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Density & Hooke's Law

01Density & Hooke's Law

Density

This section covers density, its units, volume calculations and the use of the density equation.

Density and its equation

Density is the mass per unit volume of a material. It indicates how much mass is contained within a particular volume.

ρ=mV\rho=\frac{m}{V}
Symbol Quantity SI unit
ρ\rho density kg m−3^{-3}
mm mass kg
VV volume m3^3

Comparing densities

If equal volumes are compared, the sample containing more mass has the greater density. A lower-density material therefore contains less mass in the same volume than a higher-density material.

Density may also be expressed in units such as g cm−3^{-3} when the mass is measured in grams and the volume in cubic centimetres.

lower densityhigher density

Finding the volume

When the volume of a regular solid is not supplied directly, determine it from the dimensions of the object before applying the density equation.

Shape Volume
Rectangular block V=lwhV=lwh
Cube of side dd V=d3V=d^3
Cylinder V=πr2lV=\pi r^2l
Sphere V=43πr3V=\frac{4}{3}\pi r^3

Worked example: A rectangular slab has a mass of 73 kg73\text{ kg}. Its sides measure 850 mm850\text{ mm}, 500 mm500\text{ mm} and 40 mm40\text{ mm}. Determine the density of the material.

First express the dimensions in metres:

850 mm=0.850 m850\text{ mm}=0.850\text{ m}, 500 mm=0.500 m500\text{ mm}=0.500\text{ m}, 40 mm=0.040 m40\text{ mm}=0.040\text{ m}.

The slab therefore has volume:

V=(0.850)(0.500)(0.040)=1.70×10−2 m3V=(0.850)(0.500)(0.040)=1.70\times10^{-2}\text{ m}^3

Using ρ=m/V\rho=m/V:

ρ=731.70×10−2=4.29×103 kg m−3≈4.3×103 kg m−3\rho=\frac{73}{1.70\times10^{-2}} =4.29\times10^3\text{ kg m}^{-3} \approx4.3\times10^3\text{ kg m}^{-3}

Exam Tip: Put mass and volume into compatible units before substituting into ρ=m/V\rho=m/V. When converting volumes, remember that the length conversion factor is cubed: 1 mm3=10−9 m31\text{ mm}^3=10^{-9}\text{ m}^3 and 1 cm3=10−6 m31\text{ cm}^3=10^{-6}\text{ m}^3.

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