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ACADEMIC PAPERS

Academic Papers: Text

MATHEMATICS/ENGINEERING

2020

EXPLORING CHAOS IN A DOUBLE PENDULUM

On reading "17 Equations that changed the world" by Ian Stewart and "Chaos" by James Gleick, I was inspired to write a paper exploring the wonderful intricacies of Chaos Theory. I investigated this through a deceptively simple model of a double pendulum. I began by deriving the equations for the for the double pendulum using the Lagrangian (T-V) and then used the Euler Lagrange Equations to obtain 4 differential equations. I also modified my equations to factor in a damping force using the Rayleigh Dissipation Function.

I inputed my derived equations into Python to create a program that could follow the motion of the double pendulum. Finally I compared my computer simulation to an experimental model I had built, to see if I had managed to predict even a fraction of the time that the pendulum was moving for.

 

What made the project very fascinating was the inherent nature of deterministic chaos that made any prediction beyond a few seconds almost impossible without measurement tools with extremely high accuracy.

 

Nonetheless, the findings of my exploration and the hands-on experience of building my own double pendulum and writing my own code was very fruitful.

Chaos.png

ANALYSIS OF THE BIG O NOTATION AND ITS EVALUATION OF THE TIME COMPLEXITIES OF SORTING ALGORITHMS

Having been interested in computers from a young age, I always wanted to explore how they functioned, and how they were so fast. Recently I watched a video that looked at how a single algorithm ruined an entire project because it simply wasn't efficient.

Thus I was inspired to explore the efficiency of a bedrock of modern computers - Sorting Algorithms.

To do this I explored Sorting Theory and several factors used to measure the efficiency of algorithms. In my Essay I focused on the "Big O Notations" - a form of asymptotic notation that gives programmers a general idea of the efficiency of algorithms as it measures their time complexity.

I explored the following algorithms:

  • Permutation Sort

  • Bubble Sort

  • Quick Sort

  • Merge Sort

  • Counting sort

The analysis of the data I obtained from running trials for the "time to sort" yielded some surprising results.

Sorting.png
Academic Papers: Projects

PHYSICS

2020

WINE GLASS HARMONICS AND LIQUID DENSITY

Stuck in the house during the lockdown, I did not have access to a laboratory to make precise measurements. Instead, I used my interest in music and my passion for frugality, to convert my room into a makeshift lab.

The obvious constraints that my home-lab with kitchen tools for measurement made conducting experiments more challenging, yet I did not want to explore something that had been widely researched before. Rather than looking at how the height of water in a wine glass impacts its resonant frequency, I explored how the density of liquid in the wine glass impacts it's resonant frequency.

I used the AP French Equation:

As a theoretical prediction of the frequency.

I manipulated densities by adding different salt concentrations to the liquid (water) in my wine glass. Through my investigation, I confirmed a direct linear relationship between the density of the liquid and the reciprocal of the square of the measured frequency.

AP.png
Wine glass.png
Academic Papers: Projects
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