How Do You Calculate Buoyant Force? Archimedes' Principle Guide
You calculate buoyant force by multiplying the density of the fluid by the volume of fluid the object displaces by the acceleration due to gravity: F_b = ρ × V × g. This is the direct expression of Archimedes’ principle, which states that the upward buoyant force on an object equals the weight of the fluid it pushes out of the way. So if an object displaces 2 liters of water, the buoyant force equals the weight of those 2 liters — about 19.6 newtons. The same relationship explains why a steel ship floats while a steel nail sinks: the ship displaces far more water relative to its weight.
Archimedes’ Principle Explained
Archimedes’ principle says that any object fully or partially immersed in a fluid experiences an upward force equal to the weight of the fluid displaced. The “fluid” can be a liquid or a gas, though for everyday problems it’s usually water. This principle is one of the oldest results in physics and remains central to fluid mechanics, as summarized by Encyclopaedia Britannica.
The physics behind it is a pressure difference. Pressure in a fluid increases with depth, so the bottom of a submerged object feels a greater upward push than the top feels downward. Summing those pressure forces over the whole surface gives a net upward force — the buoyant force. Its magnitude depends only on the fluid’s density and the volume displaced, not on the object’s weight or material.
That last point decides whether an object floats or sinks:
- Floats: the object’s weight is less than the buoyant force at full submersion, so it settles where the displaced weight matches its own weight.
- Sinks: the object’s weight exceeds the maximum buoyant force the fluid can provide, even when fully submerged.
- Neutral: the two are equal, so the object hovers at a given depth.
A useful way to think about it is density comparison. If an object’s average density is less than the fluid’s, it floats; if greater, it sinks. Ice (about 917 kg/m³) floats on water (about 1000 kg/m³), while most metals sink because their densities are several times higher.
The Buoyant Force Formula
The buoyant force formula is:
F_b = ρ × V × g
Each symbol has a precise meaning and unit:
| Symbol | Quantity | Typical unit |
|---|---|---|
| F_b | Buoyant force | newtons (N) |
| ρ (rho) | Density of the fluid | kg/m³ |
| V | Volume of fluid displaced | m³ |
| g | Acceleration due to gravity | m/s² (≈ 9.8) |
Physically, ρ × V gives the mass of displaced fluid, and multiplying by g converts that mass into a weight — which is exactly the buoyant force. This is why the formula is sometimes written as F_b = m_displaced × g.
Why Density Must Be the Fluid’s
A common error is plugging in the object’s density. The formula uses the fluid’s density because buoyancy comes from the fluid, not the object. The object only determines how much fluid gets displaced through its submerged volume. Keep the two separate and the math stays clean.
What g Does and Doesn’t Do
The value of g is essentially constant near Earth’s surface at about 9.8 m/s². It converts mass to weight; it does not change which objects float. Buoyancy would work the same way on the Moon — only the size of the force would shrink along with g.
Step-by-Step Calculation
Here is a worked example that shows how to apply the formula from scratch.
Problem: A solid block is fully submerged in fresh water. It displaces 0.004 m³ of water. What is the buoyant force?
Step 1 — Identify the fluid density. Fresh water has a density of about 1000 kg/m³. (Seawater is a bit higher, around 1025 kg/m³.)
Step 2 — Determine the displaced volume. The block is fully submerged, so it displaces its own volume: V = 0.004 m³.
Step 3 — Use g = 9.8 m/s².
Step 4 — Plug in and compute.
F_b = 1000 × 0.004 × 9.8 = 39.2 N
So the water pushes up on the block with 39.2 newtons, roughly the weight of a 4 kg mass.
Step 5 — Compare with the object’s weight. If the block’s weight is, say, 30 N, the buoyant force of 39.2 N exceeds it and the block would actually be pushed upward — it wouldn’t stay fully submerged on its own. If the block weighed 50 N, it would sink, because even full submersion can’t supply enough buoyancy. This comparison is the final and most important step in any buoyancy problem: compute F_b, then compare it to weight.
The same procedure appears throughout introductory physics problem sets, and The Physics Classroom walks through similar submerged-object examples.
Common Mistakes to Avoid
- Using the object’s density instead of the fluid’s. This is the single most frequent error. ρ always refers to the fluid.
- Forgetting that volume is displaced volume, not total volume. For floating objects, only the submerged portion counts.
- Mixing units. Density in kg/m³ demands volume in m³. A volume given in liters (1 L = 0.001 m³) or cm³ (1 cm³ = 1 × 10⁻⁶ m³) must be converted first.
- Forgetting g. Weight-of-fluid problems sometimes stop at ρ × V, which gives mass, not force.
Key Takeaways
- The buoyant force equals the weight of the displaced fluid: F_b = ρ × V × g, where ρ is the fluid’s density, V is the displaced volume, and g is about 9.8 m/s².
- Archimedes’ principle ties buoyancy directly to displaced fluid — not to the object’s weight or material.
- An object floats when its weight is less than the buoyant force available at full submersion, and sinks when its weight exceeds it.
- Always use the fluid’s density, keep units in kg/m³, m³, and m/s², and finish by comparing F_b to the object’s weight.
- Comparing the object’s average density to the fluid’s density is a quick shortcut for predicting float or sink.
