Solving for the Length of a Leaning Tower: A Simple Guide

[martine]

A probably very simple question, but at the moment I cannot find the sollution to it.

take for example the tower of Pisa or any other tower about to fall over. The tower fell to the left and the platform on top now dips 10degrees. Someone measured the vertical height on the right side of the tower from top to surface. What is the length of the tower on the left side?

Thanks very much

 

[gerben]

I guess you just want to know the vertical height of a tower, that had an original vertical height of H meters, after the tower has fallen X degrees with respect to vertical?

This would be H times the cosine of X.

 

[HallsofIvy]

Draw a picture of the tilted tower. Drop a vertical line from the top of the tower to the ground so that you have a right triangle. The height, Y, is the length of that vertical, opposite to the complement of angle X (since X is measured from the vertical). The "hypotenuse" of the right triangle is H, the height of the tower when it was vertical: "opposite side divided by "hypotenuse", Y/H is the definition of sine. Of course, since that angle is the complement of X, we have Y/H= cos(x)
Since Y/H= cos(X), Y= H*cos(X) just as gerben said.

That is the right side. The length of the left side will be slightly less since that side sank into the ground, causing the tilt. Exactly how much will depend on the diameter of the tower: information you didn't give.
If you know the diametero of the tower, D, you can calculate the "sink" in exactly the same way: continuing your picture below ground, you see a right triangle with angle X, near side D and U, the desired length, the opposite side. Now we have
tan(X)= U/D (definition of tangent) so U= Dtan(X).

The length of the right side, opposite the tilt, is H cos(X) and the length of the left side, above ground, is H cos(X)- D tan(X).

 

[Zurtex]

It's always useful to draw out a simple diagrams and write down everything you know on it. Usually you can then fairly simply see how to solve the problem.

 

[arivero]

It's always useful to draw out a simple diagrams and write down everything you know on it. Usually you can then fairly simply see how to solve the problem.

Or you can wait for an angel to solve it, so you do not learn.