We
are solving series of problems on the topic called expansion of solids. When
heat energy is given to a solid material, its molecules starts vibrating about
their mean position and when the heat energy is withdrawn they try to come back
to their original position due to their restoring forces among the molecules.
But because none is perfectly elastic, they fail to come back to their original
positions and hence there is change in the shape and it is called expansion.
This can happen along the length called linear expansion, along area called
areal expansion and can happen along the volume called volume expansion. To
measure this expansions, coefficient of linear expansions are defined. We can
also find the relation between them and found that they are respectively in the
ratio of 1:2:3.

**Problem**

Coefficients
of linear expansion of two different metals are in the ratio of 3:4. We need to
know the ratio of initial lengths so that for the same rise in temperature the
expansions will be the same and the problem is as shown in the diagram below.

**Solution**

We
know that the coefficient of linear expansion is defined as the ratio of
increase in the length of the rod to its original length per unit rise in the
temperature. Basing on that we can write increase in the length of the rod as
the product of initial length,coefficient of linear expansion and the change in
the temperature. As rise in temperature and and increase in the length is same,
the product initial length and coefficient of linear expansion is constant.
Thus initial lengths ratio is reciprocal to the coefficients ratio and the
solution is given as shown in the diagram below.

**Problem**

A
brass disc is having known diameter and it is having a hole of known diameter raised
to a certain temperature and we need to know the increase in the size of the
hole and the problem is as shown in the diagram below.

The
hole par t of disc expands as if there is content there and we need to use the
basic definition of linear expansion to solve the problem as shown in the
diagram below.

**Problem**

Two
different rods are having two different coefficient of linear expansions and at
all temperatures their difference in terms of lengths remains constant and we
need to find the initial lengths of both the rods.Problem is as shown in the
diagram below.

**Solution**

Their
initial lengths are different and hence there is a difference of lengths
between them. When we raise the temperature both the rods expands and it is
going to be different. For the difference between the lengths remain constant,
the increase in the length of each rod has to be same for the given
temperature. Taking this as the condition, we can solve the problem as shown in
the diagram below.

**Problem**

A
steel rod of half kilometer is used in the construction of the bridge and it
can withstand a maximum temperature of 40 degree centigrade. Coefficient of
linear expansion is given to us and we need to find the gap that has to be left
so that it can fit with out causing any problem. Problem is as shown in the
diagram below.

**Solution**

We
know that the gap that has to be left is the size to which the rod can expand.
We can measure the increase in the size of the rod as we know the initial
length, rise in temperature and coefficient of linear expansion. Problem is
solved as shown in the diagram below.

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