear modulus (G) of the metal by making it suspension wire of a torsional pendulum. The measurements were done by using dynamical method. It was experimentally found that period of oscillation of tosional pendulum was independent of the angle of twist. Further, shear modulus of the metal forming suspension wire was ~ 700 MPa or 0.7 GPa. Error analysis showed the error was within + 2.81%.

Objective

The experiment was aimed at measuring the shear modulus (G) of the metal that formed the suspension wire of a torsional pendulum. This was done by measuring the period of oscillation of the pendulum and using suitable physical equations.

Theory and Formulae:

Let us consider a long and thin metal wire (length L and radius r), clamped at one end. When a torque is applied at the free end, it twists through an angle as per the following formula

where is known as torsional rigidity and this is related to the shear modulus (G) of the suspension wire by the following relationship.

Now if a cylindrical mass is attached at the free end of the suspended wire, this forms a torsional pendulum and period of oscillation (T) of such a torsional pendulum is given as

Where I is moment of inertia of the cylindrical mass attached at the bottom end and is given by

I = (1/2)MR2, with M being mass and R being radius of the cylindrical mass.

Therefore,

Thus one can measure period of oscillation of a torsional pendulum and then calculate the shear modulus of the metal forming the suspension wire.

Materials and Procedure:

A metal wire of radius r = 1.39 mm and length L = 58 cm was used as suspension wire. Cylindrical blocks of radius R = 7.25 cm and different masses 1.41 kg, 2.82 kg, 4.21 kg and 5.59 kg were used to form torsional pendulum. A stop watch was used to measure the period of oscillation of the torsional pendulum.

The pendulum was given a twist of 180o and period of 10 oscillations were measured using stop watch and the period for 10 oscillations was found to be 53.15 s.

This was repeated for a twist of 360o and this time the period for 10 oscillations was found to be 53.46 s.

The small difference can be due to human error in time measurements and thus it was concluded that period of oscillation is independent of the angle of twist.

Period of 10 oscillations were measured for the torsional pendulum with different cylindrical masses attached to it.

The period of oscillations were recorded against the mass of the suspended cylindrical weight.

T2 was plotted against the suspended mass and the slope of this linear curve was used to calculate shear modulus of the metal forming the suspension wire.

Data and Calculations

The data pertaining to the suspension wire, attached mass and measured values of the period of oscillations are presented below.

Suspension Wire Length L = 0.58 m Radius r = 1.39 mm = 1.39*10-3 m

Suspended Mass Radius R = 0.0725 m

Period of Oscillation (T) vs Suspended Mass (m)

Mass (kg)

Period of Ten Oscillations (s)

Period of Oscillation T (s)

1.41

53.46

5.346

2.82

70.18

7.018

4.21

83.59

8.359

5.59

94.71

9.471

The chart showing T2 vs. suspended mass (M) is the following.

Now gradient of T2 vs M line is 14.64

Therefore,

Therefore, Shear Modulus (G) of the metal forming the suspension wire is

Mpa

Error Analysis

Error in measurement is due to least count of the measuring instrument and this error propagates through the calculations. Therefore, error analysis has been done by tracking all the formulae used in determining the shear modulus.

The first calculation was slope of the line (m) - T2 vs M

Error for this (m) will be given as

Next calculation was that of the torsional rigidity from this slope (m)

Error for this () will be given as

Finally error for the shear modulus G will be given as

Now, ; ; ; and

Therefore

Hence error in the measurement of the shear modulus is +2.81%.

Conclusions:

Based on these experiments it can be concluded that period of oscillation of a torsional pendulum is indipendent of the twist angle. Further it was found that dynamical method of measurement of shear modulus of a metal is very simple and elegant method. The error analysis shows that error in the measurement of shear modulus was just 2.81%. However, this does not take into account the fact that we have neglected the moment of inertial of the suspension wire which also incorporates some error in the measurement.

Objective

The experiment was aimed at measuring the shear modulus (G) of the metal that formed the suspension wire of a torsional pendulum. This was done by measuring the period of oscillation of the pendulum and using suitable physical equations.

Theory and Formulae:

Let us consider a long and thin metal wire (length L and radius r), clamped at one end. When a torque is applied at the free end, it twists through an angle as per the following formula

where is known as torsional rigidity and this is related to the shear modulus (G) of the suspension wire by the following relationship.

Now if a cylindrical mass is attached at the free end of the suspended wire, this forms a torsional pendulum and period of oscillation (T) of such a torsional pendulum is given as

Where I is moment of inertia of the cylindrical mass attached at the bottom end and is given by

I = (1/2)MR2, with M being mass and R being radius of the cylindrical mass.

Therefore,

Thus one can measure period of oscillation of a torsional pendulum and then calculate the shear modulus of the metal forming the suspension wire.

Materials and Procedure:

A metal wire of radius r = 1.39 mm and length L = 58 cm was used as suspension wire. Cylindrical blocks of radius R = 7.25 cm and different masses 1.41 kg, 2.82 kg, 4.21 kg and 5.59 kg were used to form torsional pendulum. A stop watch was used to measure the period of oscillation of the torsional pendulum.

The pendulum was given a twist of 180o and period of 10 oscillations were measured using stop watch and the period for 10 oscillations was found to be 53.15 s.

This was repeated for a twist of 360o and this time the period for 10 oscillations was found to be 53.46 s.

The small difference can be due to human error in time measurements and thus it was concluded that period of oscillation is independent of the angle of twist.

Period of 10 oscillations were measured for the torsional pendulum with different cylindrical masses attached to it.

The period of oscillations were recorded against the mass of the suspended cylindrical weight.

T2 was plotted against the suspended mass and the slope of this linear curve was used to calculate shear modulus of the metal forming the suspension wire.

Data and Calculations

The data pertaining to the suspension wire, attached mass and measured values of the period of oscillations are presented below.

Suspension Wire Length L = 0.58 m Radius r = 1.39 mm = 1.39*10-3 m

Suspended Mass Radius R = 0.0725 m

Period of Oscillation (T) vs Suspended Mass (m)

Mass (kg)

Period of Ten Oscillations (s)

Period of Oscillation T (s)

1.41

53.46

5.346

2.82

70.18

7.018

4.21

83.59

8.359

5.59

94.71

9.471

The chart showing T2 vs. suspended mass (M) is the following.

Now gradient of T2 vs M line is 14.64

Therefore,

Therefore, Shear Modulus (G) of the metal forming the suspension wire is

Mpa

Error Analysis

Error in measurement is due to least count of the measuring instrument and this error propagates through the calculations. Therefore, error analysis has been done by tracking all the formulae used in determining the shear modulus.

The first calculation was slope of the line (m) - T2 vs M

Error for this (m) will be given as

Next calculation was that of the torsional rigidity from this slope (m)

Error for this () will be given as

Finally error for the shear modulus G will be given as

Now, ; ; ; and

Therefore

Hence error in the measurement of the shear modulus is +2.81%.

Conclusions:

Based on these experiments it can be concluded that period of oscillation of a torsional pendulum is indipendent of the twist angle. Further it was found that dynamical method of measurement of shear modulus of a metal is very simple and elegant method. The error analysis shows that error in the measurement of shear modulus was just 2.81%. However, this does not take into account the fact that we have neglected the moment of inertial of the suspension wire which also incorporates some error in the measurement.