Question

Predict whether your weight in the city of Denver, which has an altitude of 1.7 km above sea level, will be the same as, more than, or less than your weight in New Orleans, a city located at sea level.

Short answer

The person's weight in Denver will be less than their weight in New Orleans due to the increased distance from the Earth's center, which results in slightly weaker gravitational force in Denver.

Step-by-step solution

Determine the Earth's radius and gravitational force at sea level

At sea level, the radius of the Earth (R) is approximately 6371 km. Using this value, we can calculate the gravitational force acting on the person in New Orleans: F_gravity_sea_level = G * (m1 * m2) / R^2

Determine the Earth's radius in Denver

Denver is at an altitude of 1.7 km above sea level, which means its effective radius is (R + 1.7) km. So, the person would be 6372.7 km away from the Earth's center in Denver.

Calculate the gravitational force in Denver

Using the radius in Denver, we can now calculate the gravitational force acting on the person: F_gravity_denver = G * (m1 * m2) / (R + 1.7)^2

Compare the gravitational forces in both locations

Due to the increased radius in Denver, the gravitational force acting on the person would be slightly weaker; thus, the person would weigh less in Denver when compared to New Orleans. Therefore, the person's weight in Denver will be less than their weight in New Orleans.

Key concepts

Gravitational Force Calculation

The gravitational force is a fundamental interaction that pulls objects with mass towards each other, and it's what gives us weight on Earth. It can be calculated using Newton's Universal Law of Gravitation, which states that the gravitational force (\(F_{gravity}\) ) between two masses (\(m_1\) and \(m_2\) ) is proportional to the product of their masses and inversely proportional to the square of the distance (\(r\) ) between their centers. The formula is represented as:
\[\begin{equation}F_{gravity} = G \times \frac{m_1 \times m_2}{r^2},\end{equation}\]
where G is the gravitational constant. While the masses involved and distance are critical, the constant ensures the calculation is consistent across all instances. In a textbook problem, the focus is on the gravitational force at sea level versus the force at a certain altitude, such as Denver's, and using the Earth's average radius to estimate weight changes.

Earth's Radius

The average radius of the Earth plays a vital role in gravitational force calculations and understanding how weight can vary at different altitudes. The Earth is not a perfect sphere; it is an oblate spheroid, which means it's slightly flattened at the poles and bulges at the equator. However, for simplification in calculations, the average radius (\(R\) ) of Earth is used, which is about 6371 kilometers (km). When considering gravitational force at sea level, this average radius is used as the distance (\(r\) ) from the center of the Earth. When moving away from sea level, to a higher altitude like Denver's 1.7 km above sea level, it's essential to adjust the Earth's radius accordingly by adding the altitude to the average radius to reflect the increased distance from the Earth's center, which impacts the gravitational force exerted on an object.

Altitude's Effect on Weight

Altitude influences a person's weight; the farther away from the Earth's center, the weaker the gravitational force due to the inverse square law of gravitation. In the context of the exercise, Denver's altitude of 1.7 km above sea level means the Earth's radius effectively becomes 6371 km + 1.7 km. The modified equation for the gravitational force in Denver is:\[\begin{equation}F_{gravity\_denver} = G × \frac{m_1 × m_2}{(R + 1.7)^2},\end{equation}\]
indicating a decrease in gravitational force as altitude increases. Simply put, your weight would slightly decrease in Denver compared to New Orleans, which is at sea level. This concept is crucial for students to understand as it demonstrates how Earth's spherical shape and gravity create varying conditions that slightly change our perceived weight depending on our location's altitude. Regular examples like these help students grasp fundamental physics concepts and relate them to the real world.