HYDRO-STATICS A-LEVEL PHYSICS

HYDROSTATICS

Pressure

The pressure at a point in a fluid is the force per unit area acting normal to an infinitesimal area taken about the point.  The units of pressure  Nm^-2 or Pascal (Pa).    The pressure in a column of fluid increase with depth. At a given point in a liquid, pressure acts in any direction hence it is a scalar quantity.

Formula for pressure in liquids

Sippose that a horizontal plate X of area A is placed at a  depth h below the liquid surface. By drawing vertica lines from the points on the perimeter of X, we can see that the force on X due to the liquid is equal to the weight of the liquid of height h and uniformm cross section A.

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Archimedes principles

When an object is immersed in a fluid, it experiences an upward force called up thrust which is equal to the weight of the fluid displaced.

Proof of Archimedes principle

Consider a uniform solid cylinder of length L, cross sectional area A. Suppose the cylinder is submerged in a liquid of density P, so that its face is a depth h, below the surface of the liquid.

Consider a solid immersed in a liquid, the pressure on the lower surface C is greater than  on the upper surface B, since the pressure at the greater depth h₂ is more than at h₁. The net horizontal force is zero.

The upward force on C = h₂rgA, where r is the liquid density.

The down wad force on side B = h₁rgA.

Thus resultant force on solid = upward force (upthrust) = (h₂– h₁)rgA.

But (h₂– h₁)A = volume, V,  of solid,

Therefore upthrust = Vrg = mg, where m = Vr.

Therefore upthrust = weight of liquid displaced.

Measurement of density or relative density using Archimedes’ principle.                                                                                          

For a solid, weigh the mass of solid in air say, m_o. Then weigh its mass when totally immersed in water say, m₁.

Then upthrust = (m_o – m₁) g = weight of water displaced.

For a liquid.

Weigh the mass of solid in air say, m₀, then weigh it when totally immersed in the liquid whose density is required say m₁ and finally weigh it when totally immersed in water say m₂.

Law of flotation

A floating body displaces its own weight of the fluid in which it floats.

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THE HYDROMETER

This is a device for comparing densities of liquids. It consists of a uniform stem having a loaded bulb at the bottom. The stem is graduated in which it is placed.

Practical hydrometers have a weighted end M for stability, a wide bulb tp produce sufficient upthrust to counterbalance the weight, and a narrow stem BL for sensitivity. If V is the whole volume of the hydrometer, a is the area of the stem and y is the length not immersed in a liquid of density,r, then upthrust = weight of liquid displaced

Examples.                                                                                                                                    1.        A cube of rubber, volume 10^-3m³, floats with half of its volume submerged in a liquid of density 1200kgm^-3. Find the depth to which the cube would be submerged in a liquid of density 1000kgm^-3.