Let's tackle these two problems step by step, ensuring we understand the concepts of thermal expansion and how they apply to each scenario.
Problem 1: Sizing the Aluminium Frame
In this situation, we need to ensure that the glass window can expand and contract without causing stress on it due to temperature changes. The key here is to calculate how much the glass and the aluminium frame will expand or contract with temperature changes.
Step 1: Calculate the change in temperature
The temperature change (ΔT) from summer to winter is:
- Summer temperature = 40°C
- Winter temperature = 0°C
- ΔT = 0°C - 40°C = -40°C
Step 2: Calculate the dimensions of the glass window
The dimensions of the glass window are:
- Width (W) = 20 cm
- Height (H) = 30 cm
Step 3: Calculate the expansion of glass
The formula for linear expansion is:
ΔL = L₀ * α * ΔT
Where:
- ΔL = change in length
- L₀ = original length
- α = coefficient of linear expansion
- ΔT = change in temperature
For glass:
- α (glass) = 9.0 x 10-6 /°C
- ΔL (width) = 20 cm * 9.0 x 10-6 /°C * (-40°C) = -0.00072 cm
- ΔL (height) = 30 cm * 9.0 x 10-6 /°C * (-40°C) = -0.00108 cm
Step 4: Calculate the expansion of aluminium
Now, let's calculate how much the aluminium frame will expand:
- α (aluminium) = 24 x 10-6 /°C
- ΔL (width) = 20 cm * 24 x 10-6 /°C * (-40°C) = -0.00192 cm
- ΔL (height) = 30 cm * 24 x 10-6 /°C * (-40°C) = -0.00288 cm
Step 5: Determine the size of the aluminium frame
To ensure that the glass does not experience stress, the frame must be larger than the glass when the temperature drops. Therefore, we need to add the expansion of the glass to its original dimensions:
- New width = 20 cm + 0.00072 cm = 20.00072 cm
- New height = 30 cm + 0.00108 cm = 30.00108 cm
Thus, the size of the aluminium frame should be approximately:
- Width = 20.00072 cm
- Height = 30.00108 cm
Problem 2: Correcting the Pendulum Clock
For the second problem, we need to find the temperature at which the pendulum clock will give the correct time at a different gravitational field strength.
Step 1: Understand the relationship between pendulum length and gravity
The period of a pendulum is given by:
T = 2π√(L/g)
Where:
- T = period of the pendulum
- L = length of the pendulum
- g = acceleration due to gravity
Step 2: Set up the equation for the two locations
At the original location:
- g₁ = 9.800 m/s²
- T₁ = 2π√(L/g₁)
At the new location:
- g₂ = 9.788 m/s²
- T₂ = 2π√(L/g₂)
Step 3: Relate the periods
Since the clock gives the correct time at 20°C, we can set T₁ = T₂:
2π√(L/g₁) = 2π√(L/g₂)
This simplifies to:
√(L/g₁) = √(L/g₂)
Squaring both sides gives:
L/g₁ = L/g₂
Thus, we can express the relationship as:
g₁/g₂ = (1 + αΔT)
Where α is the coefficient of linear expansion of steel.
Step 4: Solve for the temperature change
Rearranging gives:
ΔT = (g₁/g₂ - 1) / α
Substituting the values:
- α = 12 x 10-6 /°C
- ΔT = (9.800 / 9.788 - 1) / (12 x 10-6)
Calculating this gives:
ΔT ≈ 0.00122 / (12 x 10-6) ≈ 101.67°C
Step 5: Determine the new temperature
Since the clock was correct at 20°C, we need to add the temperature change:
New temperature = 20°C + 101.67°C ≈ 121.67°C
In summary, the aluminium frame should be approximately 20.00072 cm by 30.00108 cm to accommodate the glass window without stress, and the pendulum clock will give the correct time
