Apparent Weight Loss

When you lift a heavy rock underwater, it feels surprisingly lighter than it does in air. This is not an illusion — the rock genuinely pushes down on your hand with less force when submerged. This reduction is called apparent weight loss, and it arises directly from Archimedes' Principle: any object fully or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. Because this buoyant force acts upward, it partially cancels the downward pull of gravity, making the object seem lighter. The word 'apparent' is crucial here. The true weight of an object — the gravitational force the Earth exerts on it — does not change when you place it in water. What changes is the net downward force the object exerts on whatever is supporting it, which is what a spring balance or your muscles actually measure. If you stand on a scale inside a swimming pool, the scale reads less than your true weight because the buoyant force from the water shares the job of supporting you against gravity. The scale reading is your apparent weight. Apparent weight loss has enormous practical consequences: it explains how ships float, why astronauts train underwater to simulate microgravity, how fish control their depth using swim bladders, and why measuring the density of irregular objects is far easier using Archimedes' method than by direct calculation. Understanding it requires connecting Newton's second law, fluid pressure, and the concept of displaced volume.

Key takeaways
  • True weight versus apparent weight
  • How pressure creates the buoyant force
  • Applying Newton's first law to find apparent weight
  • Partial versus full submersion
  • Using apparent weight loss to find density

True weight versus apparent weight

The true weight of an object is W = mg, where m is its mass and g is the gravitational field strength (9.81 m s⁻² near Earth's surface). This force always acts downward and never changes just because the object is submerged. Apparent weight is what a supporting device — a spring balance, a scale, or your hand — actually measures. It equals the contact force the support must supply to keep the object in equilibrium.

How pressure creates the buoyant force

Fluid pressure increases with depth: P = ρ_fluid × g × h. Because the bottom surface of a submerged object is deeper than its top surface, the upward pressure on the bottom is greater than the downward pressure on the top. The net result of these pressure forces over the entire surface is a single upward force — the buoyant force F_B. Archimedes showed this equals the weight of the fluid occupying the same volume as the object: F_B = ρ_fluid × g × V_displaced.

Applying Newton's first law to find apparent weight

If the object is stationary (acceleration = 0), the three forces acting on it must balance: the support force T (upward), the buoyant force F_B (upward), and the true weight W (downward). So T + F_B = W, which gives T = W − F_B. The reading T is the apparent weight. The apparent weight loss is simply ΔW = F_B = ρ_fluid × g × V_displaced. Notice it depends on the fluid's density and the displaced volume, not on the object's own density.

Partial versus full submersion

For a fully submerged object, V_displaced equals the object's entire volume. For a floating object (like a boat), the object sinks only until F_B = W, meaning it displaces a volume of fluid whose weight exactly equals the object's true weight — the apparent weight is then zero (the water, not a scale, is the support). Dense objects that sink always have F_B < W, so they still have positive apparent weight, just reduced.

Using apparent weight loss to find density

By weighing an object in air (W_air) and then fully submerged in water of known density ρ_w, you can find its volume: V = (W_air − W_apparent) / (ρ_w × g). Dividing mass by this volume gives the object's density. This is Archimedes' famous method for checking whether a crown was pure gold — he compared the buoyant force on the crown with that on an equal mass of pure gold.

The Key Equation

Wapparent=Wtrue−FB=mg−ρfluid g VdisplacedW_{\text{apparent}} = W_{\text{true}} - F_B = mg - \rho_{\text{fluid}}\, g\, V_{\text{displaced}}
  • W_{\text{apparent}} — Apparent weight (N): The force registered by the supporting scale or balance — what the object seems to weigh in the fluid.
  • m — Mass of object (kg): The object's inertial mass, unchanged by submersion.
  • g — Gravitational field strength (m s⁻²): Approximately 9.81 m s⁻² at Earth's surface; appears in both weight and buoyant force.
  • \rho_{\text{fluid}} — Fluid density (kg m⁻³): Density of the surrounding fluid — 1000 kg m⁻³ for fresh water, ~1025 kg m⁻³ for sea water.
  • V_{\text{displaced}} — Displaced volume (m³): Volume of fluid pushed aside by the object; equals the object's volume when fully submerged.

Worked Example

Problem. A granite boulder has a mass of 50 kg and a volume of 0.019 m³. It is fully submerged in fresh water (density 1000 kg m⁻³). Calculate (a) its true weight, (b) the buoyant force acting on it, and (c) its apparent weight when held on a spring balance underwater. Take g = 9.81 m s⁻².

Wapparent=490.5−186.4≈304 NW_{\text{apparent}} = 490.5 - 186.4 \approx 304\text{ N}

Answer. The boulder's apparent weight underwater is approximately 304 N, a reduction of about 186 N from its true weight of 490.5 N.

Figure
Diagram
Fresh Water (ρ = 1000 kg m⁻³)Boulderm = 50 kgSpring balancereads 304 NW = 490.5 N(true weight ↓)F_B = 186.4 N(buoyant force ↑)surfaceApparent weight = W − F_B = 304 N
Origins · 1 minArchimedes of Syracuse and how this was discovered
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Archimedes of Syracuse
-287–-212 · Greek (Sicilian)

Archimedes formulated the principle bearing his name — that a body immersed in a fluid is buoyed up by a force equal to the weight of the displaced fluid — in his treatise On Floating Bodies (circa 250 BCE). This was arguably the first quantitative law in fluid statics, predating Newton's formal mechanics by nearly two millennia. He applied it to determine whether King Hiero II's crown was pure gold without melting it down, by comparing the buoyant force on the crown with that on an equal mass of pure gold. Legend holds that Archimedes leapt from his bath and ran naked through the streets of Syracuse shouting 'Eureka!' (I have found it!) upon realising that the displaced water revealed the crown's volume. Whether true or not, the story captures his genuine excitement at the insight that volume could be measured by displacement.

  • Archimedes' Principle — Apparent weight loss is a direct consequence of Archimedes' Principle; understanding the buoyant force quantitatively is essential before calculating apparent weight.
  • Buoyant Force — The buoyant force is the mechanism causing apparent weight loss; its magnitude depends on displaced fluid volume and fluid density.
  • Floatation and Sinking — Whether an object floats or sinks determines whether its apparent weight reaches zero or remains positive, directly extending the idea of apparent weight loss.
  • Fluid Pressure — The depth-dependence of fluid pressure is the microscopic origin of the buoyant force and therefore of apparent weight loss.