To determine the temperature required to cool the apparatus and bring the height down to 50mm, we need to use the principle of thermal expansion. The linear expansion coefficient is a characteristic property of a material that relates how its dimensions change with temperature.
Let's assume the apparatus has a linear expansion coefficient of α (alpha) and an initial height of h1 = 100mm at a temperature T1 = 22 degrees Celsius. The final height is h2 = 50mm, and we need to find the corresponding temperature T2.
The formula for linear expansion is:
ΔL = α * L * ΔT
Where: ΔL is the change in length, α is the linear expansion coefficient, L is the original length, and ΔT is the change in temperature.
In this case, we are dealing with a change in length (height), so we can rewrite the formula as:
Δh = α * h * ΔT
Solving for ΔT:
ΔT = Δh / (α * h)
Now we can substitute the given values:
ΔT = (h2 - h1) / (α * h1)
ΔT = (50 - 100) / (α * 100)
ΔT = -50 / (α * 100)
Since the temperature change ΔT is negative, indicating a decrease in temperature, we can rearrange the formula to find the final temperature T2:
T2 = T1 + ΔT
T2 = T1 - (50 / (α * 100))
Now, we need to know the linear expansion coefficient (α) of the apparatus material to proceed with the calculation. Different materials have different expansion coefficients, so this value would need to be provided or determined experimentally for the specific material of the apparatus.