A circular hole in an aluminium plate is 2.725 cm in diameter at 12°C. What is its diameter when the temperature of the plate is raised to 140°C?

To determine the diameter of the circular hole in the aluminum plate when the temperature increases from 12°C to 140°C, we can use the concept of thermal expansion. Materials expand when heated, and this expansion can be quantified using a formula that relates the change in temperature to the change in dimensions.

Understanding Thermal Expansion

Thermal expansion is the tendency of matter to change its shape, area, and volume in response to a change in temperature. For solids, this is often expressed in terms of linear expansion. The formula for linear expansion is:

ΔL = L₀ × α × ΔT

• ΔL = change in length (or diameter, in this case)
• L₀ = original length (or diameter)
• α = coefficient of linear expansion for the material
• ΔT = change in temperature

Applying the Formula

For aluminum, the coefficient of linear expansion (α) is approximately 22 x 10⁻⁶ /°C. Now, let’s break down the steps to find the new diameter of the hole.

Step 1: Calculate the Change in Temperature

First, we need to find the change in temperature (ΔT):

ΔT = Final Temperature - Initial Temperature = 140°C - 12°C = 128°C

Step 2: Calculate the Original Diameter

The original diameter (L₀) of the hole is given as 2.725 cm.

Step 3: Calculate the Change in Diameter

Now, we can plug these values into the linear expansion formula:

ΔL = 2.725 cm × (22 x 10⁻⁶ /°C) × 128°C

Calculating this gives:

ΔL = 2.725 cm × 22 x 10⁻⁶ × 128 = 0.0000776 cm (approximately)

Step 4: Determine the New Diameter

Finally, we add the change in diameter to the original diameter to find the new diameter:

New Diameter = Original Diameter + ΔL

New Diameter = 2.725 cm + 0.0000776 cm ≈ 2.72508 cm

Final Result

Thus, when the temperature of the aluminum plate is raised to 140°C, the diameter of the circular hole will be approximately 2.72508 cm. This small increase illustrates how even slight temperature changes can affect materials, especially metals like aluminum, which are known for their significant thermal expansion properties.