Gravity in Chicago versus Quito

GRAVITY IN CHICAGO V/S QUITO

Gravity on the surface of Earth varies due to its rotation. Earth rotates at a speed of 1040 mph or at an angular velocity of 7.27*10^-5 rad/s. Since Earth rotates, all points on its surface perform circular motion except the North and South Pole. The points at equator perform circular motion of maximum radii as compared to other points. The polar points do not perform circular motion but rotate about its own axis.

Consider two cities, Chicago, USA and Quito, Ecuador on Earth’s surface. We’ll compare acceleration due to gravity (g) between these two cities.

ASSUMPTION:

Density of earth is constant throughout

CALCULATION:

The International Gravity Formula [IGF] for Earth is,

g (φ) = 9.780327[1+0.0053024sin²φ-0.0000058sin²2φ]

Chicago: latitude [φ] = 41.8781⁰ N, height above sea level h= 181m

g (φ) = 9.780327[1+0.0053024sin²(41.8781)-0.0000058sin²(83.7562)]

g (φ) = 9.780327[1+0.0053024(0.6675)²-0.0000058(0.9940)²]

g (φ) = 9.780327[1+2.3625*10^-3-5.7306*10^-6]

g (φ) = 9.780327[1.0023682]

g (φ) = 9.80348 m/s²

Since Chicago is not exactly at sea level, we need to use the Free Air Correction [FAC] so as to find acceleration due to gravity at that height and then add it to our original answer

FAC is given by, ∆g_h = -3.086*10^-6*h

∆g_h = -3.086*10^-6*181

∆g_h = -5.58566*10^-4 m/s²

Now g (φ, h) = g (φ) + ∆g_h

g (φ, h) = 9.80348-5.58566*10^-4

g (φ, h) = 9.802921 m/s²

Quito: latitude [φ] = 0.1807⁰ S, height above sea level h = 2850m

g’ (φ) = 9.780327[1+0.0053024sin²φ-0.0000058sin²2φ]

g’ (φ) = 9.780327[1+0.0053024sin²(0.1807)-0.0000058sin²(0.3614)]

g’ (φ) = 9.780327[1+5.274*10^-8-2.3075*10^-10]

g’ (φ) = 9.780327[1.000000053]

g’ (φ) = 9.7803275 m/s²

Since Quito is way above sea level, we need to use the Free Air Correction [FAC] so as to find acceleration due to gravity at that height and then add it to our original answer

FAC is given by, ∆g_h = -3.086*10^-6*h

∆g_h = -3.086*10^-6*2850

∆g_h = -8.7951*10^-3 m/s²

Now g’ (φ, h) = g’ (φ) + ∆g_h

g’ (φ, h) = 9.7803275-8.7951*10^-3

g’ (φ, h) = 9.771532 m/s²

CONCLUSION:

On comparing g (φ, h) and g’ (φ, h) we observe that gravity in Chicago is greater than that in Quito. The difference is g (φ, h) - g’ (φ, h) = 9.802921-9.771532 = 0.031389 m/s²

For a body of mass 50Kg,

Weight in Chicago = 50* g (φ, h) = 50*9.802921 = 490.146 N

Weight in Quito = 50* g’ (φ, h) = 50*9.771532 = 488.576 N 

Thus a person in Chicago would weigh 1.57 N less in Quito.