Question

the table below provides data obtained at different geographical locations on earth. using the claim - evidence - reasoning format, write a scientific explanation to answer the following question: in which city would earths gravitational field strength be the smallest? claim: evidence: (must show at least 2 calculations for full credit) calculations: reasoning: universal gravitation rough draft universal gravitation peer review i like the way you: you might want to add: you dont need:

Explanation:

Claim:

Las Vegas, NV would have the smallest gravitational field strength.

Evidence:

Step1: Convert latitude to decimal degrees

• For Richmond, VA: \(37^{\circ}32' = 37+\frac{32}{60}\approx37.53^{\circ}\)
• For Las Vegas, NV: \(36^{\circ}10' = 36+\frac{10}{60}\approx36.17^{\circ}\)

Step2: Consider the formula for gravitational field strength \(g = g_0(1 - 2.6\times10^{-3}\sin^{2}\varphi-3.1\times10^{-7}h)\) (where \(g_0 = 9.8\ m/s^{2}\), \(\varphi\) is latitude and \(h\) is elevation)

• For Richmond, VA: \(\sin^{2}(37.53^{\circ})\approx0.35\), \(h = 30\ m\). Then \(g_{Richmond}\approx9.8(1 - 2.6\times10^{-3}\times0.35-3.1\times10^{-7}\times30)\approx9.8(1 - 9.1\times10^{-4}-9.3\times10^{-6})\approx9.79\ m/s^{2}\)
• For Las Vegas, NV: \(\sin^{2}(36.17^{\circ})\approx0.32\), \(h = 1960\ m\). Then \(g_{Las Vegas}\approx9.8(1 - 2.6\times10^{-3}\times0.32-3.1\times10^{-7}\times1960)\approx9.8(1 - 8.32\times10^{-4}-6.08\times10^{-4})\approx9.8(1 - 1.44\times10^{-3})\approx9.79\ m/s^{2}\) (but considering more accurate calculation with more terms in the formula, and also the fact that higher elevation (\(h = 1960\ m\) for Las Vegas vs lower elevations for others) and latitude (closer to equator where \(g\) is smaller) play a role. Another way, using the fact that \(g\) decreases with increasing elevation and decreasing latitude (since \(g\propto\frac{1}{r^{2}}\) where \(r\) is the distance from the center of the Earth. Higher elevation means larger \(r\), and lower latitude (closer to equator where the Earth's radius is larger due to Earth's oblate - spheroid shape) also means larger \(r\)). Las Vegas has the highest elevation (\(h = 1960\ m\)) among the given cities.

Reasoning:

The gravitational field strength \(g\) is inversely proportional to the square of the distance from the center of the Earth (\(g=\frac{GM}{r^{2}}\), where \(G\) is the gravitational constant, \(M\) is the mass of the Earth and \(r\) is the distance from the center of the Earth). Elevation (\(h\)) increases \(r\) (\(r = R_0+h\), where \(R_0\) is the mean - radius of the Earth at sea - level). Also, the Earth is an oblate spheroid (flattened at the poles and bulging at the equator), so \(r\) is larger at lower latitudes (closer to the equator). Las Vegas has the highest elevation among the listed cities, which increases \(r\). Although its latitude is not the lowest, the combined effect of its relatively low latitude (compared to some other cities like Terra Haute, IN and Salt Lake City, UT) and very high elevation leads to the largest \(r\) among the given cities. According to \(g=\frac{GM}{r^{2}}\), a larger \(r\) results in a smaller \(g\).

Answer:

Las Vegas, NV would have the smallest gravitational field strength.