Archimedes' Principle and Upthrust

Archimedes’ principle states that the upthrust (buoyant force) on an object submerged in a fluid equals the weight of the fluid displaced.

$F_{\text{upthrust}} = \rho_{\text{fluid}} \times g \times V_{\text{displaced}}$

Floating and sinking

An object floats when the upthrust equals its weight:

$\rho_{\text{fluid}} g V_{\text{submerged}} = mg$

For a floating object, the fraction submerged equals the ratio of densities:

$\frac{V_{\text{submerged}}}{V_{\text{total}}} = \frac{\rho_{\text{object}}}{\rho_{\text{fluid}}}$

An object sinks if its density is greater than the fluid’s density ($\rho_{\text{object}} > \rho_{\text{fluid}}$).

Measuring density using Archimedes’ principle

For an irregularly shaped object:

1. Weigh the object in air: $W_{\text{air}}$.
2. Weigh the object fully submerged in water: $W_{\text{water}}$.
3. Upthrust $= W_{\text{air}} - W_{\text{water}} = \rho_{\text{water}} g V$.
4. $V = \frac{W_{\text{air}} - W_{\text{water}}}{\rho_{\text{water}} g}$.
5. $\rho_{\text{object}} = \frac{m}{V}$.

Common errors and how to correct them

• Confusing the weight of fluid displaced with the weight of the object.
• Forgetting that upthrust depends on the volume submerged, not the total volume (for partially submerged objects).

How to apply this in an exam

State Archimedes’ principle. Identify the displaced volume. Calculate the upthrust. Compare upthrust with weight to determine floating/sinking.