The height of the Washington Monument is measured to be 170.00 meters on a day when the temperature is 35.000 degrees Celsius. What will its height be on a day when the temperature falls to negative 10.000 degrees Celsius? Assume that the monument’s coefficient of thermal expansion is 2.5000 times 10 to the negative sixth inverse degrees Celsius.
We can call this final height of the monument 𝐻 sub 𝑓. And we’ll consider what will happen to the height of this monument under this temperature change. We’re told that one very warm day when the temperature is 35.000 degrees Celsius, the height of the Washington Monument is measured to be 170.00 meters. We then imagine that the temperature cools down to negative 10.000 degrees Celsius.
Under these conditions, we wanna solve for the resulting height of the monument. It may come as a surprise that the height of the monument would change at all under a temperature change, but it does. And that has to do with a term called the coefficient of thermal expansion.
This term represents how much an object’s size changes with changes in temperature and it varies for different materials. For the material making up the Washington Monument, we’re told what this coefficient of thermal expansion, often represented using the Greek letter 𝛼, is.
If we want to solve for the overall change in an object’s length or dimension given some sort of thermal expansion or contraction, that’s equal to 𝛼 the coefficient of thermal expansion times the original length of the object multiplied by the temperature change it goes through.
In our scenario, since our temperature is going from a higher value to a lower value, we know that 𝐻 sub 𝑖 will be greater than 𝐻 sub 𝑓. Overall, the height of the monument will shrink as it gets colder. This means we can write that 𝐻 sub 𝑓 is equal to 𝐻 sub 𝑖 minus the change in height of the monument due to the temperature variation. And this change in height we know is equal to 𝛼 times the original height times the change in temperature Δ𝑡.
Factoring out the original height of the monument from this expression, we now have a simplified way of writing the final height of this Washington Monument. Plugging in for these values, we insert 170.00 meters for 𝐻 sub 𝑖, 25.000 times 10 to the negative sixth inverse degrees Celsius for 𝛼, and for Δ𝑡 the total temperature swing or difference is 45.000 degrees Celsius.
We see that the units of degrees Celsius cancel out. And when we calculate this whole expression, to five significant figures, it’s 169.98 meters.
Going then from one of the hottest days to one of the coldest days of the year, the height of the monument only changes about one fiftieth of a meter.