Expansion of Solids

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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 ‘l0’ 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 = t2 – t1 = t – 0 = t

 and Change in length = l l0

Experimentally it is found that the change is length ( l0) is



Directly proportional to the original length (l0)

l0   ∝  l0   ………………. (1)

Directly proportional to the change in temperature (t)

l0   ∝  t   ………………. (1)



Dependent upon the material of the rod.

From the equation (1) and (2)

l0   ∝  lt

∴   l l0   =  α lt    …………… (3)

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



Expansion of Solids linear 01

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 00c per unit rise in temperature.

From equation (3) we get

∴   l   = l0  +   α lt

∴   l   = l(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 ‘l1’ at temperature t10 °C. Let the rod be heated to some higher temperature say t °C. Let ‘l2’  be the length of the rod at temperature t2 °C. Let l0’ be the length of the rod at the temperature 0 °C. Let α be the coefficient of linear expansion, then we have

 l1  = l(1 +    α t1) ………….. (2)

 l2  = l(1 +    α t2) ………….. (2)

Dividing equation (2) by (1) we get

Expansion of Solids linear 03



  • 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 ‘A0’ 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 = t2 – t1 = t – 0 = t

 and Change in area = A – A0



Experimentally it is found that the change is area (A – A0) is

Directly proportional to the original area (A0)

A – A0   ∝  A0   ………………. (1)

Directly proportional to the change in temperature (t)

A A0   ∝  t   ………………. (1)

Dependent upon the material of the plate



From the equation (1) and (2)

A – A0   ∝  At

∴   A – A0   =  β At    …………… (3)

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



Expansion of Solids superficial 01

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 00c per unit rise in temperature.

From equation (3) we get

∴  A   = A0  +  β At

∴  A   = A(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 ‘A1’ at temperature t10 °C. Let the plate be heated to some higher temperature say t °C. Let ‘A2’  be the area of the plate at temperature t2 °C. Let ‘A0’ be the area of the plate at the temperature 0 °C. Let β be the coefficient of superficial expansion, then we have

A1  = A(1 +    β t1) ………….. (2)

 A2  = A(1 +    β t2) ………….. (2)

Dividing equation (2) by (1) we get

Expansion of Solids superficial 02

  • 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 ‘V0’ 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 = t2 – t1 = t – 0 = t

 and Change in volume = V – V0

Experimentally it is found that the change is volume ( V – V0) is

Directly proportional to the original volume (V0)

V – V0  ∝  V0   ………………. (1)

Directly proportional to the change in temperature (t)

V – V0  ∝  t   ………………. (1)

Dependent upon the material of the body.

From the equation (1) and (2)

V – V0  ∝  Vt

∴  V – V0   =  γ Vt    …………… (3)

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

Expansion of Solids cubical 01

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 00c per unit rise in temperature.

From equation (3) we get

∴   V   = V0  +  γ Vt

∴   V   = V(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 ‘V1’ at temperature t10 °C. Let the body be heated to some higher temperature say t °C. Let ‘V2’  be the volume of the body at temperature t2 °C. Let ‘V0’ be the volume of the body at the temperature 0 °C. Let γ be the coefficient of cubical-expansion, then we have

V1  = V(1 +   γ t1) ………….. (2)

 V2  = V(1 +  γ t2) ………….. (2)

Dividing equation (2) by (1) we get

Expansion of Solids cubical 02

Relation Between α and β:

  • Consider a thin metal plate of length, breadth, and area l0, b0, and Aat 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   =   A0 =   l0 b0 ,,,,,,,,,,,,,,,,  (1)

Consider linear expansion

Length,   l =   l0 (1+ αt)

Breadth, b = b0 (1 + αt)

where α = coefficient of linear expansion

Final area   =    A    =  l  b  =  l0 (1+ αt) × b0 (1 + αt)

∴   A    =  l0 b0  (1+ 2 αt + α²t²)

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

∴   A    =  A(1+ 2 αt)  ,,,,,,,,,,,,,,,,,,  (2)

Consider superficial expansion of the plate area.

A = A0( 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 l0, b0, h0, and Vat 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    =   V0 =   l0 b0 h0   ,,,,,,,,,,,,,,,,  (1)

Consider linear expansion

Length,   l =   l0 (1+ αt)

Breadth, b = b0 (1 + αt)

Height   h  = h0 (1 + αt)

where α = coefficient of linear expansion

Final volume    =    V    =  l  b h  =  l0 (1+ αt) × b0 (1 + αt)× h0 (1 + αt)

∴   V    =  l0 b0  h(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(1+ 3 αt)  ,,,,,,,,,,,,,,,,,,  (2)

Consider cubical expansion of the solid.

V = V0( 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
Physics Chemistry  Biology  Mathematics

2 Comments

  1. Shreyas Mohan Padghan

    Do add applications of arial and cubical expansion

    • Thanks for your suggestion. We have added the applications at the end of the article. We appreciate your suggestion because it is your website. your suggestions help us to improve it further. Continue suggesting. We are working to help as many students as we can. I would like to receive suggestions and corrections from your side. If you like the article share it in your group using social media. Let us make quality education free. Be the part of our movement.

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