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Forces and pressure: 6.3 Forces and pressure in liquids - density & depth factors - liquid pressure calculations using the P = hρg formula

Doc Brown's Physics exam study revision notes: What is the formula for pressure?  What causes pressure in liquids? How do you calculate pressure in a liquid?

Sub-index for physics notes on 6. Forces and Pressure in fluids and calculations

This page contains online questions only. Jot down your answers and check them against the worked out answers at the end of the page


6.3 Pressure in a liquid - density and depth factors - calculations

Density is a measure of how close the particles are together.

The more compact they are, the greater the density.

As already mentioned, in liquids the density is uniform throughout and because there is so little space between the particles the density only slightly decreases with increase in temperature with the increased kinetic energy of the particles.

However, the volume shows almost no change with increased pressure (so here you can consider liquids to be virtually incompressible).

All liquids expand on heating - observe a mercury or alcohol thermometer.

experiment to demonstrate pressure in liquids increases with depth water can with holes in sideThe pressure in a fluid varies AND increases with depth - it doesn't matter whether you are dealing with gases like the atmosphere or liquids like the water of a lake or ocean.

The greater the height/depth of fluid, the greater the weight of particles that gravity is pulling down, hence the increase in force per unit area at a particular level, hence the increase in pressure.

The pressure in a fluid acts in all directions (← → ↑  ↓) because the particles are moving and colliding with each other, and the sides of the container, at random in all directions.

Liquid pressure significantly increases with depth as the weight of the column of liquid increases.

A simple experiment can demonstrate this rule using a tall container with holes in the side. When you fill it with water, the water gushes out of the holes, but the lower the hole, the greater the water pressure, the faster the water comes out and travels a greater distance.

A note on dam construction (e.g. reservoir for water supply or hydroelectric power plant)

Since water pressure increases with depth, to resist this increase in pressure, the width of a dam must also be increased to combat the increased force on the dam wall.

The width of the dam is much greater at its base compared the top of the dam wall.

 

The pressure also increases with increase in density of the fluid - so air and water differ significantly for a pressure created at a specific depth of fluid (densities: air density 1.2 kg/m3 and water 1000 kg/m3 at room temperature).

From your own experience you may have observed:

Sometimes in a car descending or ascending a few hundred metres on a road can make your ears 'pop' and hurt slightly because of the change in pressure with height.

BUT, you only have to dive into a few metres of water to experience the same effect on your ears. As you suddenly into the water, the pressure is suddenly increased on your ear drums - the pain comes from the greater external water pressure than the internal body pressure on the other side of your ear drum. However, for most people, when under the water at shallow depths, the pressures become equal quite quickly

You can calculate the pressure at a given depth created by the weight of liquid in the earth's gravitation field using the following formula:

pressure in a liquid = depth of liquid x density of liquid x gravitational field strength

P = hρg

P, pressure in pascals (Pa);   h = depth in metres (m);   ρ = density (kg/m3),

and the gravitational field strength =  g = 9.8 N/kg (on the Earth's surface)

Unit connections

Taking the formula P = h x ρ x g 'apart' in terms of units.

pressure = force per unit area = height of column of material x density of material x gravitational constant

N / m2  =  m  x  kg/m3  x  9.8 N/kg

unit analysis: on the right the kg cancel out, m/m3 = 1/m2, you are left with N/m2 !!

 

Note: Upthrust force in fluids and flotation etc. are covered in

FORCES 7. Pressure & upthrust in liquids, why do objects float or sink in a fluid?, variation of atmospheric pressure with height

 

Example calculations involving liquid pressure  

 (the gravitational field effect is taken as 9.8 kg/N in these questions).

Q1 Divers have to be careful when working at depth in water and need to carefully control the dissolving of gases in their blood stream.

(a) Calculate the pressure created by a 30 m depth of water given the density of water is 1000 kg/m3

(b) Comment on the dangers when diving at great depths and how to avoid dangerous problems.

Worked out ANSWERS to the pressure in liquid questions

 

Q2 The density of sea water is ~1025 kg/m3, the maximum depth of the Atlantic ocean is ~8500 m (8.5 km).

(a) Calculate the water pressure at this depth.

(b) By what factor is the pressure greater at these depths compared to the ocean surface?

Worked out ANSWERS to the pressure in liquid questions

 

Q3 At what depth in water is the increased pressure five times greater than atmospheric pressure (101 kPa)?

Worked out ANSWERS to the pressure in liquid questions

 

Q4 At a depth of 12.5 m of a chemical solvent the pressure at the bottom of the storage tank due to the solvent was 306 kPa

Calculate density of the solvent.

 

Index physics Forces notes 6. Forces & pressure in fluids, calculations


Keywords, phrases and learning objectives for forces involving pressure situations

Be able to solve problems and answer calculation questions on the forces and creating pressure in liquids - be able to do calculations using the formulae P = F/A and P = hρg with the appropriate units.

Know how, and explain why, pressure in a liquid increases with increase in density or increase in depth.

Be able to describe a simple experiment that shows variation in liquid pressure with depth.


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Index physics Forces notes 6. Forces & pressure in fluids, calculations

Worked out ANSWERS to the pressure in liquid calculations

Example calculations involving liquid pressure  

 (the gravitational field effect is taken as 9.8 kg/N in these questions).

Q1 Divers have to be careful when working at depth in water and need to carefully control the dissolving of gases in their blood stream.

(a) Calculate the pressure created by a 30 m depth of water given the density of water is 1000 kg/m3 and gravity 9.8 N/kg.

P = hρg

P = 30 x 1000 x 9.8 = 294 000 Pa (2.94 x 105 Pa, 294 kPa)

(b) Comment on the dangers when diving at great depths and how to avoid dangerous problems.

Atmospheric pressure is about 101 kPa, so a diver at these depths will experience a much greater pressure than on the surface of the water

 Increase in pressure causes more gases to dissolve in the blood stream (this is a general rule for gases in contact with a liquid that can act as a solvent).

This can have serious consequences if time isn't allowed for the body pressure to adjust to the new external pressure, particularly when returning back to the surface.

The bends, also known as decompression sickness disease, occurs in divers when dissolved gases (mainly nitrogen) come out of solution in bubbles and can affect any body area including joints, lung, heart, skin and brain.

The effects can be fatal unless time is allowed for the body to adjust in a decompression chamber.

 

Q2 The density of sea water is ~1025 kg/m3, the maximum depth of the Atlantic ocean is ~8500 m (8.5 km).

(a) Calculate the water pressure at this depth.

P = hρg

P = 8500 x 1025 x 9.8 = 85 400 000 Pa (to 3 sf, 85.4 MPa, 85400 kPa, 8.54 x 107 Pa, 8.54 x 104 kPa)

(b) By what factor is the pressure greater at these depths compared to the ocean surface?

Atmospheric pressure is ~101 kPa

Pressure at bottom of ocean pressure at surface = 85400 101 = 846 (3 sf).

Note: This extraordinary increase in pressure mean to explore this 'alien' world you need a very strong submersible craft. However, evolution has allowed all sorts of creatures to live down at these depths, all fully pressure adjusted over time! If you (theoretically) brought any such creatures rapidly to the surface and exposed them to normal pressure, it would kill them!

 

Q3 At what depth in water is the increased pressure five times greater than atmospheric pressure (101 kPa)?

5 x 101 = 505 kPa, 505000 Pa, density of water 1000 kg/m3

P = hρg, rearranging gives h = P/ρg = 505000/(1000 x 9.8) = 51.5 m

Note: The pressure increase in water increases by about the value of atmospheric pressure for every 10 m.

 

Q4 At a depth of 12.5 m of a chemical solvent the pressure at the bottom of the storage tank due to the solvent was 306 kPa

Calculate density of the solvent.

P = hρg, rearranging gives ρ = P/hg = 306000/(12.5 x 9.8) = 2498 kg/m3

 

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