FR2, FR12, HR2, HR12 - Density and pressure | Eduqas GCSE Mathematics

Interpret mass per unit volume and force per unit area, use their relationships in either direction, and compare measures in consistent units. FR2/HR2 establish the concepts; FR12/HR12 apply their compound units.

Density

Does the heavier object necessarily have the greater density? A large amount of a less dense material can have more mass than a small amount of a denser material.

Density

Density is mass per unit volume. A uniform material with density $2.7\,\mathrm{g/cm^3}$ has mass $2.7\,\mathrm g$ in each cubic centimetre. Mass is measured in units such as grams or kilograms; volume in cubic units such as $\mathrm{cm^3}$ or $\mathrm{m^3}$ .

Density compares equal volumes, so size alone does not determine it. For a uniform material under the same conditions, doubling volume doubles mass while density stays the same.

Density, mass and volume

D=\frac{m}{V},\qquad m=DV,\qquad V=\frac{m}{D}.

Here $D$ is density, $m$ mass and $V$ volume. The units must agree: $\mathrm{g/cm^3}$ with $\mathrm{cm^3}$ gives grams. The relationship may be given in a question; it is not on the additional formula sheet.

Divide mass by volume to find density, rather than volume by mass.

Worked example 1: finding density

Question. A uniform solid has mass $540\,\mathrm g$ and volume $200\,\mathrm{cm^3}$ . Find its density.

Plan. Divide the mass by the number of cubic centimetres.

D=\frac{540}{200}=2.7\,\mathrm{g/cm^3}.

Answer. $2.7\,\mathrm{g/cm^3}$ .

Check. $200$ portions of $1\,\mathrm{cm^3}$ , each with mass $2.7\,\mathrm g$ , give total mass $200\times2.7=540\,\mathrm g$ .

Reverse the same relationship when mass or volume is unknown.

Worked example 2: finding mass and volume

Question. A uniform material has density $1.2\,\mathrm{g/cm^3}$ . (a) Find the mass of $350\,\mathrm{cm^3}$ . (b) Find the volume of a sample with mass $0.84\,\mathrm{kg}$ .

Plan. Multiply density by volume for (a); convert kilograms to grams before dividing by density for (b).

\begin{aligned} \text{(a)}\quad m&=1.2\times350=420\,\mathrm g\\ \text{(b)}\quad 0.84\,\mathrm{kg}&=840\,\mathrm g,\qquad V=\frac{840}{1.2}=700\,\mathrm{cm^3}. \end{aligned}

Answer. (a) $420\,\mathrm g$ . (b) $700\,\mathrm{cm^3}$ .

Check. The second mass $840\,\mathrm g$ is twice $420\,\mathrm g$ , and its volume $700\,\mathrm{cm^3}$ is twice $350\,\mathrm{cm^3}$ , as expected for the same density.

If units do not match, convert before substituting. Dividing $0.84$ kilograms by $1.2$ grams per cubic centimetre without a mass conversion mixes units.

Watch out

Density is an amount per cubic unit, not the mass of the entire object. Keep a unit attached to the final numerical answer.

Find a fresh density.

Pressure

Pressure describes how concentrated a force is over a contact area. The same downward force produces greater pressure when supported by a smaller area.

Pressure

Pressure is force per unit area. Force is measured in newtons $(\mathrm N)$ , rather than kilograms. A pressure of $2\,\mathrm{N/cm^2}$ means $2$ newtons per square centimetre of the stated area.

Use the area actually in contact with the surface, not the object's total surface area.

Pressure, force and area

P=\frac{F}{A},\qquad F=PA,\qquad A=\frac{F}{P}.

Here $P$ is pressure, $F$ force and $A$ contact area. Use $\mathrm N$ and $\mathrm{m^2}$ for $\mathrm{N/m^2}$ , or $\mathrm N$ and $\mathrm{cm^2}$ for $\mathrm{N/cm^2}$ . The relationship may be given in a question; it is not on the additional formula sheet.

Choose the form that makes the unknown the subject.

Worked example 3: finding pressure

Question. A load exerts a force of $600\,\mathrm N$ over a contact area of $0.03\,\mathrm{m^2}$ . Calculate its pressure.

Plan. Divide force by contact area; the given units produce newtons per square metre.

P=\frac{600}{0.03}=20000\,\mathrm{N/m^2}.

Answer. $20000\,\mathrm{N/m^2}$ .

Check. Pressure times area gives $20000\times0.03=600\,\mathrm N$ , recovering the force.

An unknown contact length can require a second step after finding area.

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FR2, FR12, HR2, HR12 - Density and pressure

Density

Pressure

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