A 40-cm -long icicle hangs from the eave of a house on a day when the temperature is -2.0ºC . By how many millimeters does the icicle shrink if a bitterly cold wind drops the temperature to -30ºC?

Answer 1

This is an example of Linear Thermal Expansion, a thermodynamic process where the dimensions of a solid object change due to a change in temperature. The equation describing this is as follows:

ΔL = αL₀ ΔT

ΔL = Change in temperature

α = The coefficient of linear expansion (a property of the material)

L₀ = The length of the original material

ΔT = The change in temperature.

In the description, we are given the original length, L₀ , as 40 cm, the initial temperature of -2 °C, and the final temperature of -30 °C, so therefore the change in temperature (the final temperature minus the initial temperature) is -28 °C. We want the change in length, ΔL, in mm due to that change. In order to solve this problem, we need to do a quick internet search for the coefficient of linear expansion, α, of ice. The first value I found was 50 • 10^-6 for each °C, meaning α = 50 • 10^-6 with units of 1/°C. We have to make sure that the temperatures are in Celsius as well to make them consistent.

Putting those values together, we have:

ΔL = αL₀ ΔT = (50 x 10^-6 1/°C) • (40 cm) • (-28 °C)

ΔL = -0.056 cm

The answer they want is in mm, and 1 mm = 0.1 cm, so:

ΔL = -0.56 mm

This means that the icicle shrunk by 0.56 mm!