Buoyancy Master

STUDENT READING GUIDE: ARCHIMEDES' PRINCIPLE

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1. What is Buoyant Force?

Have you ever noticed that you feel lighter when you are in a swimming pool? Or why a massive steel ship stays afloat while a small pebble sinks to the bottom? This phenomenon is due to buoyancy.

"Buoyant force (\(F_B\)) is the upward force exerted by a fluid on any object immersed in it."

This force exists because pressure in a fluid increases with depth. The bottom of an object is always deeper than the top, meaning the upward pressure on the bottom is greater than the downward pressure on the top. This difference in pressure results in a net upward force.

2. Archimedes' Principle: The Theory

Around 250 BCE, the Greek mathematician Archimedes discovered a critical relationship that explains exactly how much buoyant force acts on an object.

Object displaces its volume in fluid

The Core Statement:

"An object immersed in a fluid is buoyed up by a force equal to the weight of the fluid that the object displaces."

Crucial Distinction:

The buoyant force does not depend on the weight of the object itself; it depends only on the volume of the object that is underwater (displaced volume).

3. The Mathematical Derivation

Why is the buoyant force exactly equal to the weight of the displaced fluid? We can prove this by looking at the pressure acting on a submerged cube of area \(A\) and height \(h\).

\(F_{top} = P_{top} A\)

\(F_{bottom} = P_{bottom} A\)

VOLUME \(V\)
\((A \times h)\)

Step-by-Step Logic:

Pressure Difference: Pressure at bottom is greater than top: \(\Delta P = \rho_{f} g h\).
Net Force: The net upward force is \(F_B = F_{bottom} - F_{top}\).
Substitution: \(F_B = (P_{bottom} - P_{top}) \times A\).
The Result: \(F_B = (\rho_{f} g h) \times A\). Since \(A \times h = V\), we get:

\[ F_B = \rho_{f} V_{sub} g \]

What about the sides?

The pressure forces acting on the sides of the object cancel each other out because they are at the same depth. Only the vertical pressure difference contributes to the buoyant force.

4. Detailed Free Body Diagrams (FBDs)

SINKING

Density Object > Density Fluid

\(F_g > F_B\)

Net acceleration is downward.

NEUTRAL

Density Object = Density Fluid

\(F_g = F_B\)

Object hovers at any depth.

FLOATING

Density Object < Density Fluid

\(F_g = F_B\) (Equilibrium)

Volume sub < Volume object.

The Fraction Submerged

For a floating object, the percentage of the object that stays underwater depends entirely on the ratio of densities:

\[ \text{Fraction Submerged} = \frac{\rho_{object}}{\rho_{fluid}} = \frac{V_{submerged}}{V_{total}} \]

5. Apparent Weight and Displacement

When an object is submerged, the buoyant force acts as an "upward assist," making the object seem lighter. This "measured" weight is called the Apparent Weight (\(F_{g,app}\)).

Fundamental Equation

\[ F_{g,app} = F_g - F_B \]

If \(F_{g,app}\) becomes 0, the object is floating!

SCALE (12 N)

The Theory Diagram: Fluid Replacement

Archimedes realized that a fluid provides the exact same upward support to an object that it would have provided to the fluid that used to be in that same space.

WATER

Weight = 10 N

OBJECT

\(F_B = 10\) N (Upward)

Mastering the Math: Practice Problem

The Challenge

A solid gold crown has a mass of 1.5 kg. When it is fully submerged in water, it displaces 0.00008 m³ of water.

Using your knowledge of Archimedes' Principle and the concept of apparent weight, determine the apparent weight of the crown.

Mathematical Analysis & Procedure:

Final Result:

N

Reading Mastery Check

1. A ship is made of steel, which is denser than water, yet it floats. Explain this using the "Weight of Displaced Fluid" concept.

2. Draw the FBD for a submarine that is hovering at a constant depth of 50m.

Sketch Here

Include labels for \(F_B\) and \(F_g\). What is the relationship between their magnitudes?

3. A beach ball is held underwater. When released, it accelerates toward the surface. Draw the FBD at the moment it is released (while still fully submerged) and explain why it moves upward.

4. Summarize Archimedes' Principle in exactly one sentence using the terms "buoyant force" and "fluid weight."

Buoyancy Master Reading Guide Elaborate Revised

Buoyancy Master Reading Guide Elaborate Revised

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