A rectangular sheet of length (2a) and breadth is (a) is heated uniformly. The shaded area A1,A2,A3 ARE CHANGES IN AREAS DUE TO HEATING.THEN...

When a rectangular sheet, with dimensions of length \(2a\) and breadth \(a\), is heated uniformly, it undergoes thermal expansion. This phenomenon causes the dimensions of the sheet to increase, leading to changes in its area. Let's break down how this works and what happens to the areas \(A_1\), \(A_2\), and \(A_3\) as a result of heating.

Understanding Thermal Expansion

Thermal expansion refers to the tendency of matter to change in volume in response to a change in temperature. For solids, this expansion is generally uniform, meaning that all dimensions of the object increase proportionally. The formula for linear expansion is given by:

ΔL = L₀ * α * ΔT

Where:

• ΔL = Change in length
• L₀ = Original length
• α = Coefficient of linear expansion (specific to the material)
• ΔT = Change in temperature

Calculating the Original Area

The original area \(A_0\) of the rectangular sheet can be calculated as:

A₀ = Length × Breadth = (2a) × (a) = 2a²

Changes in Dimensions Due to Heating

When the sheet is heated, both the length and breadth will expand. If we denote the change in temperature as \(ΔT\), the new dimensions after heating can be expressed as:

• New Length (L₁) = 2a + ΔL₁ = 2a + (2a * α * ΔT) = 2a(1 + αΔT)
• New Breadth (B₁) = a + ΔL₂ = a + (a * α * ΔT) = a(1 + αΔT)

Calculating the New Area

The new area \(A_1\) after heating can be calculated as:

A₁ = New Length × New Breadth = (2a(1 + αΔT)) × (a(1 + αΔT))

Expanding this gives:

A₁ = 2a²(1 + αΔT)²

Change in Area

The change in area \(ΔA\) can then be determined by subtracting the original area from the new area:

ΔA = A₁ - A₀ = 2a²(1 + αΔT)² - 2a²

Factoring out \(2a²\), we have:

ΔA = 2a²[(1 + αΔT)² - 1]

This expression shows how the area increases due to the uniform heating of the sheet. The areas \(A_1\), \(A_2\), and \(A_3\) you mentioned likely represent different stages or aspects of this expansion process, depending on how the heating is applied or measured.

Visualizing the Expansion

To visualize this, think of a balloon. When you heat the air inside, the balloon expands uniformly in all directions. Similarly, the rectangular sheet expands uniformly, increasing its area as the temperature rises.

Real-World Applications

This principle of thermal expansion is crucial in various engineering applications. For instance, when designing bridges or railways, engineers must account for the expansion and contraction of materials due to temperature changes to prevent structural damage.

In summary, heating a rectangular sheet leads to a predictable increase in its dimensions and area, which can be calculated using the principles of thermal expansion. Understanding these changes is essential in fields ranging from material science to engineering design.