# linear expansion

The Forth railway bridge has an overall length of 2,529 m (about 1½ miles). It is made of steel (coefficient of linear expansion = 0.000012 per °C). If the greatest difference in temperature between the coldest and hottest days is 35°C the length of the bridge can change by 2529 × 0.000012 × 35 = 1.06 m. In fact, provision is made for an expansion of 1.8 m.

A brass rod exactly 100 cm long
at 15°C is heated in a steam jacket and the resulting increase
in length is found to be 0.16 cm. If the temperature of the steam
is 99°C, what is the coefficient of linear expansion of brass?

Increase in length = *L _{o} × α
× t* = 0.16 cm,

original length (

*L*) = 100 cm,

_{o}rise in temperature (

*t*) = 99 - 15 = 84°C.

Hence, 0.16 = 100 × α × 84,

so that α = 0.000019 per °C.

A steel bridge is 100 m long when the temperature is 2°C. What will be its length when the temperature is 27°C? Unless provision is made for the increase in length linear expansion, as it is called – the bridge could buckle disastrously on a hot day. So before the bridge is built, engineers must be able to calculate the increase in length. In this example, it is just over 2½ cm.

The increase in length depends upon three things. First, as might be expected,
it is proportional to the **original length** of the structure.
For example, a steel bridge 100 m long will expand 10 times as much
as a steel rail 10 meters long when both are heated equally. Secondly, the
increase in length is proportional (or practically so) to the **increase
in temperature** which the structure undergoes. For example, if a
particular specimen expands by 1 mm when its temperature is raised through
25°C, it will expand by 4 mm when its temperature is raised through
11°C. Thirdly, the increase in length depends on the **material** from which the structure is made.

As usual, when a quantity is proportional to a number of factors it is also
proportional to the factors multiplied together (i.e., the product of the
factors). Hence the increase in length of a specimen is proportional to
the original length × the increase in temperature. To find the actual
increase in length, however, the product is multiplied by a number, which
depends on the material, called the **coefficient of linear expansion**.
The increase in length then equals the original length × coefficient
of linear expansion × increase in temperature. This equation can be
written as:

Increase in length = *L _{o}* ×

*α*×

*t*

where *L _{o}* is the original length,

*α*is the coefficient of linear expansion, and

*t*is the increase in temperature.

With the aid of this equation it is easy to solve the bridge problem we
posed earlier. The original length (*L _{o}* of the bridge
is 100 m, the coefficient of linear expansion of steel is 0.000012 per °C,
and the rise in temperature (

*t*) is 27 - 2 = 25°C. The increase in length is therefore 100 × 0.000012 × 25 m = 0.03 m = 3 cm.

The same equation can be used to find the *decrease* in length of
a solid when the temperature decreases. What will be the length of the bridge
when the temperature drops to *minus* 13°C (a temperature decrease
of 15°C)? The answer is 100 × 0.000012 × 15 = 0.018 m = 1.8
cm, and the length of the bridge at -13°C is 100 - 0.018 m = 99.982
m.

## Linear, superficial, and cubical expansion

Often the equation for linear expansion is given as:

*L _{t}* =

*L*(1 +

_{o}*αt*)

where *L _{t}* is the new length after expansion,

*α*the coefficient of linear expansion, and

*t*the rise of temperature.

A similar equation for calculating the increase in *area* (superficial
expansion) is *A _{t}* =

*A*(1 +

_{o}*βt*), where

*A*is the new area after expansion,

_{t}*A*the original area, and

_{o}*β*the coefficient of superficial expansion. β is equal to twice the coefficient of linear expansion, i.e.,

*β*= 2

*α*.

The equation for calculating the increase in *volume* (cubical
expansion) is *V _{t}* =

*V*(1 +

_{o}*γt*), where

*V*is the new volume after expansion,

_{t}*V*the original volume, and

_{o}*γ*the coefficient of cubical expansion. γ is equal to three times the coefficient of linear expansion, i.e.,

*γ*= 3

*α*.