Thermal expansion of a metal plate

[Rohin.T.Narayan]

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

 

[Andrew Mason]

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)$$A'= A + A\alpha T$$
(2)$$A'-r' = (A-r) + (A-r)\alpha T$$

Subtracting (2) from (1):
$$r' = A + A\alpha T - ((A-r) + (A-r)\alpha T)$$
$$r' = r + r\alpha T$$

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

 

[thepatientmental]

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