Thermal expansion of a metal plate

In summary: ALIn summary, the conversation discusses the phenomenon of circular holes increasing in diameter when a metal plate is heated. The mathematical proof is demonstrated using the differential equations of coefficients of expansion, showing that the radius of the hole increases at the same linear rate as the metal.
  • #1
Rohin.T.Narayan
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Hi guys ,I am stuck up with the following problem,can you please give me a solution.

When a metal plate with a circular hole at its centre, is heated, definitely along with the areal expansion of the plate the diametre of the circular hole also increases .But can you give a mathematical proof for this using the differential equations of coefficients of expansions
 
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  • #2
Rohin.T.Narayan said:
When a metal plate with a circular hole at its centre, is heated, definitely along with the areal expansion of the plate the diametre of the circular hole also increases .But can you give a mathematical proof for this using the differential equations of coefficients of expansions
You don't need to worry about differential equations. It is a linear expansion.

The sides of the plate are A and B. Divide it into quarters. The length of the sides with the quarter hole taken out of them are A-r and B-r. Heat it up. The A side and holed A side will be:

(1)[tex]A'= A + A\alpha T [/tex]
(2)[tex]A'-r' = (A-r) + (A-r)\alpha T [/tex]

Subtracting (2) from (1):
[tex]r' = A + A\alpha T - ((A-r) + (A-r)\alpha T) [/tex]
[tex]r' = r + r\alpha T [/tex]

Do the same thing for the B and B-r sides.

So the hole radius increases at the same linear rate as the metal.

AM
 
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  • #3


Hello, thank you for reaching out with your question. The thermal expansion of a metal plate is a well-studied phenomenon in materials science and engineering. It is a result of the increase in atomic vibrations and spacing within the material when it is heated, leading to an increase in its physical dimensions. This effect is quantified by the coefficient of thermal expansion (CTE), which is a material-specific constant that relates the change in length or area of a material to the change in temperature.

In the case of a metal plate with a circular hole at its center, the CTE of the plate will determine the change in its dimensions when heated. Let us denote the initial diameter of the circular hole as D and the initial area of the plate as A. When the plate is heated, its area will increase by a factor of (1+αΔT), where α is the CTE and ΔT is the change in temperature. This means that the final area of the plate will be (1+αΔT)A.

Similarly, the diameter of the circular hole will also increase by a factor of (1+αΔT). To understand this mathematically, we can use the formula for the area of a circle, A=πr^2, where r is the radius of the circular hole. The initial radius of the hole is D/2, so the initial area of the hole is π(D/2)^2. When the plate is heated, the radius of the hole will increase by a factor of (1+αΔT), making the final radius (1+αΔT)(D/2). Plugging this into the formula for area, we get the final area of the hole as π[(1+αΔT)(D/2)]^2.

Expanding this equation and simplifying, we get (1+αΔT)^2(πD^2/4) as the final area of the hole. Comparing this to the initial area of the hole, we see that the final area is (1+αΔT)^2 times larger. This is the same factor by which the area of the plate increased, showing that the diameter of the hole also increases by (1+αΔT).

In conclusion, the mathematical proof for the increase in diameter of a circular hole in a metal plate due to thermal expansion is based on the material's CTE and the
 

FAQ: Thermal expansion of a metal plate

1. What is thermal expansion?

Thermal expansion is the tendency of a material to expand or contract in response to changes in temperature. This occurs because heat causes the atoms in a material to vibrate and take up more space, resulting in an increase in volume.

2. How does thermal expansion affect metal plates?

When a metal plate is heated, the atoms within the metal gain kinetic energy and vibrate more, causing the plate to expand. Conversely, when the plate is cooled, the atoms lose energy and vibrate less, resulting in contraction. This expansion and contraction can cause changes in the dimensions and shape of the metal plate.

3. Are all metal plates affected by thermal expansion?

Yes, all metals are affected by thermal expansion to some degree. However, different metals have different coefficients of thermal expansion, meaning they expand and contract at different rates. This can be important to consider when working with metal plates in applications where precise measurements are necessary.

4. How is thermal expansion measured in metal plates?

Thermal expansion in metal plates is typically measured using a device called a dilatometer. This instrument measures the change in length or volume of a material as it is heated or cooled. By recording these changes at different temperatures, the coefficient of thermal expansion for a specific metal can be determined.

5. Can thermal expansion cause damage to metal plates?

In extreme cases, thermal expansion can cause damage to metal plates. If a metal plate is heated or cooled too quickly, it can cause uneven expansion or contraction, leading to warping, cracking, or even breaking. It is important to take into account the coefficient of thermal expansion and properly manage temperature changes when working with metal plates to prevent damage.

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