Friday, January 2, 2015

Expansion of Solids and Applications

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 When the heat energy is withdrawn the molecules are supposed to come back to their original position and a body is also supposed to get its original state.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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