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19 June 2008

The Italian construction industry / How far does it lean, but PIZZAQ! How long do cannonballs take to hit the ground?

Yes yes, by all means click on the image.

Okay, received today this photograph of my nephew Alex and his friend Kelly, guess where (no Pizza offered), photographer unknown, possibly a robot, maybe a nice guy from Alberta named Norman.

VAMRI, the Vleeptron Advanced Mathematics Research Institute, has analyzed the digital image and concluded that la Torre pendente di Pisa leans 1.81660792334491º from the vertical.

To be charitable to VAMRI and the photographer, Alex & Kelly just wanted to show the folks back home that they were indeed in Pisa, and did not set up the photo for a trigonometry demonstration to see how much the Tower leans from the vertical. (It was designed to stand straight up, but shit happens.)


The Leaning Tower of Pisa (Torre pendente di Pisa) or simply The Tower of Pisa (La Torre di Pisa) is the campanile, or freestanding bell tower, of the cathedral of the Italian city of Pisa. It is situated behind the Cathedral and it is the third structure by time in Pisa's Piazza del Duomo (Cathedral Square).

Although intended to stand vertically, the tower began leaning to the southeast soon after the onset of construction in 1173 due to a poorly laid foundation and loose substrate that has allowed the foundation to shift direction. The tower presently leans to the southwest.

The height of the tower is 55.86 m (183.27 ft) from the ground on the lowest side and

56.7 m (186.02 ft)

on the highest side. The width of the walls at the base is 4.09 m (13.42 ft) and at the top 2.48 m (8.14 ft). Its weight is estimated at 14,500 tonnes. The tower has 296 or 294 steps; the seventh floor has two fewer steps on the north-facing staircase. The tower leans at an angle of 3.97 degrees. This means that the top of the tower is 3.9 meters from where it would stand if the tower were perfectly vertical.


Galileo Galilei is said to have dropped two cannon balls of different masses from the tower to demonstrate that their descending speed was independent of their mass. This is considered an apocryphal tale, and the only source for it comes from Galileo's secretary.

In 1934 Benito Mussolini ordered that the tower be returned to a vertical position, so concrete was poured into its foundation.

However, the result was that the tower actually sank further into the soil.

Well, I've always liked the story that Galileo dropped cannonballs of different masses/weights from the Tower of Pisa.

Until he performed this experiment, Europe's premier authority on Physics,
Aristotle, said the heavier cannonball would hit the ground first, and everybody accepted the old Macedonian gasbag's opinion as Truth for about 2000 years.

The heavier cannonball hits the ground at the exact instant the lighter cannonball hits the ground.

In a vacuum -- like on the Moon -- a feather and an anvil dropped from the same height at the same instant will hit the ground at the same instant.

But Galileo did a lot more experimenting with falling objects, and worked out the math to describe precisely how they fall toward the Earth as time passes. (He rolled balls down ramps to slow down the process so he could observe it more precisely.)

Okay, 2 slices of Pizza, with Pepperoni and garlic.

Assuming G did indeed drop the cannonballs from the very top of the Tower of Pisa, how long did it take the cannonballs to hit the ground? (Gimme 3 decimal places of accuracy please.)

I find it amazing that grown women and men who don't know how to solve this problem are allowed out unsupervised.

It is very possible that the next president of the United States could not solve this problem if his life depended on it. (The U.S. Naval Academy at Annapolis is heavy on math, but McCain was graduated 5th from the bottom of his class of 899 midshipmen. I don't know if Obama could win the Pizza either.)

I attended public schools in Washington DC and we had to do this fucking headache over and over again with massive objects dropped from the top of the Washington Monument, which -- entirely from painful memory -- is

555 feet 5 and 1/8 inches


1 more slice: How long does it take the besbol to hit the ground if you drop it from the top of the Washington Monument? (Also 3 decimal places please.)

Pisa and Washington DC are enclosed in vacuums, so air resistance is not a factor in the calculations.


abbas halai said...

s = u.t + 1/2(a.t^2) where s is distance, u is initial velocity, t is time and a is acceleration.

u.t = 0 since initial velocity = 0.

therefore s = 1/2(a.t^2)
therefore 2s/a = t^2
therefore t = sqrt[2s/a]

Vleeptron Dude said...

Okay, abbas, you're ALMOST there!

Scroll back and I've given (in extra-large centered type) the Height of the Tower = s

You're gonna have to find

a = constant of acceleration near the Earth's surface


Then crank out the value of t (in seconds).

Then do it again if you want the extra slice for the Washington Monument. For both, 3 digits of precision.

But you got Galileo's equations on the nose! You have earned the right to wander around unsupervised!

I smell an English Public School education. Which one? You still got your tie?

Don't tarry, we got Mike and Amy and RamanuJohn and occasionally Dame RheLynn, and the Parson & Amateur Astronomer sometimes gives these a whack.

Jim Olson said...

Yeah, I got other things to do this week. Though, I did have fun this week staring at the full moon and saturn through a very, very nice privately owned telescope that belongs to a wealthy friend. He has a small observation dome at his very nice home, and we all got a chance to look through the telescope after a superb dinner. The 25 year old port we were sipping while waiting our turn wasn't bad either. Perhaps if I get another church job, I will purchase another small telescope myself. Viewing celestial objects in rural Vermont away from the light pollution here in the city was lovely. said...

Surely, the dude is absolutely fair.

mainlymilitary said...

The dude is completely just, and there is no suspicion.