Abstract:

The aim of a trifilar experiment is to evaluate the moment of inertia of a cylindrical/circular plate using apparatus set up in the trifilar suspension, then to compare it to theoretical results in order to judge accuracy, to then investigate how various objects placed in different locations on the system affect the period of oscillation of the plate.

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Introduction

Inertia is defined as a measure of the resistance of a mass/body to angular acceleration. A key part is that the resulting moment administers the analysis of rotational dynamics with use of the equation M=I? which grants us further knowledge of things like angular acceleration, torque etc. When we talk about the polar moment of inertia we are referring to the measure of a body’s resistance to torsion and can use this to calculate things like angular displacement and periodic time of a body under simple harmonic motion. The inertia of any mechanical component that will encounter any rotational motion must be analyzed as part of designing the system.

This information can be useful in many aspects of engineering, from wind turbines all the way to something as simple as a flywheel, these findings are crucial in calculating the efficiencies and aid in the comparisons of different prototypes before going into mass production. In this particular example ( Trifilar suspension ) the moment of inertia that is used is of a body about an axis passing through that body’s mass center, which is perpendicular the plane of motion.

Theory:
To obtain the moment of inertia of a solid object you must integrate the second moment of mass about an axis. The formula most likely used for inertia is Ig=mk2

Ig being Inertia in Kgm2 about the center of mass, m being the Mass in Kg, k being the radius of gyration about a mass center in m. To calculate the inertia of an assembly, Ig must be by mh2 (local mass in Kgh-being the distance between the parallel axis which goes through the local mass center and the mass center for the full system. The Parallel axis theorem must be put to use to each member of the system, therefore I= (Ig+mh2)

In this experiment, an array of three solid objects are placed on the platform, which is hung from three chains in order to create a trifilar suspension. The periodic time for small oscillations about a vertical axis is related to the Moment of Inertia

Equipment

The equipment we used was a stopwatch app on a mobile phone.

Three 2 meter long chains attached to a ceiling mount.

A circular platform
Three different masses to place on said platform ( Hollow square, hollow cylinder, and solid cylinder)

The three chains would be attached to three evenly spread points across the circumference of the wooden circular platform. The objects would then be placed on top of the circular platform and then the experiment would be ready to commence.

Procedure

The trifilar suspension was set up and the length of the chains was measured with a measuring tape coming in at two meters long each. In order to have multiple takes on the experiment while maintaining the same force, a tangential reference line should be created using a marker pen or pencil onto the circular platform and mark another point of reference on the table. Firstly we measured the results of the table without any of the objects on it and noted the average oscillation time. Then the experiment is repeated with the objects loaded onto the platform this time, and following the same procedures, we find the average time per oscillation.

Results and discussion

Objects placed on the circular platform and their respective Inertia are as follows;

Cylindrical solid
I=mr^2/2=11.65×10^-3 kg.m^2

Cylindrical tube
I=m/2(ro^2+ri^2)=2.898×10^-5 kg.m^2

Square hollow
I=m/6(ao^2+ai^2)=6.939×10^-3kg.m^2

Circular platform
I=(mr^2)/2 =0.1215 kg.m^2

Therefore;

Total Inertia
Itot = 0.14299 kg.m^2

The best way to define the moment of inertia would be as the integral of the second moment about a particular axis. The trifilar suspension experiment works by using this factor to calculate the polar moment of inertia of the system about the vertical axis z by employing the parallel axis theorem. For more minute oscillations about a vertical axis, the periodic time has a relation to the moment of inertia and therefore a suitable value can be obtained from the said relationship by investigating the ratio between experimental/theoretical time and the ratio of lm can be of further use.

Theoretical results vs Experimental results

The accuracy of our stopwatch was to around about a tenth of a second but due to human reaction time this accuracy would have been reduced to a further 0.25 seconds (https://backyardbrains.com/experiments/reactiontime)

The time observed from our experiment come to be 1.8 seconds but taking into account reaction time this number becomes 1.8 ±0.25

Another factor that affects accuracy within the trifilar suspension experiment is that the reference lines will not have been 100% accurately lined up meaning the force applied would be inconsistent. There is also a degree of inaccuracy while using a tape measure to measure the chain length, meaning that the platform could have been uneven i.e leaning at an angle, meaning the torque applied may not have been a pure force component acting perpendicular to the axis of rotation. There will have also have been a loss of momentum due to the moment of inertia as the system got into its latter oscillations.

Conclusion

This experiment helps us prove the correlation regarding the polar moment of inertia and the formulae for calculating periodic time as derived from the equation of the simple harmonic motion of the system. The experiment allows us a greater understanding that the period from both hypothetical and realistic approaches is directly proportional to the ratio of I’m even when taking into account our small errors in measurement.

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