Figure 1.1 Torsional Pendulum

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1 Torsional Pendulum Torsional Pendulum Torsional Pendulum (without masses) (with masses at d1 (with masses at d close to the axis) away from the axis) Figure 1.1 Torsional Pendulum Department of Physical Sciences, Bannari Amman Institute of Technology, Sathyamangalam 1

2 Rigidity Modulus - Torsional Pendulum Expt. No. : Date: AIM To find the moment of inertia of the disc and the rigidity modulus of the material of the suspension wire subjected to torsional oscillations. GENERAL OBJECTIVE To assess the shear elastic behavior of a given material using torsional pendulum SPECIFIC OBJECTIVES 1. To measure the time period of the torsional pendulum. To calculate the moment of inertia of the disc 3. To measure the radius of the wire using screw gauge 4. To determine the rigidity modulus of the wire using the formula APPARATUS REQUIRED Metallic disc Brass/steel wire Symmetrical masses Stop clock Metre scale Screw gauge Stand FORMULA 1. Moment of inertia of the disc I T0 m( d d1 ) ( T T ) (kg m ) 1 Department of Physical Sciences, Bannari Amman Institute of Technology, Sathyamangalam

3 TABLE I L To find T O and ( T T0 T 1 ) S.No Length of the suspension wire L (10 - m) Distance between the masses (10 - m) Time for 10 oscillations(s) Trial I Trial II Mean Period T (s) L / T (m/s ) T ( T - T1 ) No mass T0 = 1 d1 = T1 = d = T = No mass T0 = d1 = T1 = d = T = L Mean = m/s T O Mean ( T T0 T 1 = ) Department of Physical Sciences, Bannari Amman Institute of Technology, Sathyamangalam 3

4 . Rigidity modulus of the material of the wire n 8 I L 4 r T (N/m ) 0 Symbol Explanation Unit I Moment of inertia of the disc kg m m Value of one of the two masses placed on the disc kg d 1 Closest distance between the center of mass and the wire m d Farthest distance between the center of mass and the wire m T 0 Time period without any mass placed on the disc s T 1 Time period when two masses are placed at a distance d 1 s T Time period when two masses are placed at a distance d s n rigidity modulus of the material of the suspension wire N/m L length of the suspension wire m r radius of the suspension wire m PREREQUISITE KNOWLEDGE 1. Torsional oscillation It is the periodic oscillation produced by twisting the wire. Torsional pendulum A body suspended from a rigid support by means of a thin elastic wire is called a torsional pendulum 3. Rigidity modulus Rigidity modulus is defined as ratio of shearing stress to shearing strain 4. Moment of Inertia It is a measure of a body s tendency to resist the angular acceleration 5. Torque The product of moment of inertia and angular acceleration (or) the rate of change of angular momentum Department of Physical Sciences, Bannari Amman Institute of Technology, Sathyamangalam 4

5 LEAST COUNT FOR SCREW GAUGE Least Count (LC) = Pitch Number of head scale divisions Pitch = Distance moved Number of rotations given = 5 mm 5 = 1mm LC = 1 mm 100 = 0.01mm TABLE - II To measure the radius of the wire using screw gauge Zero Error (ZE) : division Zero Correction (ZC) : mm S. No. 1. Pitch Scale Reading PSR ( 10-3 m) Head Scale Coincidence HSC (divisions) Observed Reading OR = PSR + ( HSC LC ) (10-3 m) Correct Reading CR = OR ± ZC ( 10-3 m) Mean diameter (d) = m Radius (r = d/) = m Department of Physical Sciences, Bannari Amman Institute of Technology, Sathyamangalam 5

6 PROCEDURE 1. One end of the material of the wire is clamped using a vertical chuck. A metallic disc is attached to the other end of the wire.. The length of the suspension wire is fixed to a particular value. The disc is slightly twisted so that the disc executes torsional oscillations. 3. The time taken for ten oscillations is noted using a stop clock. Two trials are taken for each length. The mean time period T 0 is found. 4. Now two equal masses are placed on either side of the center of the disc close to the suspension wire. 5. The closest distance d 1 from the center of the mass to the center of the suspension wire is found. 6. Now the disc is made to execute torsional oscillations. The time taken for 10 oscillations is measured and the time period T 1 is calculated. 7. Now the two equal masses are placed at the edges of the disc. The farthest distance d from the center of the mass to the center of the suspension wire is found. 8. Now the disc is made to execute torsional oscillations. The procedure is repeated and the time period T is calculated. 9. The radius of the wire(r) is found using a screw gauge. Department of Physical Sciences, Bannari Amman Institute of Technology, Sathyamangalam 6

7 OBSERVATION Value of one of the masses placed on the disc m = kg The closest distance between the suspension wire and the center of the mass d 1 = m The farthest distance between the suspension wire and the center of the mass d = m Radius of the suspension wire r = m Mean value of L/T 0 = m/s Mean value of T0 /( T - T1 ) =. CALCULATION Moment of inertia of the disc I T0 m( d d1 ) ( T T ) kg m 1 Department of Physical Sciences, Bannari Amman Institute of Technology, Sathyamangalam 7

8 RESULT The moment of inertia of the disc I = kg m The rigidity modulus of the material of the given wire n = N/m APPLICATIONS Torsional pendulum clocks, shafts in automobiles VIVA VOCE QUESTIONS 1. Why is it called torsion pendulum?. What happens to the period of oscillation when the length of suspension wire is decreased? 3. Define moment of inertia. 4. If the radius of the wire is doubled, what happens to the moment of inertia? 5. Is it possible to determine the rigidity modulus other than solids? STIMULATING QUESTIONS 1. Why is a hollow shaft stronger than a solid shaft of same mass and material?. Identify the role of moment of inertia of the clothes from the beginning to end of cycle in washing machine. Department of Physical Sciences, Bannari Amman Institute of Technology, Sathyamangalam 8

9 Rigidity modulus of the material of the wire n 8 I L 4 r T N/m 0 Department of Physical Sciences, Bannari Amman Institute of Technology, Sathyamangalam 9

10 Department of Physical Sciences, Bannari Amman Institute of Technology, Sathyamangalam 10

Figure 3.1 Young s modulus - Non-uniform bending

Figure 3.1 Young s modulus - Non-uniform bending Figure 3.2 Model Graph Department of Physical Sciences, Bannari Amman Institute of Technology, Sathyamangalam 21 Young s Modulus - Non-uniform Bending Expt.