 ### An Engineering Feat that Works on Archimedes’ Principle

By TK NG

Archimedes’ Principle

Archimedes’ principle states that the upward buoyant force exerted on a body immersed in a fluid, whether fully or partially, is equal to the weight of the fluid that the body displaces. Reasoning is straight forward and is elucidated in the ensuing paragraphs.

Referring to Fig 1 below, when a cube of volume V and side H is submerged in a fluid with density ρ, the fluid pressure acts on all its surfaces. The magnitude of the pressure varies with the depth from the fluid surface. The fluid pressure acts on the top surface is P. The pressure acting on the bottom surface is higher by ρgH Note due to the additional depth H. The downward force is hence P x H2 and the upward force is (P + ρgH) x H2. The forces act on the vertical opposite sides of the cube due to fluid pressure cancel each other. The resulting buoyant force is (P + ρgH) x H2 – P x H2 = ρgH3 = ρgV, i.e. weight of the fluid displaced by the cube.

If the cube is only partially immersed in the fluid at a depth of h, the submerged volume v is h x H2. The side thrusts on the vertical opposite sides again cancel each other while the upward buoyant force is ρgh x H2 = ρgv, i.e. weight of the fluid displaced by the submerged portion of the cube. If the cube stays floating in the fluid, the buoyant force must be equal to the weight of the cube (neglecting the effect of the fluid surface tension). By the same token, a vessel floating in water displaces an amount of water equal to its own weight.

Thanks to the Archimedes’ Principle, an ingenious and green design is made possible for a rotating boat lift in Scotland linking two canals with a level difference of 35 metres.

Boat Lift in Scotland – The Falkirk Wheel

The world’s only rotating boat lift, the Falkirk Wheel (Fig 2) in Scotland links the Forth & Clyde Canal to the Union Canal 35 metres above, allowing transfer of vessels between them. It is a green engineering design that only takes around four minutes and 1.5 kWh to complete every lifting operation, given that the two gondolas of the Wheel can each hold 250 tonnes of water and vessels together. Before that, vessels had to go through a flight of 11 staircase locks, which took nearly a day, for transit from one canal to the other.

The two gondolas hold the same amount of water when there is no vessel inside. According to the Archimedes’ Principle, when vessels sail into the gondolas, water amounting to their own weights is displaced. Since the water levels inside the gondolas are kept constant under all situations, the combined weight of the vessel(s) and water in each gondola is same as that of the water alone when there is no vessel inside. A balance of weight is hence maintained between the two gondolas irrespective of the number and sizes of vessels inside them. It is because of this weight balance that only minimal power is required to turn the Falkirk Wheel.

This is a good example of people making use of an age-old scientific principle to come up with an ingenious and green solution for overcoming a formidable engineering challenge. The following is a YouTube video showing how the Falkirk Wheel works. Hope you enjoy it!

Note
The pressure at the bottom of a fluid column of height H, base area A and density ρ
= weight of the fluid column divided by the base area
= density x gravity acceleration x volume of fluid column / base area
= ρ x g x (H x A) / A
= ρgH 