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SPI Physics · Fluid Mechanics and Properties of Matter

Density, Buoyancy and Flotation

A liquid pushes up on anything dipped in it with a force equal to the weight of the liquid pushed aside, and a body floats when that push can match its whole weight.

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Why this matters

Four questions in the bank, one EASY, two MODERATE and one HARD (from the 2022 paper): the relative density of copper, a body weighed in air and in water on two balances, a cylinder lifted out of a liquid, and the luggage a wooden raft can carry. Each is one idea from Archimedes, applied in kilograms or newtons.

Concept 1 of 3: Density and relative density

A kilogram of cotton and a kilogram of iron weigh the same, but the iron takes far less room. Density says how much mass is packed into each unit of volume, and relative density compares that with water.

Definition

  • Density == mass ÷\div volume. Its SI unit is kg/m³; water's density is 1000 kg/m31000\ \text{kg/m}^3 (1 g/cm³).
  • Relative density == density of the substance ÷\div density of water.
  • Relative density is a ratio of two densities, so it has no unit. It equals the density in g/cm³ as a plain number.
  • A body with relative density less than 1 floats in water; more than 1, it sinks.

Density and relative density

ρ=mVrelative density=ρsubstanceρwater\rho = \frac{m}{V} \qquad \text{relative density} = \frac{\rho_{\text{substance}}}{\rho_{\text{water}}}

Worked example

A metal block has a mass of 540 g and a volume of 200 cm³. Find its density and its relative density.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Fluid Mechanics and Properties of Matter · Density, Buoyancy and Flotation

If density of water is 103 kg/m310^3\ \text{kg/m}^3 and density of copper is 8.9×103 kg/m38.9 \times 10^3\ \text{kg/m}^3. What is the relative density of copper?

Giving relative density a unit

Relative density divides one density by another, so the units cancel. Writing kg/m³ after it is wrong.

Concept 2 of 3: Buoyant force and the loss of weight in a liquid

Lift a friend in a swimming pool and she feels light. The water is pushing up on her. That upward push is the buoyant force, and Archimedes found exactly how big it is.

Definition

  • Archimedes' principle: a body wholly or partly in a liquid feels an upward force, the buoyant force, equal to the weight of the liquid it displaces.
  • So the body seems lighter: apparent weight == true weight −- buoyant force.
  • The more of the body is under the liquid, the more liquid it displaces and the larger the buoyant force. As a body is lifted out, it feels heavier and heavier, until it is fully out and weighs its full weight.
  • The liquid pushes up on the body, and the body pushes down on the liquid with an equal force (Newton's third law). So a vessel of water on a balance reads more by exactly the weight the body seems to lose.

Archimedes' principle

FB=weight of liquid displaced=Vρliquidgapparent weight=W−FBF_B = \text{weight of liquid displaced} = V \rho_{\text{liquid}} g \qquad \text{apparent weight} = W - F_B

Worked example

A metal piece weighs 12 N in air and has a volume of 0.0005 m³. Find its weight when it is fully under water. Take ρ=1000 kg/m3\rho = 1000\ \text{kg/m}^3 and g=10 m/s2g = 10\ \text{m/s}^2.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Fluid Mechanics and Properties of Matter · Density, Buoyancy and Flotation

Consider the following figures 1 and 2. In figure 1, the mass of the body A in air is 0.2 kg. The mass of the water container (with water in it) is 0.5 kg. In figure 2, when A is completely immersed in the water of the container B, the readings of the respective balances change as shown. What is the value of X?

Forgetting the push back on the water

When a hanging body is dipped in water, the water's balance reading goes UP by the weight the body loses. The total held by the two balances does not change.

Thinking the buoyant force depends on the body's weight

The buoyant force depends on the volume under the liquid and the liquid's density, not on how heavy the body is. Two bodies of the same volume, fully under water, get the same push.
Drill 1 more on buoyant force and the loss of weight in a liquid

Concept 3 of 3: Floating and the extra load a floating body can carry

A raft can keep taking load until it is pushed right down to the water line. At that moment the water it displaces is as much as it can ever displace, and that sets the most it can hold up.

Definition

  • A body floats when the buoyant force equals its weight, that is, when it displaces its own weight of liquid.
  • For a floating body, the fraction under the liquid equals its density divided by the liquid's density. Ice, of density about 900 kg/m³, floats with nine tenths under water.
  • The most a body can hold up is the weight of liquid equal to its whole volume. Mass of water it can displace == volume ×\times 1000 kg/m³.
  • So the extra load it can carry == (mass of water equal to its volume) −- (its own mass). Subtract every load already on it to find what is left.
  • An iron ship floats though iron is denser than water: its hollow shape displaces much more water than a solid lump of the same iron.

Flotation

V=mρgreatest load (in kg)=V×1000−mvolume under watertotal volume=ρbodyρwaterV = \frac{m}{\rho} \qquad \text{greatest load (in kg)} = V \times 1000 - m \qquad \frac{\text{volume under water}}{\text{total volume}} = \frac{\rho_{\text{body}}}{\rho_{\text{water}}}

Worked example

A wooden block of mass 30 kg has a density of 500 kg/m³. What is the greatest extra mass it can hold up in water without sinking?
Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q84Hard

Example 3 · Fluid Mechanics and Properties of Matter · Density, Buoyancy and Flotation

A raft of wood (density 600kg/ m3\text{m}^3) of mass 120kg floats in a pond. Rahul whose mass is 37kg, has to travel on it. How much additional luggage can he carry with him without sinking the raft?

Stopping at the total load

The greatest load a raft can hold includes the people on it. If a person is already aboard, take his mass away too before giving the luggage he can bring.

Using the body's density for the water

The water displaced is the body's volume times the density of WATER, 1000 kg/m³. Multiplying the volume by the body's own density only gives its own mass back.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

  • Density and relative density

    Density and relative density

    ρ=mVrelative density=ρsubstanceρwater\rho = \frac{m}{V} \qquad \text{relative density} = \frac{\rho_{\text{substance}}}{\rho_{\text{water}}}
  • Buoyant force and the loss of weight in a liquid

    Archimedes' principle

    FB=weight of liquid displaced=Vρliquidgapparent weight=W−FBF_B = \text{weight of liquid displaced} = V \rho_{\text{liquid}} g \qquad \text{apparent weight} = W - F_B
  • Floating and the extra load a floating body can carry

    Flotation

    V=mρgreatest load (in kg)=V×1000−mvolume under watertotal volume=ρbodyρwaterV = \frac{m}{\rho} \qquad \text{greatest load (in kg)} = V \times 1000 - m \qquad \frac{\text{volume under water}}{\text{total volume}} = \frac{\rho_{\text{body}}}{\rho_{\text{water}}}

Watch out for (5)

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