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- Whenever there is an increase in the dimensions of a body due to heating, then the body is said to be expanded and the phenomenon is known as expansion.
- Solids undergo three types of expansions a) Linear (Longitudinal) expansions, b) Superficial expansions (Arial) and c) Cubical expansions (Volumetric)

**Linear Expansion of Solid:**

- Whenever there is an increase in the length of a body due to heating then the expansion is called as linear 0r longitudinal expansion.

**Expression for the Coefficient of Linear Expansion of a Solid:**

- Consider a metal rod of length ‘
*l*_{0}’ at temperature 0 °C. Let the rod be heated to some higher temperature say t °C. Let ‘*l*’ be the length of the rod at temperature t °C.

∴ Change in temperature = t_{2} – t_{1} = t – 0 = t

and Change in length = *l *– *l*_{0}

Experimentally it is found that the change is length ( *l *– *l*_{0}) is

Directly proportional to the original length (*l*_{0})

*l *– *l*_{0} ∝ *l*_{0 }………………. (1)

Directly proportional to the change in temperature (t)

*l *– *l*_{0} ∝ t_{ }………………. (1)

Dependent upon the material of the rod.

From the equation (1) and (2)

*l *– *l*_{0} ∝ *l*_{0 }t

*∴ l *– *l*_{0} = α *l*_{0 }t …………… (3)

Where ‘α’ is a constant called a coefficient of linear expansion

This is an expression for the coefficient of linear expansion of a solid.

- The coefficient of linear-expansion is defined as the increase in length per unit original length at 0
^{0}c per unit rise in temperature.

From equation (3) we get

*∴ l * = *l*_{0 }+ α *l*_{0 }t

*∴ l * = *l*_{0 }(1 + α t) ………….. (4)

This is an expression for length of rod at t °C

#### Note:

- The magnitude of the coefficient of linear expansion is so small that it is not necessary to take initial temperature as 0 °C.
- Consider a metal rod of length ‘
*l*_{1}’ at temperature t_{1}0 °C. Let the rod be heated to some higher temperature say t °C. Let ‘*l*’ be the length of the rod at temperature t_{2}_{2}°C. Let*l*_{0}’ be the length of the rod at the temperature 0 °C. Let α be the coefficient of linear expansion, then we have

* l _{1} * =

*l*

_{0 }(1 + α t

_{1}) ………….. (2)

* l _{2} * =

*l*

_{0 }(1 + α t

_{2}) ………….. (2)

Dividing equation (2) by (1) we get

- The coefficient of linear expansion is different for different material

**Superficial Expansion of Solid:**

- Whenever there is an increase in the area of a solid body due to heating then the expansion is called as superficial or arial expansion.

**Expression for the Coefficient of Superficial Expansion of a Solid:**

- Consider a thin metal plate of area ‘A
_{0}’ at temperature 0 °C. Let the plate be heated to some higher temperature say t °C. Let ‘A’ be the area of the plate at temperature t °C.

∴ Change in temperature = t_{2} – t_{1} = t – 0 = t

and Change in area = A* *– A_{0}

Experimentally it is found that the change is area (A* *– A_{0}) is

Directly proportional to the original area (A_{0})

*A *– A_{0} ∝ A_{0 }………………. (1)

Directly proportional to the change in temperature (t)

*A *– *A*_{0} ∝ t_{ }………………. (1)

Dependent upon the material of the plate

From the equation (1) and (2)

A – A_{0} ∝ A_{0 }t

*∴ A *– A_{0} = β A_{0 }t …………… (3)

Where ‘β’ is a constant called a coefficient of superficial expansion

This is an expression for the coefficient of superficial expansion of a solid.

- The coefficient of superficial expansion is defined as the increase in area per unit original area at 0
^{0}c per unit rise in temperature.

From equation (3) we get

*∴ A * = A_{0 }+ β A_{0 }t

*∴ A * = A_{0 }(1 + Βt) ………….. (4)

This is an expression for the area of the plate at t °C

#### Note:

- The magnitude of the coefficient of superficial expansion is so small that it is not necessary to take initial temperature as 0 °C.
- Consider a thin metal plate of area ‘A
_{1}’ at temperature t_{1}0 °C. Let the plate be heated to some higher temperature say t °C. Let ‘A’ be the area of the plate at temperature t_{2}_{2}°C. Let ‘A_{0}’ be the area of the plate at the temperature 0 °C. Let β be the coefficient of superficial expansion, then we have

*A _{1} * =

*A*

_{0 }(1 + β t

_{1}) ………….. (2)

* A _{2} * =

*A*

_{0 }(1 + β t

_{2}) ………….. (2)

Dividing equation (2) by (1) we get

- The coefficient of superficial expansion is different for different material

**Cubical Expansion of Solid:**

- Whenever there is an increase in the volume of the body due to heating the expansion is called as cubical or volumetric expansion.

**Expression for the Coefficient of Cubical Expansion of a Solid:**

- Consider a solid body of volume ‘V
_{0}’ at temperature 0 °C. Let the body be heated to some higher temperature say t °C. Let ‘V’ be the volume of the body at temperature t °C.

∴ Change in temperature = t_{2} – t_{1} = t – 0 = t

and Change in volume = V – V_{0}

Experimentally it is found that the change is volume ( V – V_{0}) is

Directly proportional to the original volume (V_{0})

V – V_{0} ∝ V_{0 }………………. (1)

Directly proportional to the change in temperature (t)

V – V_{0} ∝ t_{ }………………. (1)

Dependent upon the material of the body.

From the equation (1) and (2)

V – V_{0} ∝ V_{0 }t

*∴ V – V _{0}* = γ V

_{0 }t …………… (3)

Where ‘γ’ is a constant called a coefficient of cubical expansion

This is an expression for the coefficient of cubicalexpansion of a solid.

- The coefficient cubical expansion is defined as an increase in volume per unit original volume at 0
^{0}c per unit rise in temperature.

From equation (3) we get

*∴ *V* * = V_{0 }+ γ V_{0 }t

*∴ V * = V_{0 }(1 + γ t) ………….. (4)

This is an expression for volume of the body at t °C

#### Note:

- The magnitude of the coefficient of cubical expansion is so small that it is not necessary to take initial temperature as 0 °C.
- Consider a solid body of volume ‘V
_{1}’ at temperature t_{1}0 °C. Let the body be heated to some higher temperature say t °C. Let ‘V’ be the volume of the body at temperature t_{2}_{2}°C. Let ‘V_{0}’ be the volume of the body at the temperature 0 °C. Let γ be the coefficient of cubical-expansion, then we have

*V _{1} * = V

_{0 }(1 + γ t

_{1}) ………….. (2)

* V _{2} * = V

_{0 }(1 + γ t

_{2}) ………….. (2)

Dividing equation (2) by (1) we get

#### Relation Between Coefficients of Expansions

#### Relation Between the Coefficient of Linear Expansion and the Coefficient of Superficial Expansion of Solid:

- Consider a thin metal plate of length, breadth, and area
*l*_{0}, b_{0}, and A_{0 }at temperature 0 °C. Let the plate be heated to some higher temperature say t °C. Let*l*, b and A be the length, breadth, and area of the plate at temperature t °C.

Then original area = A_{0} = l_{0} b_{0} ,,,,,,,,,,,,,,,, (1)

Consider linear expansion

Length, l = l_{0} (1+ αt)

Breadth, b = b_{0} (1 + αt)

where α = coefficient of linear expansion

Final area = A = l b = l_{0} (1+ αt) × b_{0} (1 + αt)

∴ A = l_{0} b_{0} (1+ 2 αt + α²t²)

Now α is very small hence α2 is still small, hence quantity α²t² can be neglected

∴ A = A_{0 }(1+ 2 αt) ,,,,,,,,,,,,,,,,,, (2)

Consider superficial expansion of the plate area.

A = A_{0}( 1+ βt) ,,,,,,,,,,,,,,,,, (3)

From (2) and (3)

β = 2α

Thus the coefficient of superficial expansion is twice coefficient of linear expansion.

#### Relation Between the Coefficient of Linear Expansion and the Coefficient of Superficial Expansion of Solid:

- Consider a thin rectangular parallelopiped solid of length, breadth, height, and volume
*l*_{0}, b_{0}, h_{0}, and V_{0 }at temperature 0 °C. Let the solid be heated to some higher temperature say t °C. Let*l*, b, h and V be the length, breadth, height, and volume of the solid at temperature t °C.

Then original volume = V_{0} = l_{0} b_{0} h_{0 },,,,,,,,,,,,,,,, (1)

Consider linear expansion

Length, l = l_{0} (1+ αt)

Breadth, b = b_{0} (1 + αt)

Height h = h_{0} (1 + αt)

where α = coefficient of linear expansion

Final volume = V = l b h = l_{0} (1+ αt) × b_{0} (1 + αt)× h_{0} (1 + αt)

∴ V = l_{0} b_{0} h_{0 }(1+ 3 αt + 3 α²t² + α³t³ )

Now α is very small hence α2 is still small, hence quantity α²t², α³t³ can be neglected

∴ V = V_{0 }(1+ 3 αt) ,,,,,,,,,,,,,,,,,, (2)

Consider cubical expansion of the solid.

V = V_{0}( 1+ γt) ,,,,,,,,,,,,,,,,, (3)

From (2) and (3)

γ = 3α

Thus the coefficient of cubical expansion is thrice coefficient of linear expansion.

#### Notes:

We have β = 2α hence α = β/2 ………………… (1)

We have γ = 3α hence α = γ/3 ………………… (2)

From relations (1) and (2) we get

α = β/2 = γ/3

Hence 6 α = 3 β = 2γ

**Applications of Linear Expansion of Solids:**

**Gap is kept between two successive rails**

- The rails of a railway expand in summer and contract in winter. Therefore gaps are kept between successive rails to allow for their expansion. If there are no gaps, the increase in temperature will cause the rails to expand and they will overlap one another or dislodge from the position. This will be dangerous to the trains and may result in a severe accident.

**To put an iron tyre on wooden wheel of bullock cart, the iron tyre is heated first**

- The diameter of the iron tyre is always slightly less than the wooden wheel of the bullock cart. When the iron tyre is heated, its diameter increases. Due to increase in the diameter the circumference of the iron tyre increases. Hence it easily slides over the wooden wheel of the bullock cart. Then water is poured on the red-hot tyre. It contracts and grips the wheel firmly.

**The clocks regulated with pendulum require adjustment during summer and winter.**

- The pendulums of clocks expand in summer and contract in winter, therefore, they lose time in summer and gain time in winter. In order that the clocks should give correct time, the pendulums are made from invar which is an alloy having a very small coefficient of linear expansion in some clocks, compensating pendulums are used.
- A compensating pendulum is made of a number of iron and brass rods joined in such a way that the length of the pendulums remains constant even if there is a change in temperature. Such clocks show accurate time. If this is not possible the clocks require the adjustment in summer and winter.

**Bimetallic strips are used as temperature controlling devices.**

- The bending of the bimetallic strip is due to the difference in the coefficient of linear expansion of two different metals used in the bimetallic strip.
- A bimetallic strip consists two strips of different metals fixed to each other lengthwise An increase in temperature causes bending of the strip in such a way that the metal of greater linear coefficient of expansion lies on the outer side. A lowering of temperature again bends the strip, but with the metal of smaller linear coefficient of expansion on the outer side. Such strips can be used in electric iron, electric oven, refrigerators to control temperature.

**When the hot glass of a lamp is touched with a cold knife, it sometimes cracks. **

- When the hot glass of a lamp is touched with a cold knife, the part of the glass, which is in contact with the knife contracts. Due to this local contraction, While remaining portion remains in the expanded condition. Due to uneven expansion in the glass, the glass cracks.

**When hot milk is poured into a thick walled glass vessel it cracks.**

- When milk is poured into a thick-walled glass vessel at room temperature, the inner surface of the vessel gets expanded, while the outer surface remains at room temperature. Thus there is no expansion on the outer surface. Due to uneven expansion in the vessel, the glass cracks

#### The unit of the coefficient of expansion the same for linear, superficial as well as a cubical expansion:

- The coefficient of expansion is the ratio of two similar quantities divided by temperature. Therefore, its unit is the same as the reciprocal of the unit of temperature for any kind of expansion.
- If the temperature is measured on the Celsius scale the unit is per degree Celsius and if the temperature is measured on the Kelvin scale, the unit is per degree Kelvin.

**Applications of Superficial Expansion of Solids:**

#### Removing Tight Lids of Glass Jar:

- To open the lid of a glass jar that is tight enough, it is immersed in hot water for a minute or so. Metal cap expands and becomes loose. It would now be easy to turn it to open.
- The high-temperature water causes the metal lid and glass jar to expand. But the glass has a low coefficient of expansion than the material of the lid. Hence the lid expands more than the glass jar and the lid can be easily removed.

#### To Pass a Nail Through Hole in Metal Plate:

- To pass nail through a metal plate having a hole of diameter slightly less than that of the nail, the plate is heated. so that the diameter of hole increases and the nail can easily pass through it.

**Applications of Volumetric Expansion of Solids:**

#### Use of Mercury or Alcohol in Thermometer:

- A thermometer measures temperature by measuring a temperature dependent property. Expansion of liquid is temperature dependent property and they are in direct proportion. Thus measuring the change in volume of mercury we can find the change in temperature.
- Mercury has a high boiling point, and a highly predictable and shows a uniform response to changes in temperature. Mercury has a very high coefficient of volumetric expansion than the glass. Hence expands at faster rate than glass. The expansion of the glass is negligible.
- In a typical mercury thermometer, mercury is placed in a long, narrow sealed tube called a capillary. Because it expands at a much faster rate than the glass capillary, the mercury rises and falls with the temperature. A thermometer is calibrated with one of the temperature scales.

#### Riveting of two Metal Plates:

- Two steel plates can be jointly tightly together by a process called riveting. Rivets are heated to red hot condition and are forced through coaxial holes in the two plates. The end of hot rivets is then hammered and shaped. On cooling, the rivets contract and bring the plates tightly gripped to each other.

#### Design of Air Craft:

- The aircraft expands by 15-25centimetress during its flight due to increase in temperature on account of heat created by friction with the air. Designers used rollers (separators) to isolate the cabin and passenger area from the body of the aircraft so that the expansion does not rip the plane apart.

#### Overflow Tanks of Coolant in Automobiles:

- The efficiency of internal combustion engine depends on heat rejected to the surroundings. Thus efficient cooling of engine ensures the efficiency of the engine. Coolants are used to cool the engine. But during the process coolant itself undergoes volumetric expansion. If overflow tanks are not provided, there is a possibility of the bursting of coolant line. Hence automobiles have coolant overflow tanks.

#### Use of Thick Bottles for Soft Drinks:

- To avoid bursting of soft drink bottles containing gas, due to thermal expansion, their walls are made very thick.

Science > Physics > Expansion of Solids > You are Here |

Physics |
Chemistry |
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Do add applications of arial and cubical expansion

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