Properties of matter for NDA: density, pressure, buoyancy and fluids

The NDA physics syllabus opens with matter: mass, weight, volume, density and specific gravity, Archimedes' principle, pressure and the barometer. Its "how things work" list then adds the siphon, pumps, the hydrometer and the balloon. All of them rest on the few ideas in this guide.

This is the most "everyday" chapter in the paper. Questions ask why a camel walks easily on sand, why hot water cleans better than cold, why a fountain pen leaks in an aircraft, why a sudden fall in the barometer warns of a storm. The numericals are simple too: pressure at a depth, a hydraulic lift, a fraction submerged. If you can say which principle is at work, the answer usually follows.

Typical question types are:

• density and relative density (specific gravity), and whether a body floats;
• pressure = force/area, with everyday examples;
• pressure in a liquid and why it depends only on depth;
• atmospheric pressure, barometers, straws, siphons and pumps;
• Pascal's law and hydraulic machines;
• Archimedes' principle and flotation;
• elasticity, surface tension, capillarity, viscosity and Bernoulli's principle.

Density and relative density

Density ρ = mass/volume, in kg/m³. Water is about 1,000 kg/m³ (1 g/cm³); ice about 920 kg/m³; mercury about 13,600 kg/m³.

Relative density (specific gravity) = density of the substance ÷ density of water. It is a pure number with no unit. Mercury's relative density is 13.6. A body floats in a liquid if its average density is less than the liquid's. A hydrometer measures the relative density of liquids; a lactometer is a hydrometer used to test the purity of milk.

Pressure

Pressure = force ÷ area, measured in pascal (1 Pa = 1 N/m²). The same force spread over a larger area gives less pressure, and concentrated on a small area gives more.

• Knives, needles and nails are sharp so that a small force produces a large pressure.
• School bags have broad straps, and heavy trucks and tractors have wide tyres, to reduce pressure.
• A camel's broad feet and snow shoes spread weight so they do not sink.
• Lying on a bed of nails is possible because hundreds of points share the weight.

Pressure in liquids

A liquid at rest exerts pressure in all directions. At a depth h below the surface:

P = hρg (plus atmospheric pressure acting on the surface, if the question wants total pressure).

The pressure depends only on depth and density, not on the shape or width of the vessel or the total amount of liquid. Points at the same depth in the same liquid have the same pressure. That is why water finds its own level in connected vessels.

• Dams are thicker at the base because water pressure grows with depth.
• Deep-sea divers need pressure suits, and submarines need thick hulls.
• Water tanks are placed on rooftops or towers so that taps below get a good pressure head.

Atmospheric pressure

The weight of the air above us presses on every surface. At sea level it is about 1.013 × 10⁵ Pa, equal to the pressure of a 76 cm column of mercury or about 10.3 m of water. Torricelli measured it with a mercury barometer; a water barometer would need a tube over 10 m tall, which is why mercury is used.

• Pressure falls with altitude. An aneroid barometer calibrated in height works as an altimeter in aircraft. Fountain pens can leak on flights because the air trapped inside the pen stays at ground pressure while the pressure outside falls, pushing ink out.
• Weather: a sudden, sharp fall in the barometer reading signals a storm; a gradual rise suggests fair weather.
• Drinking straw and syringe: sucking lowers the pressure inside; atmospheric pressure on the liquid surface pushes the liquid up.
• Siphon: a tube that carries liquid from a higher container over a rim to a lower outlet. It works because of the pressure difference, and it needs the outlet below the liquid surface in the source. It would not work in a vacuum.
• Lift (suction) pump: atmospheric pressure pushes water up the pipe, so it can never lift water more than about 10 m, and less in practice. Deeper wells need a submersible or force pump.

Pascal's law and hydraulic machines

Pressure applied at any point of an enclosed liquid is transmitted equally and undiminished in all directions. In a hydraulic lift, a small force F₁ on a small piston of area A₁ produces a pressure F₁/A₁, which acts on the large piston of area A₂ to give:

F₂ = F₁ × (A₂/A₁)

Hydraulic brakes, hydraulic jacks and presses, and the lifts at car service stations all use this. The larger force is paid for with distance: the small piston must move much farther than the large one.

Archimedes' principle and flotation

A body wholly or partly immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. Its apparent weight in the fluid is its true weight minus that buoyant force.

A body floats when it can displace its own weight of fluid before going fully under (the law of flotation). The fraction of a floating body under the surface = its density ÷ the liquid's density. So about nine-tenths of an iceberg lies below water.

• An iron ship floats because its hollow shape displaces a large volume of water.
• A ship floats higher in sea water than in fresh water, because sea water is denser. Load lines on a ship's hull mark safe loading in different waters.
• A submarine dives by filling ballast tanks with water and rises by blowing them out.
• A hot-air or helium balloon rises because the air it displaces weighs more than the balloon.
• When an ice cube floating in water melts, the water level does not change.

The full treatment, with relative density from weighing in air and water, is in Archimedes' principle and flotation.

Elasticity

A body that regains its shape when the deforming force is removed is elastic. Hooke's law: within the elastic limit, stress is proportional to strain. The constant of proportionality for stretching is Young's modulus.

Steel is more elastic than rubber. For the same stress, steel stretches far less, so its Young's modulus is much greater, and in physics a larger modulus means more elastic. Stretch a material beyond its elastic limit and it deforms permanently.

Surface tension and capillarity

Surface tension makes a liquid surface behave like a stretched membrane that tries to shrink to the least area.

• Raindrops and mercury droplets are spherical, because a sphere has the least surface area for a given volume.
• A dry needle or razor blade can float on water; some insects walk on it.
• Soap and detergents lower surface tension, so water spreads and wets greasy cloth better.
• Surface tension falls as temperature rises, which is one reason hot water cleans better.

Capillarity is the rise or fall of a liquid in a narrow tube. Water rises in glass (it wets glass and forms a concave meniscus); mercury falls (it does not wet glass and forms a convex meniscus). The narrower the tube, the greater the rise. Examples: oil climbing a lamp wick, ink soaking into blotting paper, a towel drying the body, water rising through soil. Farmers plough fields to break the fine capillaries in the soil, which slows the loss of moisture.

Viscosity and Bernoulli's principle

Viscosity is a fluid's internal friction, its resistance to flowing. Honey and glycerine are more viscous than water. The viscosity of a liquid decreases when heated (warm honey pours easily); that of a gas increases. Lubricating oils are chosen for the right viscosity. A body falling through a fluid reaches a steady terminal velocity when drag balances its weight, which is why raindrops and parachutists fall at a safe steady speed.

Bernoulli's principle: where a fluid flows faster, its pressure is lower. Examples:

• An aircraft wing is shaped so air moves faster over the top, giving lower pressure above and a lift force.
• Strong winds can lift tin roofs off, because fast air above lowers the pressure there.
• Sprayers and atomisers draw liquid up a tube into a fast jet of air.
• Two boats moving side by side close together tend to be drawn towards each other.

Worked NDA-style MCQs

Q1. The pressure due to water alone at a depth of 10 m (ρ = 1,000 kg/m³, g = 10 m/s²) is:
(a) 10³ Pa (b) 10⁴ Pa (c) 10⁵ Pa (d) 10⁶ Pa

P = hρg = 10 × 1,000 × 10 = 10⁵ Pa, about one atmosphere. Answer: (c).

Q2. In a hydraulic lift, the small piston has an area of 10 cm² and the large one 500 cm². A force of 100 N on the small piston can support a load of:
(a) 500 N (b) 1,000 N (c) 5,000 N (d) 50,000 N

F₂ = 100 × 500/10 = 5,000 N. Answer: (c).

Q3. A block of wood of density 600 kg/m³ floats in water. The fraction of its volume under water is:
(a) 40% (b) 100% (c) 6% (d) 60%

Fraction = 600/1,000 = 0.6. Answer: (d).

Q4. A body weighs 50 N in air and 30 N when fully immersed in water. Its relative density is:
(a) 2.5 (b) 1.67 (c) 0.6 (d) 5

Loss of weight = 20 N = weight of water displaced. RD = 50/20 = 2.5. Answer: (a).

Q5. A person of mass 60 kg stands on a floor, with a total contact area of 0.03 m² (g = 10 m/s²). The pressure on the floor is:
(a) 2,000 Pa (b) 18 Pa (c) 200 Pa (d) 20,000 Pa

P = 600 N ÷ 0.03 m² = 20,000 Pa. Standing on one foot would double it. Answer: (d).

Q6. Which statement is correct?
(a) Surface tension of water falls when detergent is added (b) Viscosity of a liquid increases on heating (c) Rubber is more elastic than steel (d) Liquid pressure depends on the shape of the vessel

Steel is more elastic; liquid viscosity falls on heating; liquid pressure depends only on depth and density. Answer: (a).

Common mistakes

• Treating relative density as having a unit. It is a ratio.
• Thinking a wider dam or tank changes pressure at the base. Only depth and density matter.
• Calling rubber more elastic than steel. Easy to stretch is not the same as elastic.
• Saying a siphon or suction pump works by "suction". Atmospheric pressure does the pushing.
• Guessing that the level rises when floating ice melts. It stays the same.

Practice set

1. Hydraulic brakes work on: (a) Archimedes' principle (b) Bernoulli's principle (c) Hooke's law (d) Pascal's law
2. Mercury in a glass capillary tube: (a) rises (b) falls (c) stays level (d) rises then falls
3. It is easier to swim in sea water than in river water because sea water: (a) is warmer (b) has waves (c) is denser and gives more buoyancy (d) is less viscous
4. An object of volume 0.002 m³ is fully immersed in water (g = 10 m/s²). The buoyant force on it is: (a) 20 N (b) 2 N (c) 200 N (d) 0.2 N
5. When a liquid is heated, its viscosity: (a) increases (b) decreases (c) stays the same (d) becomes zero
6. A sudden sharp fall in barometer reading indicates: (a) fair weather (b) snowfall only (c) a storm (d) no change
7. Raindrops are spherical because of: (a) viscosity (b) gravity (c) surface tension (d) air resistance
8. Normal atmospheric pressure supports a mercury column of about: (a) 76 cm (b) 76 mm (c) 10.3 m (d) 1 m

Answers:

1. (d). Pressure applied at the pedal is carried undiminished to every brake cylinder.
2. (b). Mercury does not wet glass.
3. (c). A denser liquid gives a larger upthrust for the same volume displaced.
4. (a). 1,000 × 0.002 × 10 = 20 N.
5. (b). For gases, it increases.
6. (c). A gradual rise suggests fair weather.
7. (c). A sphere has the least surface for its volume.
8. (a). Equal to about 10.3 m of water.

What to do next

• Make one line per principle (Pascal, Archimedes, Bernoulli, surface tension, capillarity, viscosity) with three everyday examples each.
• Solve ten numericals on hρg, hydraulic lifts and fraction submerged, with g = 10.
• Work through Archimedes' principle and flotation for the full set of buoyancy puzzles.
• Then move on to heat for NDA.