Buoyancy, Viscosity & Elasticity

Matter is anything that has mass (weight) and occupies space, and it exists in three common states whose shape and volume behaviour distinguish them.

• Matter exists in three states — solid, liquid and gas.

A solid is the state of matter that has both a definite shape and a definite volume, because its molecules are very closely packed and tightly bound.

• Definite shape and volume — solids have a fixed shape and a fixed volume.
• Rigidity — solids are rigid and cannot be compressed easily.
• High density — solids are generally more dense than liquids and gases.
• Incompressibility — solids are almost incompressible.
• Low diffusion — diffusion of particles in solids is very slow.
• Strong intermolecular force — molecules are held together by strong intermolecular forces.

These two opposite properties describe whether a body returns to its original shape after a deforming force is removed.

• Quartz and phosphor bronze are almost perfectly elastic bodies.
• Plasticity — the property by which a body does NOT regain its original configuration after the deforming force is removed.
• Plastic deformation: original shape → force applied → after force removed, body does not regain its original shape.

When a deforming force acts on a body it produces both strain (the change it causes) and stress (the internal force that resists it).

• Strain has no unit (it is a pure ratio).

• Stress is the internal restoring force acting per unit area of cross-section of a deformed body.
• Stress is of two types — normal stress and tangential stress.

Within the elastic limit, stress and strain are directly proportional — this is Hooke's law, and the constant of proportionality is the modulus of elasticity.

• Young's modulus = stress / strain (for change in length); it measures a material's stiffness.

Every material can stretch only so far before it stops behaving elastically and eventually breaks.

• Breaking stress — the minimum stress required to break a wire.
• Elastic limit — the limit of stress and strain up to which a wire remains elastic.
• Plastic behaviour — if a wire is stretched beyond the elastic limit, the strain increases much more rapidly.
• If the stretching force is then removed, the wire does NOT come back to its natural length (permanent set).
• Fracture point — if deformation is increased further into the plastic region, the wire breaks at the fracture point.

How a material behaves between its elastic limit and its fracture point decides whether we call it ductile or brittle.

• Brittle — the wire breaks soon after the elastic limit is crossed (e.g. glass, cast iron).

Plotting stress (σ) on the Y-axis against strain (ε) on the X-axis gives the tension-test curve, whose key points are asked directly in exams.

• Up to B → elastic behaviour, Hooke's law applicable.
• Between B and C → plastic (permanent) deformation.
• Beyond C → necking starts and the wire finally breaks at D.

Repeated loading slowly tires out an elastic body and makes it less elastic over time.

• Due to elastic fatigue, bridges become less elastic after long use and are declared unsafe.
• A new bridge is more elastic, while an old bridge under repeated load becomes less elastic and unsafe.

A fluid is anything that can flow — both liquids and gases are fluids — and a key property of any fluid is its density.

• Examples — Liquids: water, oil. Gases: air, LPG.

• Density is the ratio of mass to volume (mass present in unit volume).
• Formula: Density (ρ) = Mass (m) / Volume (V).
• SI unit = kg/m³; density is a scalar quantity.
• Density of water = 1000 kg/m³.
• Density of water is maximum at 4°C (about 1000 kg/m³); at other temperatures it is lower (about 997 kg/m³).

• Relative density = density of substance / density of water; it has no unit.
• RD of water = 1.

• Hydrometer — an instrument used to measure the density or relative density of a liquid.
• Its working is based on the law of floatation.
• Parts: graduated stem, liquid-level reading, weight.
• It floats in the liquid; the denser the liquid, the less it sinks, so its density can be read off.

The normal force (thrust) a fluid exerts per unit area is its pressure, and the air above us exerts atmospheric pressure on every surface.

• Formula: P = F / A (thrust / area).
• Unit = N m⁻² or pascal (Pa).
• For a liquid column of height h: P = h ρ g.

• Atmospheric pressure is the pressure exerted by the atmosphere on the Earth's surface.
• Aneroid barometer — used to measure atmospheric pressure and the height of a place; parts: pointer, scale.
• Other units of atmospheric pressure are torr and bar.

Pressure applied to an enclosed fluid is passed on equally in every direction — this is Pascal's law, the basis of all hydraulic machines.

• Hydraulic lift, hydraulic press and hydraulic brakes all work on Pascal's law.
• For two pistons: F₁/A₁ = F₂/A₂, so a small force on a small area produces a large force on a large area.

When a body is dipped in a fluid, the fluid pushes it up — this upward force is the buoyant force (upthrust), and the property is buoyancy.

• Buoyant force = weight of the liquid displaced by the submerged part of the body.
• The buoyant force acts at the centre of gravity of the displaced liquid, called the centre of buoyancy.
• Formula: Fᵦ = ρₗ × V_d × g (ρₗ = density of liquid, V_d = volume displaced).
• Upthrust (Fᵦ) acts upward, weight (W) acts downward, through the centre of buoyancy.

• A body floats when its weight = weight of the fluid it displaces, i.e. upthrust balances weight.

Archimedes' principle gives the exact size of the upthrust and explains the apparent loss of weight in fluids.

• The loss in weight equals the weight of the liquid displaced by the submerged part of the body.
• Loss in weight = Fᵦ = weight of liquid displaced = ρₗ × V_d × g.
• Apparent weight = W − Fᵦ (true weight minus upthrust).
• Upthrust (Fᵦ) acts upward; true weight (W) acts downward.

The free surface of a liquid behaves like a stretched elastic skin that tries to shrink to the smallest possible area — this is surface tension.

• The minimum surface area for a given amount of liquid is a sphere — therefore raindrops are spherical.
• The surface acts like a stretched skin tending to minimise area.
• Examples: a needle can float on water; a soap film forms a stretched film.

• The free surface of a liquid at rest is always horizontal.
• In a container of any shape, the liquid takes the shape of the container, but its free surface stays horizontal.

• Temperature — surface tension of a liquid decreases as temperature increases.
• Surface tension becomes zero at the critical temperature.
• Soluble impurities — less soluble impurity → surface tension decreases; highly soluble impurity → surface tension increases.

• Soap, detergent, dettol, phenyl mixed in water decrease its surface tension.
• Adding salt to water increases its surface tension.
• Oil spreading over water decreases surface tension.
• Kerosene sprinkled on water lowers surface tension, so mosquito larvae sink and die.
• Warm soup is tasty because high temperature lowers surface tension, letting it spread over the tongue.
• Antiseptics like dettol have low surface tension, so they reach tiny cracks of a wound and clean germs.
• Soap solution has lower surface tension than pure water, so it cleans greasy stains better.

The rise or fall of liquid in a very fine tube is capillarity, decided by the tug-of-war between adhesion and cohesion.

• In water: adhesion > cohesion, so the liquid RISES in the tube.
• In mercury: cohesion > adhesion, so the liquid FALLS in the tube.

• Blotting paper soaks ink because its pores act as capillary tubes.
• Oil rises up the wick of a lamp by capillary action of the threads.
• Root hairs of plants draw water from the soil by capillary action.
• Farmers loosen the soil to prevent loss of water by capillary action.
• In a satellite, water rises to the other end of the tube because of its zero apparent weight, however long the tube.
• A towel soaks up water from the body by capillary action of the cotton.
• Melted wax rises up a candle's wick by capillary action.

Capillarity and surface tension arise from two kinds of intermolecular attraction — between like molecules, and between unlike molecules.

• Adhesive force — attraction between molecules of DIFFERENT substances (e.g. paper–gum, paper–ink).
• Cohesive force keeps molecules of the same substance together; adhesive force makes different substances stick.

A fluid resists motion between its own layers — this internal friction is the viscous force, and the property is viscosity.

• Oil between machine parts — viscous force of oil opposes motion and reduces wear and tear.
• Air on a parachute — air offers viscous force that slows the parachute's fall.
• Moving in water — water offers viscous force opposing the swimmer's motion.
• Flow of honey — honey is more viscous, so it flows slowly due to large viscous force.

• Viscosity — the property of a fluid by which it opposes relative motion between its layers (internal friction).
• Viscosity is a property of both liquids and gases.
• Viscosity of a liquid is due to cohesive force between molecules.
• Viscosity of a gas is due to diffusion of molecules from one layer to another.
• Viscosity of gases is much less than that of liquids.
• There is no viscosity in solids, because their particles are fixed in position.
• Viscosity of an ideal fluid is zero.
• With rise in temperature, viscosity of liquids DECREASES while that of gases INCREASES.
• Viscosity is measured by the coefficient of viscosity; SI unit = decapoise (kg/m·s) or pascal second (Pa·s).

Stokes' law gives the viscous drag on a small sphere moving through a fluid in terms of viscosity, size and speed.

• Formula: Fᵥ = 6 π η r v — Fᵥ = viscous force, η = coefficient of viscosity, r = radius of sphere, v = velocity.
• The viscous force acts opposite to the direction of motion, with Fᵥ ∝ η, Fᵥ ∝ r and Fᵥ ∝ v.

• Honey flows slowly because it is more viscous; water flows quickly because it is less viscous.
• Engine oil reduces friction in machines by its viscosity.
• Brake oil helps brakes work smoothly due to its viscosity.
• Blood has viscosity which helps regulate its flow in the body.