Calculating the Volume of Water Displaced by an Object: The Archimedes Principle and Practical Applications

This article delves into the fundamental concept of the volume of water displaced by an object, applying the principles outlined by the ancient Greek philosopher, Archimedes. We will explore the mathematical formulas and practical applications of these principles, explaining how to calculate the displaced volume for both regular and irregularly shaped objects.

Theoretical Foundation: Archimedes' Principle

According to Archimedes’ principle, the volume of fluid displaced by an object is equal to the volume of the submerged part of the object. Mathematically, this can be expressed as:

Vd Vs

Where:

Vd is the volume of water displaced Vs is the volume of the submerged portion

For a fully submerged object, the displaced volume equals the total volume of the object. However, if the object is partially submerged, the calculation requires determining the volume of the underwater portion.

Applying the Formula for Regular Geometric Shapes

For simple, geometrically shaped objects, the calculation of the volume can be straightforward. Below are some common shapes and their volume formulas:

1. Cube

V a3 where 'a' is the length of a side.

2. Cylinder

V πr2h where 'r' is the radius and 'h' is the height.

3. Sphere

V (4/3)πr3 where 'r' is the radius.

For these regular solids, you can directly apply these volume formulas to find the displaced water volume.

Practical Applications: Weight and Density Method

When dealing with irregularly shaped objects, practical methods can offer valuable insights. If the object's density and the density of the fluid (usually water) are known, the formula:

Vd W/ρ

can be used, where:

Vd is the volume of water displaced W is the weight of the object ρ is the density of the fluid. For water, ρ ≈ 1 gm/cc.

This method works particularly well for irregularly shaped objects, where direct volume calculation is complex or impossible.

Volume of Water Displaced in Floating Objects

For floating objects, a key principle is that the weight of the object equals the weight of the water displaced. This relationship is crucial in fields such as naval engineering and material science. The weight of the object in cubic centimetres (cc) is equal to the volume of water it displaces:

Volume of water Weight of floating object (in cc)

For regularly shaped solids:

V L × W × (H - h)

Where:

V is the volume of the floating object. L is the length of the object. W is the width of the object. H is the total height of the object. h is the height of the portion of the object above the water surface.

For irregularly shaped solids:

There is no specific formula. Visual or physical methods are often used to determine the displaced volume.

Understanding these principles and formulas is crucial for engineers, scientists, and anyone dealing with the principles of fluid mechanics. The ability to calculate the volume of water displaced can provide insights into buoyancy, stability, and the behavior of objects in water.