Question

If the variation of the acceleration of gravity, in \(\mathrm{m} / \mathrm{s}^{2}\), with elevation \(z\), in \(\mathrm{m}\), above sea level is \(g=9.81-\left(3.3 \times 10^{-6}\right) z\), determine the percent change in weight of an airliner landing from a cruising altitude of \(10 \mathrm{~km}\) on a runway at sea level.

Short answer

The percent change in weight is 0.336%.

Step-by-step solution

Identify the given variables

Given: Gravitational acceleration at sea level, g at sea level, \( g_0 = 9.81 \frac{m}{s^2} \) Elevation, z = 10,000 m Formula for gravitational acceleration: \(g = 9.81 - (3.3 \times 10^{-6})z \)

Calculate the gravitational acceleration at cruising altitude

Substitute the value of z into the formula: \( g(10,000) = 9.81 - (3.3 \times 10^{-6}) \times 10,000 \) g(10,000) = 9.81 - 0.033 = 9.777 \frac{m}{s^2}

Determine the change in gravitational acceleration

Calculate the change in gravitational acceleration from z = 10,000 m to z = 0 m. \(\text{Change in } g = 9.81 - 9.777 = 0.033 \frac{m}{s^2}\).

Calculate the percent change in weight

Percent change can be calculated as follows: \((\frac{\text{Change in } g}{g_0}) \times 100 = (\frac{0.033}{9.81}) \times 100 \text{Percent change in weight} = 0.336\text{%}\)

Key concepts

percent change in weight

When calculating the percent change in weight due to the change in altitude, it's essential to understand the concept of percent change itself. Percent change is a way to express how much a quantity has changed, as a percentage of its original value. Given the formula:

• Original value (weight related to gravitational acceleration at sea level) is denoted as \( g_0 = 9.81 \frac{m}{s^{2}} \)
• Change in the value of gravity at 10 km altitude \( g_{\text{alt}} = 9.777 \frac{m}{s^2} \)

To determine the percent change in weight, we first calculate the difference in gravitational acceleration between the two altitudes. Then, we compare this change to the original value and convert it into a percentage. So we have:\( \text{Percent change} = \frac{ \text{Change in value} }{ \text{Original value} } \times 100 \)This results in:\( \text{Percent change in weight} = \frac{0.033}{9.81} \times 100 = 0.336\text{%} \)Thus, by descending from 10 km altitude to sea level, the weight change due to gravity is approximately 0.336%.

elevation effect on gravity

Gravitational acceleration varies with elevation. At higher elevations, the effect of Earth's gravity diminishes slightly. This reduction is because gravity decreases with the square of the distance between the two masses.

For engineering purposes, we use a simplified linear formula to approximate this change:\( g = 9.81 - (3.3 \times 10^{-6})z \)

Where:

• \( g \) is the gravitational acceleration at elevation \( z \).
• \( z \) is the elevation in meters.

In our exercise, we analyzed how gravity changes with a 10,000-meter (10 km) change in elevation. By substituting \( z = 10,000 \text{m} \) into the formula, we calculated the gravitational acceleration at this altitude as:\( g(10,000) = 9.81 - (3.3 \times 10^{-6}) \times 10,000 \)

This gives us \( 9.777 \frac{m}{s^2} \), showing that gravity decreases as we ascend.

Understanding this concept is crucial for fields such as aerospace engineering, where knowing the exact behavior of gravity at various altitudes can impact flight dynamics and the design of aircraft and spacecraft.

engineering problem solving

Solving engineering problems often involves breaking down complex real-world phenomena into manageable calculations. Let's summarize an effective approach:

1. Identify Given Variables: Begin by extracting all the known data from the problem statement. In our case, this includes the gravitational constant at sea level and the cruising altitude.
2. Implement Formulas: Use relevant equations to determine the unknowns. Here, we substituted the elevation value into the gravity formula.
3. Calculate Changes: After finding the new value (gravitational acceleration at 10 km), calculate the change compared to the original value.
4. Determine Percent Change: Finally, translate this change into a percentage to understand its impact on the system, which in our example pertains to the airliner's weight change.

Approaching problems with clear, structured thinking helps avoid errors and ensures accurate solutions. Always double-check units and calculations, and ensure each step logically leads to the next. Properly documenting and understanding each stage builds a robust foundation for tackling more complex problems in engineering.