It Started with a Bottle of Soap

For a long time, every time I took a shower, I had this recurring thought. “When I use some of the soap from this bottle, the amount of soap in the bottle decreases. Therefore, the centre of mass of the bottle changes. How does it change?”

I knew that the way it changed would have certain characteristics. For a cylindrical bottle, when it was full, its centre of mass would be at its centre. And when it was empty, the centre of mass would once again be at its centre. I also realised that when it was full and mass started flowing out, the height of the centre of mass would decrease. This in turn led to another question. “When is the centre of mass at its lowest?”

These were all questions I knew I needed the answer to. Being a physics major, I was equipped with the necessary tools. And so, about a month ago, I set off to get my answers. It didn’t take too long and in the end my curiosity was satisfied. Those of you interested can check out my results here.

Thanks for reading!

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8 thoughts on “It Started with a Bottle of Soap

  1. Interesting.. however, in the absence of units, it is difficult to fully understand the results. For example, the density of glass is 2.5 in relation to that of water, which is 1. How does the density factor of 0.25 come into the fore?

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    1. Hello, uncle! Thank you for your comment. One of the reasons I left out the units was because all my parameters were in the same units. For example, if all my distance parameters (height, radius) were in metres, my result would also be in metres. You will also see that the mass units in the numerator and denominator cancel out. With respect to the density, while doing my initial analysis I noticed that what matters in the end is the relative density of the two materials. For example, as you stated, glass to water is 2.5 to 1. Please note that the data I gave (H = 2, r = 1 and rho = 0.25) are all just numbers I randomly selected. I did not think of any particular materials like glass and water. My goal was to make my formulation as general as possible. To be fair, I could have been a little clearer in the way I wrote some of the parts and it is something I will work on in future works.

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  2. Nice response. thank you. However, it is also important to realize that a material, which has lower density than the liquid (since you used 1, I assumed it is water), can not hold the liquid.

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  3. good question. I thought mercury is contained in special containers, details of which I do not know. I will check when I get a bit of time. If you do find the answer, plz share. thanks; btw, i read your blog about universities. Hope you applied to Cambridge or Oxford. These are exceptional places to study.

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    1. Thank you for your response! Yes, Cambridge and Oxford are most definitely exceptional places to study! At the same time they are expensive and not easy to get in. Also, for studying physics you are required to have gotten at least 95 in mathematics. Unfortunately, I got 94. I completed my first year studying at the University of Hong Kong. Due to certain circumstances, I will be headed (albeit online) to the University of Groningen in the Netherlands this September.

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