Pressure in Liquids മർദം
Where does the upward push of water – the buoyant force – actually come from? Water presses on everything in it, and it presses harder the deeper you go. An object under water has its bottom deeper than its top, so the water pushes up on the bottom more strongly than it pushes down on the top. The difference is the buoyant force. This page explains pressure, shows how liquid pressure grows with depth, and then uses it to work out the buoyant force on a block, step by step.
- Force, weight and W = m × g (page 1).
- Volume (page 2) and density ρ = m/V (page 3).
1. Pressure = force ÷ area
Push a drawing pin into a board: its sharp tip goes in easily, but its flat head does not hurt your thumb. The force is the same at both ends; what differs is the area it is spread over. The force acting on each unit of area is called pressure (മർദം):
The same force on a smaller area gives a bigger pressure. That is why a knife is sharpened, why nails are pointed, and why it is hard to walk on soft sand at Kovalam beach in narrow heels but easy in flat slippers.
Each brick weighs 30 N. Stand it on its smallest face: the area is a quarter of the largest face, so the pressure is 4 times bigger and the brick sinks deeper. Stack two bricks: double the force on the same area gives double the pressure.
Worked example 1 · Standing on two feet
Q. A girl weighing 450 N stands on the floor. The total area of her two soles touching the floor is 0.03 m². What pressure does she exert on the floor?
The force is her weight. Divide it by the area: 450 ÷ 0.03 = 45 000 ÷ 3.
2. Liquid pressure grows with depth
Water has weight. Every layer of water presses down on the layers below it. So the deeper you go, the more water there is above you, and the greater the pressure. Picture a column of water of height h standing on an area A:
- its volume is V = A × h, so its mass is m = ρ × A × h, and its weight is ρ × A × h × g;
- this weight presses on the area A, so the pressure is (ρ A h g) ÷ A:
Two things decide it: the depth h and the density ρ of the liquid. (The air above the water also presses on it – atmospheric pressure. We leave that out here, because it presses on every side of an object equally and does not change the up-and-down difference.)
The jet from the deepest hole comes out fastest and lands farthest, because the pressure behind it is greatest. Lower the water level and every jet weakens; a hole above the water gives no jet at all.
3. Pressure acts in all directions
A solid block presses only down on the floor. A liquid is different: at any point in it, the pressure pushes equally in every direction – down, up and sideways. That is why water squirts sideways out of the holes in the bottle, and why a swimmer feels it on every part of the body. At the same depth in the same liquid, the pressure is the same everywhere.
The eight arrows are always equal: at one point, the pressure is the same in all directions. Go deeper and they all grow. The graph is a straight line through the origin, so double the depth gives double the pressure. A denser liquid gives a steeper line.
The Idukki arch dam is much thicker at its base than at its top, because the water presses hardest at the bottom of the reservoir. Divers who go down for pearl oysters or to inspect the hull of a ship at Cochin Shipyard feel the pressure on their ears grow as they go deeper. The bottom of a full overhead water tank must be strong for the same reason.
Worked example 2 · Pressure under water
Q. Find the pressure due to the water at a depth of 2 m in a pond. (ρ of water = 1000 kg/m³, g = 9.8 m/s²)
Use p = h ρ g with h = 2 m.
4. Why a block under water is pushed up
Now put a block in the liquid. The liquid presses on all six faces. The pushes on the left and right faces are equal and opposite, and so are the pushes on the front and back faces: they cancel. But the top face is at a smaller depth than the bottom face. So the downward force on the top is smaller than the upward force on the bottom. The difference is an upward force – the buoyant force (പ്ലവക്ഷമബലം).
The block's top face measures 10 cm × 10 cm (A = 0.01 m²). Lower it fully under, then keep going deeper: both the top and bottom forces grow, but their difference stays exactly the same. Lift it so part is above the surface (negative depth): there is no force on the top, and the upward force shrinks. It is always ρ × (volume under the liquid) × g.
Why the net force is ρ V g – step by step
Liquid pressure increases with depth (p = h ρ g) and acts in all directions. On an object in a liquid, the upward push on its lower surface is bigger than the downward push on its upper surface. The net upward force is F = ρliquid × Vunder liquid × g. Once the object is fully under, this does not change with depth.
Worked example 3 · A cube under water
Q. A cube of side 10 cm is held fully under water with its top face 20 cm below the surface. Find (a) the force on the top face, (b) the force on the bottom face, (c) the net upward force. Check that (c) equals ρ V g. (ρ = 1000 kg/m³, g = 9.8 m/s²)
A = 0.1 m × 0.1 m = 0.01 m². The bottom face is 10 cm deeper than the top, at 0.3 m. Force = pressure × area, with p = h ρ g.
9.8 N is the weight of 1 kg of water – exactly the water the cube pushes aside. Hold on to this: on page 7 it becomes Archimedes' principle (ആർക്കിമിഡീസ് തത്വം): the buoyant force equals the weight of the liquid displaced.