We know that matter exists in three different states. They
are solid, liquid and gaseous states. Whenever heat energy is supplied to any
of the states there is a increase in the size and it is called expansion. The
basic reason for the expansion is simple. The different states of the matter
will have molecules and with a certain separation between them. When the
temperature is increased and there is a permanent increase in the distance
between the molecules which leads to the expansion.

In the case of the solid material the force of attraction
between the molecules is very strong and the molecules are very tightly packed
with each other. When the heat energy is given the molecules absorb the heat
energy and starts vibrating about the mean position. When the heat energy is
withdrawn the molecules are supposed to come back to their original position
and a body is also supposed to get its original state. But due to the
non-harmonic nature of the molecule vibration, there is a permanent increase in
the distance between the molecules. Because of lack of perfect elasticity, the
molecules are unable to come back to their original positions which lead to a
permanent expansion.

A solid material can expand along its length, area and along
its volume. To measure the expansion along the length, we have coefficient of
linear expansion. To measure the expansion along the area, we have coefficient
of year expansion. To measure the expansion along the volume, we have
coefficient of volume expansion.

Coefficient of linear expansion is defined as the ratio of
increase the length of the material to its original length the per 1°C rise in
temperature. It can be measured with the unit of 1°C or per degree Kelvin.

Coefficient of areal expansion is defined as the ratio of
increase in the area of the material to its original area per 1°C rise in
temperature.

Coefficient of volume expansion is defined as the ratio of
increase in the volume of the material to its original volume per 1°C rise in
temperature.

All these coefficients of expansion depend on the nature of
the material but is independent of physical dimensions of the body. It can be
clearly notice that coefficient of volume expansion is three times the
coefficient of areal expansion is two
times that of the coefficient of linear expansion.

We can derive the relation between them in a simple format as
shown below. To derive the relation we have considered a body in a cube shape with
the unit dimensions. Let us rise the temperature of this body by only 1°C and
we can write the equation for the final length, final area and the final volume
as shown.

We have also derived the equation further ratio of change in
the moment of inertia to its original moment of inertia in terms of coefficient
of linear expansion in the above equation. During this derivation we have
assumed that the change in the radius is small and hence we have applied the
concept of approximation.

**Applications of expansion of solids**

Expansion of the solids is used in our daily life that so
many occasions.

1. For example telephone and electric wires are arranged
little bit loosely between any of the two poles. It is simply because with the
decrease in temperature in winter season, the wire contracts and if they were
arranged tightly that will become further tight and they may break. To avoid
this problem, they were arranged loosely.

2.Between the two rails on a railway track always a small gap
is left. In the summer with the increase in temperature they expands as this
materials are made up of metals. One end of the rail is fixed and other end is
having a small gap with the next one. The expansion can happen in the gap
therefore the track will remain intact.

3. In electric glass bulbs tungsten is used as a filament.
The linear expansion coefficient of the glass is close to that of the tungsten
therefore with the increase in the temperature both of them expands equally so
that the use is not going to fail for a long time.

4. When water is sprinkled on a hot chimney glass only at the
particular points the glass contracts and that is the reason why the glasses
going to break.

**The loss or gain of a time with the pendulum clock**

1.The pendulum clock consists of the pendulum which is made
up of a metal.

2.In the summer when temperature is raised the length of the
pendulum increases and hence it’s time period also increases.

3.It means it is going to take more time to make one
oscillation.

4.It implies it is going to make less number of oscillations
in a given time. It means it is going to show less time than required when the
temperature is raised.

5. In the winter season the temperature decreases and hence
the pendulum makes more number of oscillations than required and hence it shows
again of time.

6. We can measure the loss or gain of the time using a
mathematical equation. We can write the equation for the time period of a
pendulum using a concept that time period is directly proportional to Squire
route of its length.

7. The equation further difference in the time with respect
to the variation of the temperature is as shown below.

In the place of the time period we can write the number of
the seconds per day when we are calculating the loss/gain of the time with
respect to the pendulum per day. Depending on the question we can write any
number of the seconds as per the demand of the problem. For example if we have to
calculate the difference in the time for half today we have to write the number
of the seconds per half-day only.

**Thermal stress**

Assume that a wire of certain modulus of elasticity is
arranged between two rigid supports. When the temperature of the system is
reduced the wire tries to contract. As it is permanently fixed, it cannot
contract. As a result stress is developed on the wire and the stress and is
called thermal stress. In the above diagram we have derived the equation for
this thermal stress also.

**Problem and solution**

A pendulum clock loses 10 seconds per day at 40°C and gains
five seconds per day at 20°C. At what temperature the pendulum clock shows
correct time?

We know it very clearly that pendulum clock shows less time
with a higher temperature and more time at a lower temperature.

It implies that the pendulum clock shows the correct time at
a temperature less than 40°C and more
than 20°C.

We can write the equation for the loss of the time in the
first case another gain of the time in the second case as shown below.

We can design a pendulum with a bimetallic strip so that it
can always show correct time.

The pendulum shall consists of two different materials of
similar sizes arranged together.

One metal with the increase of the temperature shall be
arranged in such a way that it moves in the upward direction.

Simultaneously the other metal shall move in the downward
direction.

We shall arrange such
that the expansion of the first metal in the upward direction so the length of
the pendulum increases.

And the expansion of the second material shall happen in
downward direction the pendulum length decreases.

If they are arranged in such a way that the increase in the
length of both the metals is same then the length of the pendulum comes back to
its original value and hence it always shows the correct time.

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