Question

The weight of bodies may change somewhat from one location to another as a result of the variation of the gravitational acceleration \(g\) with elevation. Accounting for this variation using the relation in Prob. \(1-8,\) determine the weight of an 80 -kg person at sea level \((z=0),\) in Denver \((z=1610 \mathrm{m})\) and on the top of Mount Everest \((z=8848 \mathrm{m})\)

Short answer

Question: Calculate the weight of an 80-kg person at sea level, in Denver, and on top of Mount Everest using the gravitational acceleration formula given in Problem 1-8. Answer: The weight of an 80-kg person at different elevations is as follows: - At sea level: Weight = 784.56 N - In Denver: Weight ≈ 783.6 N - On top of Mount Everest: Weight ≈ 781.76 N

Step-by-step solution

Gravitational acceleration formula

Recall the relation of gravitational acceleration given in Problem 1-8, which relates gravitational acceleration (\(g\)) to the elevation (\(z\)) as follows: \(g = g_0\left(1-\frac{2z}{R + z}\right)\) where \(g_0\) is the gravitational acceleration at sea level (9.807 m/s²), \(z\) is the elevation above sea level, and \(R\) is the Earth's radius (approximately 6.37 x 10^6 m).

Calculate the weight at sea level

To calculate the weight of an 80-kg person at sea level, we first need to determine the gravitational acceleration at sea level (\(z=0\)). Using the formula: \(g = g_0\left(1-\frac{2(0)}{R + (0)}\right)\) \(g = g_0\) Since at sea level, gravitational acceleration is equal to \(g_0\), we have \(g = 9.807 \mathrm{m/s^2}\). Weight is determined by the formula: \(W = mg\) So for the 80-kg person at sea level: \(W = (80 \mathrm{kg})(9.807 \mathrm{m/s^2})\) \(W = 784.56 \mathrm{N}\)

Calculate the weight in Denver

Next, we need to calculate the weight of the 80-kg person in Denver (\(z=1610 \mathrm{m}\)). First, determine the gravitational acceleration: \(g = g_0\left(1-\frac{2(1610)}{R + (1610)}\right)\) \(g = 9.807 \left(1-\frac{2(1610)}{6.37 \times 10^6 + (1610)}\right)\) \(g \approx 9.795 \mathrm{m/s^2}\) Now, calculate the weight: \(W = (80 \mathrm{kg})(9.795 \mathrm{m/s^2})\) \(W \approx 783.6 \mathrm{N}\)

Calculate the weight on the top of Mount Everest

Finally, we need to calculate the weight of the 80-kg person on the top of Mount Everest (\(z=8848 \mathrm{m}\)). Determine the gravitational acceleration: \(g = g_0\left(1-\frac{2(8848)}{R + (8848)}\right)\) \(g = 9.807 \left(1-\frac{2(8848)}{6.37 \times 10^6 + (8848)}\right)\) \(g \approx 9.772 \mathrm{m/s^2}\) Now, calculate the weight: \(W = (80 \mathrm{kg})(9.772 \mathrm{m/s^2})\) \(W \approx 781.76 \mathrm{N}\)

Present the final results

After calculating the weight of the 80-kg person at different elevations, we found the following results: - At sea level: Weight = 784.56 N - In Denver: Weight ≈ 783.6 N - On top of Mount Everest: Weight ≈ 781.76 N