To determine the appropriate size of the aluminium frame for the glass window, we need to account for the thermal expansion of both materials as the temperature changes. The goal is to ensure that the glass remains unstressed when the temperature drops in winter. Let's break this down step by step.
Understanding Thermal Expansion
Thermal expansion refers to how materials change in size when their temperature changes. Each material has a coefficient of linear expansion, which quantifies how much it expands or contracts per degree change in temperature. For our scenario:
- Coefficient of linear expansion for glass: α_glass = 0.000009 /°C
- Coefficient of linear expansion for aluminium: α_aluminium = 0.000024 /°C
Calculating the Change in Dimensions
We need to calculate how much both the glass and the aluminium frame will expand or contract between the two temperatures: 40°C and 0°C.
1. Change in Temperature
The change in temperature (ΔT) is:
ΔT = Final Temperature - Initial Temperature = 0°C - 40°C = -40°C
2. Change in Size of the Glass
The dimensions of the glass window are 20 cm by 30 cm. We can calculate the change in size using the formula:
ΔL = L × α × ΔT
For the width (20 cm):
ΔL_glass_width = 20 cm × 0.000009 /°C × (-40°C) = -0.0072 cm
For the height (30 cm):
ΔL_glass_height = 30 cm × 0.000009 /°C × (-40°C) = -0.0108 cm
3. New Dimensions of the Glass
The new dimensions of the glass at 0°C will be:
- Width: 20 cm - 0.0072 cm = 19.9928 cm
- Height: 30 cm - 0.0108 cm = 29.9892 cm
4. Change in Size of the Aluminium Frame
Now, we calculate how much the aluminium frame will contract:
For the width (assuming the frame is initially the same size as the glass):
ΔL_aluminium_width = 20 cm × 0.000024 /°C × (-40°C) = -0.0192 cm
For the height:
ΔL_aluminium_height = 30 cm × 0.000024 /°C × (-40°C) = -0.0288 cm
5. New Dimensions of the Aluminium Frame
The new dimensions of the aluminium frame at 0°C will be:
- Width: 20 cm - 0.0192 cm = 19.9808 cm
- Height: 30 cm - 0.0288 cm = 29.9712 cm
Determining the Frame Size
To ensure that the glass does not experience stress, the frame must be larger than the glass at the lowest temperature. Therefore, we need to add the changes in size of the glass to the original dimensions:
- Required width of the frame = Width of glass at 0°C + Change in width of glass
- Required height of the frame = Height of glass at 0°C + Change in height of glass
Calculating the required dimensions:
- Required width = 19.9928 cm + 0.0072 cm = 20 cm
- Required height = 29.9892 cm + 0.0108 cm = 30 cm
Final Frame Dimensions
Thus, the aluminium frame should be designed to be slightly larger than the glass dimensions at the lowest temperature to accommodate the contraction of the glass. The final size of the aluminium frame should be:
- Width: 20 cm
- Height: 30 cm
This ensures that the glass window will fit perfectly without any stress, even when the temperature drops to 0°C. Always remember that proper allowances for thermal expansion are crucial in construction and design to avoid material failure.
