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

- 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 α and β:

- 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 α and γ:

- 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

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

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Do add applications of arial and cubical expansion

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