Upthrust & Archimedes’ Principle

1. Definitions (Must Know)

A. Upthrust (buoyant force), U

Upthrust (buoyant force), U, is the upward force on a body in a fluid due to a pressure difference between its lower and upper surfaces.

Its magnitude equals the weight of fluid displaced:

U = ρ_fluidgV_displaced

B. Archimedes’ principle

Archimedes’ principle: a body partially or totally immersed in a fluid experiences an upthrust equal to the weight of fluid displaced.

C. Floating, sinking, neutral buoyancy

• floats (in equilibrium): U = W
• sinks: U < W
• neutral buoyancy (fully immersed, no rise/fall): U = W while fully submerged

2. Core qualitative reasoning

• Upthrust acts upward because fluid pressure is greater on the lower surface than on the upper surface.
• A floating object at rest still has weight; upthrust balances it.
• A sinking object can experience upthrust, but its downward forces produce a downward resultant.
• Upthrust and weight act on the same object, so they are not a Newton’s third-law pair.

3. Optional quantitative model

• Upthrust depends on fluid density and displaced volume (not the object’s mass directly).
• If the object is fully submerged: V_displaced = V_object.
• If the object floats: U = W and only part of the object is submerged.
• For a floating object: V_displaced/V_object = ρ_object/ρ_fluid
• Apparent weight reduces in a fluid because upthrust acts upwards: Wₐₚₚₐᵣₑₙₜ = W - U
• In a uniform-density fluid, upthrust depends on V_displaced (not on depth).

4. Detailed explanations

A. Exam workflow (most questions)

1. Draw an FBD: weight W = mg down, upthrust U up (and tension if there’s a string).
2. Identify the displaced volume V_displaced.
3. Use U = ρ_fluidgV_displaced.
4. Apply equilibrium or ∑ F = ma.

B. Floating: why the fraction submerged depends on density

For a floating object in equilibrium:

U = W

ρ_fluidgV_displaced = ρ_objectgV_object

Cancel g:

V_displaced/V_object = ρ_object/ρ_fluid

So a lower-density object needs to submerge less volume before upthrust balances its weight.

C. Apparent weight / tension in a string

If an object is fully submerged and held by a string at rest:

• forces: U up, tension T up, weight W down
• equilibrium: T + U = W ⇒ T = W-U

This is the usual “tension in string” question.

5. Common mistakes

• Using the object’s density instead of the fluid’s density in U = ρ gV.
• Using the object’s full volume when it is floating (use V_displaced, not V_object).
• Forgetting unit conversion (e.g. 1 cm³ = 10⁻⁶ m³).
• Writing “upthrust = weight” for a sinking object (only true when it has reached terminal conditions / equilibrium).
• Calling upthrust and weight a third-law pair; both forces act on the immersed object.
• Treating the quantitative Archimedes model as a required 9478 formula.

6. Exam tips

• State clearly whether the object is fully submerged or floating.
• If it floats and is at rest: start with U = W (fastest).
• If asked for “apparent weight”, the scale reads the normal reaction; in a fluid, it is usually reduced because of upthrust.
• In multi-step questions, write the force balance line explicitly (it’s method marks).

6. Worked Examples

7. Mind Stretchers