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Tower

Can we calculate the time when the Tower will lean to the extreme point that its center of mass will be out of its balance...?

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  • $\begingroup$ Do you mean calculate the maximum angle of lean before the tower falls? $\endgroup$ – John Rennie Aug 3 at 8:13
  • $\begingroup$ but i am more interested in "time" $\endgroup$ – Sudipto Sarker Aug 3 at 8:19
  • $\begingroup$ Well the time is infinite now because the foundations have been reinforced to stop it leaning any farther i.e. the rate of change of the angle is now zero. $\endgroup$ – John Rennie Aug 3 at 8:40
  • $\begingroup$ oh it used to lean 5 degrees at a time.. $\endgroup$ – Sudipto Sarker Aug 3 at 8:43
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    $\begingroup$ Not sure if this question can be answered, since we have no idea how much the tower is leaning per unit time. $\endgroup$ – Yejus Aug 3 at 8:59
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In theory would be possible, but to do it you'll need to know the precise position of the centre of mass of the tower, which if it was a homogeneous cylinder would be pretty easy, and the angular velocity at which the tower leans.

So, suppose that we know the position of the centre of mass, with which we can evaluate the maximum angle of lean before the tower falls, call it $\theta_M$. Moreover we know the angular velocity $\omega$ and we suppose that is constant, then $$\theta(t) = \omega t\implies \theta(t_M) = \theta_M = \omega t_M$$ and therefore we simpy have $$t_M = \frac{\theta_M}{\omega}$$

This is all well and good, but the real problem is to find the centre of mass and the angular velocity. The latter could be experimentally found just by measuring the angular position of the tower at least two times at two different moments, and so by interpolating with a simple line. This would imply that we're taking the angular velocity as constant over time, which won't be true since, due to gravity, the tower is slowly accelerating towards the ground. A better way of doing it would be taking lots more measurement at many different times and then finding a curve of best fit. This is a rigid body problem and can be solved using common techniques. If you find the angular acceleration $\alpha$ only due to gravity, then you'll need to solve a second order equation for $\theta_M$ $$\theta(t_M)\equiv \theta_M = \omega t_M + \frac{1}{2}\alpha t_M^2$$

All of this is done supposing that nobody prevents it to fall down, which is clearly not true.

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    $\begingroup$ Also, while the stone work of the respective stories was strong enough to support the static and dynamic load of an upright tower, the lean was putting a very large load on the low side. One by one stones developed cracks. So the actual tower would collapse due to the stone work buckling under the stress way before the center of mass would be out of balance. This does not detract from your answer; the submitted question is about an abstract cilinder, not about the actual tower. $\endgroup$ – Cleonis Aug 3 at 9:28
  • $\begingroup$ @Cleonis You seem to know very much about the leaning tower of Pisa! A lot more than me, and I'm italian! Very cool answer and comment. $\endgroup$ – Davide Morgante Aug 3 at 9:35
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    $\begingroup$ Don't take my word for it. I'm a tertiary source. (As in: my sources are secondary) The tower of Pisa is the icon of being iconic. It's been many years since I read about the tribulations of the tower, but it has stuck in my mind. (I first wrote my answer, then I googled. It's not too bad, but that's why there are no dates in my answer.) $\endgroup$ – Cleonis Aug 3 at 10:04
  • $\begingroup$ The leaning tower problem is sophisticated system. What you have proposed is similar to a ball falling under action of gravity, but there is no air resistance. How on earth can somebody do that! For leaning tower problem, one should account for composition of soil underneath, how was foundation laid out, and many more factors which a person who as done Majors in Architecture can answer better. $\endgroup$ – Kartik Chhajed Aug 3 at 11:41
  • $\begingroup$ @Cleonis why I should think that tower a cylinder....while that is too easy to solve... $\endgroup$ – Sudipto Sarker Aug 4 at 5:39
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The tower of Pisa started leaning because of uneven settling of the soil underneath it. (It could be that it wasn't just a process of compression, but that there was also reflow. I don't know the details.)

Decades ago it was discovered that this uneven settling was accelerating. That is, the lean wasn't increasing linear with time, it was speeding up.

Study after study was commissioned, but for many years city council after city council was not able to make a decision (I assume that italian city politics, like the national politics, is very volatile.)

At one point in time many hundreds of tons of giant lead ingots were stacked in a cilinder section around the base, to increase the weight at the high side, in the hope that would decelerate the rate of increasing lean. It worked partially only.

Finally the danger of collapse was so acute that the city council went ahead with a plan.

There are machines that can drill narrow bore holes at a sharp angle. Machines like that are used to create conduits underneath roads and so on, whithout need to tear up the road.

That machinery was used to slowly remove soil from underneath the high side of the tower, allowing the soil underneath the high side to to undergo additional subsidence.

The concern, of course, was that that subsidence might occur in fits, rather than gradually.

To everyone's relief the plan was succesful. The progress of the lean came to a halt, and was even reversed a little.


Incidentally, there is the story of why the increase of the lean started accelerating.

What used to be the ground floor at the time of construction had sunk pretty deep in to the ground. It did not match the street level. At some point in time a huge amount of surrounding soil was removed to make that original ground floor accessible again. My understanding is that the removal of such a large amount of mass and ground pressure destabilized the foundations.

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