Short on time? This page is the recap. For the derivations, diagrams and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- Newton’s laws apply to any small piece of fluid. Its forces are the pressure forces on its faces, its weight, and pushes from walls.
- A fluid’s velocity changes only when the forces on it do not balance. In a level pipe, that means a pressure difference: the fluid accelerates from high to low pressure.
- In a fluid at rest, each layer holds up the fluid above it. That is why P = P₀ + ρgh.
- The buoyant force is the net upward force from all the fluid’s pressure forces. Pressure is greater on the bottom of an object than on its top.
- Its size equals the weight of the displaced fluid. It does not depend on the object’s mass or material, or on depth once the object is fully under.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| F_b = ρ_fluid V_disp g | ρ in kg/m³, V in m³, F in N | find the buoyant force |
| F_b = (P_bottom − P_top)A | P in Pa, A in m² | derive or check F_b from pressures |
| floating: F_b = mg | N | find a submerged depth or volume |
| V_sub / V = ρ_object / ρ_fluid | no unit | find the fraction under the surface |
| scale under water: T = mg − F_b | N | find F_b, V or ρ from two readings |
| ρ_object / ρ_fluid = mg / F_b | no unit (fully submerged) | density from two scale readings |
| ma_y = F_b − mg (+y up) | m/s² | rising or sinking objects, ignoring drag |
| a = (P₁ − P₂) / (ρL) | m/s² | acceleration of a slug of fluid of length L |
Assumptions behind the numbers
- The fluid is ideal: incompressible (constant density at all depths) and with no viscosity.
- Drag on a moving object is ignored unless a question gives it.
- The buoyant force from air is ignored for solids and liquids weighed in air.
- g = 9.8 m/s²; density of water 1000 kg/m³.
Mistakes to avoid
- Using the object’s density in F_b. Use the fluid’s density.
- Using the whole volume for a floating object. Use only the submerged volume.
- Saying F_b grows with depth. Not for a fully submerged object in an ideal fluid.
- Drawing F_b and the pressure forces together. That counts the fluid twice.
- “Floats because it is light.” Compare densities, not masses.
- “Weighs less in water.” The weight is the same; the scale reading is less.
- “A fluid needs a push to keep moving.” A pressure difference changes velocity; it is not needed for constant velocity.
Quick self-check
- A block reads 12.0 N on a spring scale in air and 9.0 N fully under water. What is the buoyant force, and what is the block’s volume? (3.0 N; 3.1 × 10⁻⁴ m³)
- A plastic block of density 920 kg/m³ floats in water. What fraction of it is above the surface? (0.080, or 8%)
- A ball of density 400 kg/m³ is held under water and released. What is its acceleration just after release, ignoring drag? (14.7 m/s² ≈ 15 m/s² upward)
Next: practice questions.