When an object is placed in a liquid, the liquid pushes upward on it.
This upward force is called the buoyant force.
Buoyancy explains why some objects float, some sink, and some remain suspended in a fluid.
1. Archimedes’ principle
Archimedes’ principle states that the buoyant force on an object equals the weight of the fluid displaced by the object.
\[
\begin{aligned}
F_b &= \rho_{\mathrm{liq}}gV_{\mathrm{sub}}.
\end{aligned}
\]
Here:
- \(F_b\) is the buoyant force.
- \(\rho_{\mathrm{liq}}\) is the density of the liquid.
- \(g\) is gravitational field strength.
- \(V_{\mathrm{sub}}\) is the submerged volume, which is also the displaced liquid volume.
2. Object weight
If the object has density \(\rho_{\mathrm{obj}}\) and total volume \(V\), then its mass is
\[
\begin{aligned}
m &= \rho_{\mathrm{obj}}V.
\end{aligned}
\]
Its weight is
\[
\begin{aligned}
W &= mg \\
&= \rho_{\mathrm{obj}}Vg.
\end{aligned}
\]
3. Floating, sinking, or neutral buoyancy
The object’s behavior is mainly determined by comparing the object density with the liquid density.
| Condition |
Behavior |
Meaning |
| \(\rho_{\mathrm{obj}}<\rho_{\mathrm{liq}}\) |
Floats |
The object needs only partial submersion to displace enough liquid. |
| \(\rho_{\mathrm{obj}}=\rho_{\mathrm{liq}}\) |
Neutral |
The object can remain fully submerged without rising or sinking in the ideal model. |
| \(\rho_{\mathrm{obj}}>\rho_{\mathrm{liq}}\) |
Sinks |
Even full submersion cannot produce enough buoyant force to balance the weight. |
4. Fraction submerged for a floating object
A floating object is in vertical equilibrium, so
\[
\begin{aligned}
F_b &= W.
\end{aligned}
\]
Substitute the buoyant force and weight formulas:
\[
\begin{aligned}
\rho_{\mathrm{liq}}gV_{\mathrm{sub}}
&= \rho_{\mathrm{obj}}gV.
\end{aligned}
\]
Cancel \(g\):
\[
\begin{aligned}
\rho_{\mathrm{liq}}V_{\mathrm{sub}}
&= \rho_{\mathrm{obj}}V.
\end{aligned}
\]
Divide by \(\rho_{\mathrm{liq}}V\):
\[
\begin{aligned}
\frac{V_{\mathrm{sub}}}{V}
&= \frac{\rho_{\mathrm{obj}}}{\rho_{\mathrm{liq}}}.
\end{aligned}
\]
This is one of the most useful buoyancy results.
It says that the submerged fraction of a floating object depends only on the density ratio.
5. Apparent weight
When an object is fully submerged, the liquid reduces the apparent weight.
The apparent weight is
\[
\begin{aligned}
W_{\mathrm{app}}
&= W-F_b.
\end{aligned}
\]
If \(W_{\mathrm{app}}>0\), the object still feels heavy underwater and tends to sink.
If \(W_{\mathrm{app}}=0\), the object is neutrally buoyant.
If \(W_{\mathrm{app}}<0\), the object has a net upward tendency when fully submerged.
6. Maximum buoyant force
The maximum buoyant force happens when the whole object is submerged:
\[
\begin{aligned}
F_{b,\max}
&= \rho_{\mathrm{liq}}gV.
\end{aligned}
\]
If \(F_{b,\max}W\), the object can float with only part of its volume submerged.
7. Common volume formulas
| Shape |
Volume formula |
Submerged depth for floating |
| Cube |
\(V=a^3\) |
\(d_{\mathrm{sub}}=fa\) |
| Cuboid |
\(V=LWH\) |
\(d_{\mathrm{sub}}=fH\) |
| Vertical cylinder |
\(V=\pi R^2H\) |
\(d_{\mathrm{sub}}=fH\) |
| Sphere |
\(V=\frac{4}{3}\pi R^3\) |
Use spherical-cap volume |
In the table, \(f=V_{\mathrm{sub}}/V\) is the submerged fraction.
8. Floating sphere depth
For a sphere, submerged depth is not simply \(f(2R)\), because the cross-sectional area changes with height.
The submerged part is a spherical cap. If \(d\) is the submerged depth measured from the lowest point up to the liquid surface, then
\[
\begin{aligned}
V_{\mathrm{cap}}
&= \frac{\pi d^2(3R-d)}{3}.
\end{aligned}
\]
For a floating sphere, solve
\[
\begin{aligned}
V_{\mathrm{cap}}
&= fV.
\end{aligned}
\]
This calculator solves that equation numerically.
9. Worked example: wooden block in water
A wooden block has density
\[
\begin{aligned}
\rho_{\mathrm{obj}} &= 650\ \mathrm{kg/m^3}.
\end{aligned}
\]
Water has density
\[
\begin{aligned}
\rho_{\mathrm{liq}} &= 1000\ \mathrm{kg/m^3}.
\end{aligned}
\]
Since the object floats, its submerged fraction is
\[
\begin{aligned}
\frac{V_{\mathrm{sub}}}{V}
&= \frac{\rho_{\mathrm{obj}}}{\rho_{\mathrm{liq}}} \\
&= \frac{650}{1000} \\
&= 0.65.
\end{aligned}
\]
So \(65\%\) of the block’s volume is submerged and \(35\%\) remains above the water.
At equilibrium, the buoyant force equals the block’s weight:
\[
\begin{aligned}
F_b &= W.
\end{aligned}
\]
10. Formula summary
| Goal |
Formula |
Use |
| Buoyant force |
\(F_b=\rho_{\mathrm{liq}}gV_{\mathrm{sub}}\) |
Find upward force from displaced liquid |
| Object weight |
\(W=\rho_{\mathrm{obj}}Vg\) |
Find downward gravitational force |
| Floating fraction |
\(V_{\mathrm{sub}}/V=\rho_{\mathrm{obj}}/\rho_{\mathrm{liq}}\) |
Find how much of a floating object is underwater |
| Maximum buoyant force |
\(F_{b,\max}=\rho_{\mathrm{liq}}gV\) |
Check whether full submersion can support the object |
| Apparent weight |
\(W_{\mathrm{app}}=W-F_b\) |
Find how heavy the object seems in the liquid |
| Spherical cap volume |
\(V_{\mathrm{cap}}=\pi d^2(3R-d)/3\) |
Find floating depth for a sphere |
11. Assumptions
- The liquid is at rest.
- The object has uniform average density.
- The liquid density is constant.
- Surface tension and capillary effects are ignored.
- Wave motion, fluid drag, and object rotation are ignored.
- The object is treated as rigid and non-absorbing.
Key idea: the buoyant force equals the weight of the displaced liquid. Floating happens when the object displaces just enough liquid for \(F_b=W\).
