Hydrostatics: Archimedes’ principle.

Archimedes’ principle states that:

If a solid body floats, or is submerged, in a liquid, the liquid exerts an upthrust on the body equal to the gravitational force on the liquid displaced by the body.

In other words, if a solid body is immersed in a liquid, the apparent loss of weight is equal to the weight of liquid displaced.

If V is the volume of the body below the surface of the liquid, then the apparent loss of weight W is given by:

W = Vω = Vρg

where ω is the specific weight (i.e. weight per unit volume) and ρ is the density.

If a body floats on the surface of a liquid all of its weight appears to have been lost. The weight of liquid displaced is equal to the weight of the floating body.

Problem 8. A body weighs 2.760 N in air and 1.925 N when completely immersed in water of density 1000 kg/m3. Calculate (a) the volume of the body (b) the density of the body and (c) the relative density of the body. Take the gravitational acceleration as 9.81 m/s2.

(a) The apparent loss of weight is 2.760 N – 1.925 N = 0.835 N. This is the weight of water displaced, i.e. Vρg, where V is the volume of the body and ρ is the density of water,

Problem 9. A rectangular watertight box is 560 mm long, 420 mm wide and 210 mm deep.

It weighs 223 N. (a) If it floats with its sides and ends vertical in water of density 1030 kg/m3, what depth of the box will be submerged? (b) If the box is held completely submerged in water of density 1030 kg/m3, by a vertical chain attached to the un- derside of the box, what is the force in the chain?

(a) The apparent weight of a floating body is zero.

That is, the weight of the body is equal to the weight of liquid displaced. This is given by:

Vρg

where V is the volume of liquid displaced, and ρ is the density of the liquid.

(b) The volume of water displaced is the total volume of the box. The upthrust or buoyancy of the water, i.e. the ‘apparent loss of weight’, is greater than the weight of the box. The force in the chain accounts for the difference.

Volume of water displaced,

The force in the chain

= weight of water displaced – weight of box

= 499.1 N – 223 N = 276.1 N

Now try the following Practise Exercise

Practise Exercise 119 Further problems on Archimedes’ principle

Take the gravitational acceleration as 9.8 m/s2, the density of water as 1000 kg/m3 and the den- sity of mercury as 13600 kg/m3.

1. A body of volume 0.124 m3 is completely immersed in water of density 1000 kg/m3. What is the apparent loss of weight of the body? [1.215 kN]

2. A body of weight 27.4 N and volume 1240 cm2 is completely immersed in water of specific weight 9.81 kN/m3. What is its apparent weight? [15.25 N]

3. A body weighs 512.6 N in air and 256.8 N when completely immersed in oil of density 810 kg/m3. What is the volume of the body?

[0.03222 m3]

4. A body weighs 243 N in air and 125 N when completely immersed in water. What will it weigh when completely immersed in oil of relative density 0.8? [148.6 N]

5. A watertight rectangular box, 1.2 m long and 0.75 m wide, floats with its sides and ends vertical in water of density 1000 kg/m3. If the depth of the box in the water is 280 mm, what is its weight? [2.47 kN]

6. A body weighs 18 N in air and 13.7 N when completely immersed in water of density 1000 kg/m3. What is the density and relative density of the body?

[4.186 tonne/m3, 4.186]

7. A watertight rectangular box is 660 mm long and 320 mm wide. Its weight is 336 N. If it floats with its sides and ends vertical in water of density 1020 kg/m3, what will be its depth in the water? [159 mm]

8. A watertight drum has a volume of 0.165 m3 and a weight of 115 N. It is completely submerged in water of density 1030 kg/m3, held in position by a single vertical chain attached to the underside of the drum. What is the force in the chain? [1.551 kN]

Measurement of pressure

As stated earlier, pressure is the force exerted by a fluid per unit area. A fluid (i.e. liquid, vapour or gas) has a negligible resistance to a shear force, so that the force it exerts always acts at right angles to its containing surface.

The SI unit of pressure is the pascal, Pa, which is unit force per unit area, i.e. 1 Pa = 1 N/m2

The pascal is a very small unit and a commonly used larger unit is the bar, where 1 bar = 105Pa

Atmospheric pressure is due to the mass of the air above the Earth’s surface being attracted by Earth’s gravity. Atmospheric pressure changes continuously. A standard value of atmospheric pressure, called ‘standard atmospheric pressure’, is often used, having a value of 101325 Pa or 1.01325 bars or 1013.25 millibars. This latter unit, the millibar, is usually used in the measurement of meteorological pressures. (Note that when atmospheric pressure varies from 101325 Pa it is no longer standard.)

Pressure indicating instruments are made in a wide variety of forms because of their many different applications. Apart from the obvious criteria such as pressure range, accuracy and response, many mea- surements also require special attention to material, sealing and temperature effects. The fluid whose pressure is being measured may be corrosive or may be at high temperatures. Pressure indicating devices used in science and industry include:

(i) barometers (see Section 22.6),

(ii) manometers (see Section 22.8),

(iii) Bourdon pressure gauge (see Section 22.9), and

(iv) McLeod and Pirani gauges (see Section 22.10).