Center Of Mass Vs. That Of Gravity In The Tallest Buildings

Center of mass vs. that of gravity in the tallest buildings

Acceleration due to gravity is not really a constant like most people think, it is dependent on the distance from the center of the Earth. The formula for gravity can be obtained using Newton’s law of universal gravitation (F=Gm1m2r2) because acceleration is force divided by mass (g=GmEr2). Since the distance between the center of theEarth and a body normally varies very little, it is considered to be a constant value (the radius of Earth at sea level on the equator) and thus gravity is also constant.
The center of gravity of a body is the point where the weight of it acts upon, since gravity is considered to be constant; the center of gravity is the center of mass. In these cases we will take into consideration the changeof gravity due to height (since they are very tall structures) and thus analyze where the center of gravity really is and if it should be taken into account or not.
Since the Earth is not a perfect sphere, the radius of Earth is not constant. To find the radius at each location the World Geodetic System will be used.
Frontal area will be approximated to a rectangle in every case so that it iseasier, so the center of mass will be located at the middle of the height. The center of gravity is that point where the weight is concentrated; in these cases we consider it the height where the masses above and below weight the same. Since gravity is a function of height, we cannot say that these points are the same, but we can find them. In the next equations it is considered that the width ofthe building and the density are constant and uniform.
Note: In all of these equations r is the distance from the core of Earth to ground level
Note: In all of these equations r is the distance from the core of Earth to ground level

The formula above only works when the building is perfectly rectangular, which in most cases it is not. Therefore, I decided to try and find the relationshipbetween the difference in gravity from ground level to the centroid and the difference between the center of mass and gravity, and found that gggc=cmcg2. I proved this by using excel and randomizing the height of the building. Using the formula of gravity and some algebra we arrive at the equation: cm+rr=cmcg, so we can say that for any body the center of gravity will be cg=rcmcm+r
Five of the tallestbuildings in the world are Tapei 101, the Petronas Towers on Kuala Lempur, the Sears Tower in Chicago, the Empire State in New York and the Canary Wharf in London. This study will analyze these, plus two buildings under construction that are to be even higher. The analysis on each building will be done separately, and in each we will see how much the point lowers. The x and z coordinates will notbe taken into account, as they don’t vary.
Canary Wharf, London
* Altitude: 5 m
* Height: 235 m
* Latitude: 51°30'20.14"N
* Radius: 6,365,080 m
* Distance from core to ground: 6,635,085 m
The Canary Wharf is a building erected on the city of London that has a very rectangular shape with a pyramid pinnacle. This pyramid is 40 m high, while the front face is approximately 55 mlong and 195 m high.
The area of the front figure can be divided in two main parts: a rectangle and a triangle, each with its own centroid. The centroid of the rectangle would be at 97.5 m of height, while the triangle would be at 208.33 m of height. To find the centroid of the complete figure we use the formula

A uniform distribution of mass will be considered, as most buildings work thisway and because if not it would be impossible to compute without the specifics of the building. This means that the center of mass is also at a height of 113.07 m.

We can see that in this building the change is minimal, so it doesn’t need to be take into account. As it was so little, the next building to analyze is the tallest, because if in the tallest building the change is still negligible,...