Density, pressure & gravity

Learning goals

What students should be able to explain, calculate and investigate after completing Density, Pressure and Gravity.

Goals marked Higher Level are required only for Higher Level students; the remaining goals are for all students.

• Define density and calculate it from mass and volume.
• Define pressure and calculate it for a force acting on a surface.
• Explain why pressure in a liquid increases with depth, and calculate it.
• State Newton's law of gravitation and distinguish gravitational field strength from weight.
• Higher Level: Calculate escape velocity from a planet or other celestial body.
• Describe experiments to measure density of solids and liquids, and to investigate pressure in liquids.

Key definitions

Precise terms, symbols and SI units used in Density, Pressure and Gravity.

Density

Density describes how much mass is packed into a given volume.

Density compares mass with the volume occupied. It is a property of a material under stated conditions, not a measure of how heavy the whole object is. Two objects made from the same uniform material have the same density even if their masses and volumes differ. For irregular solids, volume can be obtained by water displacement, provided the object does not dissolve or trap air.

Pressure

Pressure is force per unit area: the same force produces greater pressure when concentrated over a smaller area.

Pressure describes how concentrated a normal force is over a surface. The same force produces a greater pressure when it acts over a smaller contact area. Force is measured in newtons and area in square metres, so one pascal is one newton per square metre.

Pressure in Liquids

Liquid pressure increases with vertical depth because a deeper point supports a taller column of liquid above it.

Imagine a vertical liquid column of area A and height h. Its volume is Ah, so its mass is ρAh and its weight is ρAhg. Pressure is force divided by area, so p = ρAhg/A = ρgh. Here h is vertical depth below the liquid surface. At the same depth, pressure is greater in a denser liquid and is independent of container shape. This is the pressure due to the liquid column; absolute pressure also includes pressure at the surface, usually atmospheric pressure.

• Greater vertical depth h gives greater pressure because there is more liquid above.
• Greater density ρ gives greater pressure at the same depth.
• Container shape does not change the pressure at a given depth.
• For absolute pressure in an open container: total pressure = atmospheric pressure + ρgh.

Atmospheric pressure is caused by the weight of the air above us. In a mercury barometer, atmospheric pressure supports a vertical mercury column; a greater atmospheric pressure supports a greater height.

Buoyancy (Upthrust)

Buoyancy is the upward force a fluid exerts on an immersed object.

The pressure in a fluid increases with depth, so the force on the lower surface of an immersed object is greater than the force on its upper surface. This produces a resultant upward force. An object floats when this buoyant force balances its weight.

Gravity and Gravitational Field Strength

Every mass creates a gravitational field around it, and every other mass in that field experiences a force.

Gravitational field strength, g, is the force per unit mass at a point, and near the Earth's surface g ≈ 9.8 N kg⁻¹ (equivalent to 9.8 m s⁻², the acceleration due to gravity for a freely falling object). Newton's law of gravitation, F = Gm₁m₂/r², shows that this force decreases with the square of the distance between the two masses — doubling the distance reduces the force to one quarter. Weight: Fg = mg. Since mass can also be written as density × volume (m = ρV), weight can equivalently be written Fg = ρVg. Some books use W for weight, but W is also used for work done — always check the unit: weight is measured in newtons, work is measured in joules. Full treatment of circular orbits, satellites and Kepler's Third Law is covered in the Circular Motion topic; this lesson covers only the underlying gravitational force, field strength, and escape velocity.

Escape Velocity

Escape velocity is the minimum launch speed needed at a planet’s surface to escape its gravitational field without further propulsion, reaching an infinite distance with zero final speed in the ideal model.

It depends on the mass and radius of the planet: a more massive or more compact (smaller radius) planet has a higher escape velocity. Escape velocity does not depend on the mass of the escaping object itself.

Diagrams and formulae

Key diagrams and equations for reviewing Density, Pressure and Gravity.

• The same force produces a much greater pressure when concentrated over a small contact area than when spread over a large contact area.
• Pressure in a liquid increases with depth: at greater depth h, the liquid must support more weight above it, so p = ρgh increases.
• Gravitational field lines point radially inward towards the centre of a spherical mass such as the Earth, showing the direction of the force on a small test mass placed in the field.

The same force produces a much greater pressure when concentrated over a small contact area than when spread over a large contact area.

Pressure in a liquid increases with depth: at greater depth h, the liquid must support more weight above it, so p = ρgh increases.

Gravitational field lines point radially inward towards the centre of a spherical mass such as the Earth, showing the direction of the force on a small test mass placed in the field.

Worked Example

A step-by-step calculation applying the central relationships in Density, Pressure and Gravity.

• (a) p (water) = ρgh = 1000 × 9.8 × 5 = 49 000 Pa = 4.9 × 10⁴ Pa

Practical / Investigation

Experimental planning, measurement and evaluation for Density, Pressure and Gravity.

Experiment: Measuring the density of a solid and of a liquid

• For a regular solid: measure mass with a balance; measure dimensions with a vernier callipers or metre stick to find volume; calculate ρ = m/V.
• For an irregular solid: find volume by water displacement in a measuring cylinder or eureka can.
• For a liquid: measure the mass of an empty measuring cylinder, then the mass with a known volume of liquid added; find the liquid's mass by subtraction and calculate ρ = m/V.
• Sources of error: air bubbles trapped on the object increase the apparent displaced volume, making the measured volume too large and the calculated density too low; parallax error when reading the meniscus.
• Improvement: tap the container to release trapped air bubbles; read the meniscus at eye level.

• Measure mass and volume directly; no independent variable needs to be changed through a range.
• For displacement, fully submerge the object without trapping bubbles and read the lower meniscus at eye level. The object must not dissolve or absorb water.
• Repeat dimension/volume readings and report density with a unit and precision justified by the measuring instruments.

Collect primary data for at least one model. Use supplied secondary data to test all four models, calculate g and compare the values with 9.81 m s⁻².

• For primary data, repeat measurements, use a wide range of the independent variable and include uncertainties.
• For secondary data, identify the variables and units. Use a best-fit gradient for the pendulum and free-fall models, and calculate g from supplied planetary or liquid-pressure data for the other two models.
• Explain differences using air resistance, timing or distance uncertainty, model assumptions and the precision of the data.

Quick-Check Questions

Short retrieval questions covering the essential ideas in Density, Pressure and Gravity.

1. Define pressure and give its SI unit.

2. Why does pressure in a liquid increase with depth?

4. Distinguish gravitational field strength from weight.

5. What happens to the gravitational force between two masses if the distance between them is doubled?

6. Does escape velocity depend on the mass of the object being launched? Explain.

Common mistakes and exam tips

Common misconceptions and exam-technique checks for Density, Pressure and Gravity.

• Avoid: Treating liquid pressure as dependent on container shape. Correct idea: At a given depth, p = ρgh depends on liquid density, gravitational field strength and depth, not container shape.
• Avoid: Using distance between surfaces in Newton’s law of gravitation. Correct idea: Use the distance between the centres of the two masses.

• Keep density in kg m⁻³ and depth in metres when using p = ρgh.
• Distinguish gravitational field strength in N kg⁻¹ from weight in N.

Exam-style questions

Five exam-style questions with worked answers on Density, Pressure and Gravity.

Higher Level — Required Content

Required Higher Level content for this topic.

Escape velocity is the minimum launch speed needed to escape a body's gravitational field without further propulsion.

• Set the initial kinetic energy equal to the gravitational energy needed to reach a very large distance.
• The mass of the escaping object cancels; r is measured from the centre of the body.

End-of-unit revision

Formulae, definitions, experiments and mixed revision questions for Density, Pressure and Gravity.

• ρ = m/V
• p = F/A
• p = ρgh
• Fg = mg = ρVg (weight — not to be confused with work done, W = Fd)
• F = Gm₁m₂/r²
• g = F/m
• vₑ = √(2GM/r) (escape velocity)

• Density
• Pressure
• Pressure in a liquid
• Gravitational field strength
• Newton's law of gravitation
• Escape velocity

• Force spread over different areas showing p = F/A
• Pressure increasing with depth in a liquid
• Radial gravitational field lines around a spherical mass

• Measuring the density of a regular solid, an irregular solid (by displacement), and a liquid
• Investigating how pressure in a liquid varies with depth

• Density vs weight confusion
• Believing container shape/width affects liquid pressure at a given depth
• Confusing weight (W = mg) with work done (W = Fd)
• Mixing up g (N kg⁻¹) and weight (N)

1. Calculate the pressure due to water at a vertical depth of 3.0 m. Use ρ = 1000 kg m⁻³ and g = 9.8 m s⁻².

2. Explain why a nail has a sharp point, in terms of pressure.

3. Two point masses are 2 m apart and attract each other with a gravitational force of F. What would the force be if the distance were increased to 6 m?

4. Describe how you would find the density of an irregularly shaped object that sinks in water.

1. p = ρgh = 1000 × 9.8 × 3.0 = 2.94 × 10⁴ Pa. This is the pressure due to the water; atmospheric pressure would be added for absolute pressure.

2. A sharp point has a very small area, so for a given applied force, the pressure (p = F/A) is very high, allowing the nail to pierce the material.

3. Distance increases by a factor of 3, and F ∝ 1/r², so the new force = F ÷ 3² = F/9.

4. Find its mass using a balance. Fill a measuring cylinder partway with water and record the volume; lower the object in fully and record the new volume; the volume of the object is the difference. Calculate density using ρ = m/V.

The table below shows which equations from this unit appear in the official State Examinations Commission Formulae and Tables book, and where.

Note: the new Leaving Certificate Physics specification's own equation list confirms density, pressure, pressure in a liquid, weight, Newton's law of gravitation, gravitational field strength and escape velocity — but does not include Boyle's Law or thrust on a submerged surface; these two are old-syllabus material and are flagged accordingly above.