\[ \]
\[ \]
\[ \]
Issue No 61, 6 May 2024
By: Anthony O. Ives
Boats or building a set of floats is a simple, fun way to use mathematics to build something accurately to do exactly what you want it to do. Archmedes principle defines how and why a objects floats. Archimedes principle states that for an object to float it must displace a volume of fluid equivalent to the weight of object which is partially or fully submerged in the fluid [1]. Archimedes principle does not just apply to boats it also applies lighter than air aircraft such as airships and balloons.
Another definition of archimedes principle is that weight of fluid displaced is equal to buoyancy force acting on the object. After all, just because an object has bouyancy does not mean it is floating, if its weight is greater than the bouyancy force then it will sink. As with any mathematical or physics principle it can be quite confusing to work out what is means when defining it in words, its a lot easier to get what it means by using an equation and also easier to use the equation to calculate what you want to know.
The equation used to calculate bouyancy force based on archimedes principle is quite simple as seen below:
\[F_{B}=\rho g V\]
Where FB is the bouyancy force, ρ is the density of fluid which the object is submerged in, g is the acceleration of freefall asssumed to 9.81 ms-1, V is the volume of the displaced fluid. The volume of displaced fluid can be the most complicated part of the equation to calculate. Depending on what the floating or submerged object is, the displaced volume is usually calculated from the shape of the boat hull, float, etc which is submerged under water. Complicated shapes can use intergration [2] to calculate their submerged volumes, this is something that will be the topic of a future article looking at the practical uses of intergration calculus.
However, the buoyancy force can be calculated for a simple shape to demonstrate its use. A good simple example would be if you want to build a set simple floats for a small RC helicopter weighting approximately 0.5 kg. The floats will be a simple design using a cylinder with two spherical ends as in the picture below:
The volume of the middle part and majority of the float would calculated similar to how you calculate a cylinder volume as in the equation below:
\[V = \frac{\pi D^2}{4} L \]
Where D is the diameter of the cylinder, π is the mathematical constant and L is the length of the cylinder without the spherical ends. D and L is also defined in the diagram. This equation should be familiar as simply the area of a circle [3] multiplied by the length of a cylinder. If you consider that there are two floats which you assume each is half submerged and assume the spherical ends do not make much of contribution to the buoyancy force then your bouyancy can be calculated for the floats as below:
\[F_{B}=W=2 \times 0.5 \times \rho g \frac{\pi D^2}{4} L \]
\[W = \rho g \frac{\pi D^2}{4} L \]
Where W is the weight of the helicopter. However, if you want to size the diameter of the floats for a particular weight then you would rearrange the equation to the following:
\[D=\sqrt{\frac{4 W}{\rho g \pi L}}\]
The length of the floats could also be sized for an assumed diameter but as you know the length of the helicopter skids it makes more sense to assume those as the length of the floats without the spherical ends.
Using the following values, L = 18cm = 0.18m, ρ = 997 kgm-3, g=9.81 ms-1 and m=0.5kg (therefore W =mg=0.5×9.81) then the diameter of an individual float would calculated as below:
\[D=\sqrt{\frac{4(0.5)9.81}{(997) 9.81 \pi (0.18)}}\]
\[D=0.06m=6cm\]
Substituting the value for diameter back into the equation for bouyancy force you should get correct weight and mass of the helicopter:
\[W = 997 (9.81) \frac{\pi (0.06)^2}{4} 0.18 \]
\[W = 4.98 N\]
\[m = \frac{4.98}{9.81} = 0.5kg\]
Now just as a check you can consider the bouyancy of spherical ends to see if they make much of a difference. Its typical practice in engineering to design using the parameters that will have big influence and neglect those that you think will only have a small influence especially when sizing an initial concept. However, its also good practice just to confirm that those parameters do not have more of an influence than you originally thought. Therefore the volume is calculated as below:
\[V = \frac{4}{24} \pi D^3 \]
Considering two spherical ends make up one sphere but the two spherical ends are half submerged on two floats then on substituting the volume of a sphere into the equation for bouyancy force then the bouyancy contribution of the spherical ends is as follows:
\[F_{B}=1\times0.5\times2\times\rho g \frac{4}{24} \pi D^3\]
\[F_{B}=997(9.81)\frac{4}{24}\pi (0.06)^3\]
\[F_{B}=1.11 N\]
The calculation shows that the spherical ends increases the bouyancy by about 22% so if you needed the floats to sit exactly half submerged they may need to be resized by iteratively reducing the float diameter. Alternatively the bouyancy equation for the sphere and cylinder could be combined and rearranged to form a polynomial equation [4,5] in terms of the float diameter the polynomial could then solved to gave the required diameter.
Submerged depth of a float or boat determines stability and likehood of capsize however this is a detailed topic which would need a separate article. Water density as seen from the bouyancy equation also determines buoyancy. Therefore the bouyancy force is larger in salt water as its more dense than fresh water which contains little or no salt.
On the topic of bouyancy and hydrodynamics it is worth mentioning John Philip Holland who made major contributes towards hydrodynamics. John Philip Holland [6] was a self-taught Irish engineer and inventor who did a lot of work to develop the modern submarine mainly for U.S. Navy. Having developed the submarine mainly for war he was rebuked by Miss Clara Barton for creating such a dangerous and effective war machine. Miss Clara Barton was first president of the American Red Cross and probably a pacificist who been invited onboard one of Mr Holland's latest submarine. Miss Barton's comment certainly had a big impact on Mr Holland as from that point he tried to find peaceful means for submarines and then towards his last years worked on trying to develop an aircraft. It just goes to show you, you have to be carefully what you create. As you could create something without considering what it is really going to get used. However, thankfully submarines for are and can be used for peaceful means. Unmanned sunmarines especially can used for search and rescue, and various other important peaceful operations. So hopefully Mr Holland's work can be used for more peaceful applications in the future. Mr Holland's life and legacy was personal inspiration to me as an engineer though most of his work goes unrecognised especially in his home country of Ireland.
Please leave a comment on my facebook page or via email and let me know if you found this blog article useful and if you would like to see more on this topic. Most of my blog articles are on:
Mathematics
Helicopters
VTOL UAVs (RC Helicopters)
Sailing and Sailboat Design
If there is one or more of these topics that you are specifically interested in please also let me know in your comments this will help me to write blog articles that are more helpful.
References:
[1] Mechanics of Fluids, B. S. Massey, John Ward-Smith, 7th Edition, 1998, CRC Press
[2] http://www.eiteog.com/EiteogBLOG/No6EiteogBlogRange.html
[3] http://www.eiteog.com/EiteogBLOG/No6EiteogBlogRange.html
[4] http://www.eiteog.com/EiteogBLOG/No6EiteogBlogRange.html
[5] http://www.eiteog.com/EiteogBLOG/No6EiteogBlogRange.html
[6] John P. Holland 1841-1914 Inventor of the Modern Submarine, Richard Knowles Morris, 1998, Unversity of South Carolina Press
Disclaimer: Eiteog makes every effort to provide information which is as accurate as possible. Eiteog will not be responsible for any liability, loss or risk incurred as a result of the use and application of information on its website or in its products. None of the information on Eiteog's website or in its products supersedes any information contained in documents or procedures issued by relevant aviation authorities, manufacturers, flight schools or the operators of aircraft, UAVs.
For any inquiries contact: [email protected] copyright © Eiteog 2023