The acceleration due to gravity at sea level is g = 32.2 ft/s^2 and decreases by 0.003 ft/s^2 per 1000 ft of elevation. Find the weight (in lbf) of a 165 lbm man atop the 8849 meters Mt. Everest relative to this point? Full solution.
The Correct Answer and Explanation is:
To find the weight (in pounds-force, lbf) of a 165 lbm (pound-mass) man atop Mt. Everest, we account for the reduction in gravitational acceleration with altitude.
Given:
- Sea-level gravity, g0=32.2 ft/s2g_0 = 32.2 \, \text{ft/s}^2g0=32.2ft/s2
- Gravity decreases by 0.003 ft/s² per 1000 ft
- Elevation of Mt. Everest: 8849 meters
- 1 meter = 3.28084 feet
- Mass of man: m=165 lbmm = 165 \, \text{lbm}m=165lbm
Step 1: Convert elevation to feet
8849 m×3.28084 ftm=29,032 ft (approx.)8849 \, \text{m} \times 3.28084 \, \frac{\text{ft}}{\text{m}} = 29,032 \, \text{ft (approx.)}8849m×3.28084mft=29,032ft (approx.)
Step 2: Calculate reduction in gravitational acceleration
Reduction=(29,0321000)×0.003=29.032×0.003=0.0871 ft/s2\text{Reduction} = \left(\frac{29,032}{1000}\right) \times 0.003 = 29.032 \times 0.003 = 0.0871 \, \text{ft/s}^2Reduction=(100029,032)×0.003=29.032×0.003=0.0871ft/s2
Step 3: Gravity at the summit
gEverest=32.2−0.0871=32.1129 ft/s2g_{\text{Everest}} = 32.2 – 0.0871 = 32.1129 \, \text{ft/s}^2gEverest=32.2−0.0871=32.1129ft/s2
Step 4: Weight on Mt. Everest
Weight in pounds-force is given by:W=m×gEverest/g0×g0=m×(gEverestg0)×g0W = m \times g_{\text{Everest}} / g_0 \times g_0 = m \times \left(\frac{g_{\text{Everest}}}{g_0}\right) \times g_0W=m×gEverest/g0×g0=m×(g0gEverest)×g0
But since lbf is defined as lbm × (local g) / standard g, we can simplify as:W=m×(gEverest32.2)=165×(32.112932.2)W = m \times \left(\frac{g_{\text{Everest}}}{32.2}\right) = 165 \times \left(\frac{32.1129}{32.2}\right)W=m×(32.2gEverest)=165×(32.232.1129)W≈165×0.9969=164.5 lbfW \approx 165 \times 0.9969 = 164.5 \, \text{lbf}W≈165×0.9969=164.5lbf
Final Answer:
The man’s weight atop Mt. Everest is approximately 164.5 lbf.
Explanation:
Weight is a measure of force, which depends on both mass and the local acceleration due to gravity. While a person’s mass remains constant regardless of location, their weight can vary depending on the gravitational pull. At higher elevations like Mt. Everest, gravity weakens slightly because the distance from Earth’s center increases.
Here, we calculated how gravity changes with altitude based on the given rate of decrease. After converting the mountain’s height from meters to feet, we determined the total reduction in gravity. By subtracting this from the standard gravity at sea level, we obtained the adjusted gravitational value at the summit. Finally, multiplying this value by the man’s mass in pound-mass and adjusting proportionally, we found his weight in pound-force.
This problem shows how physics must account for environmental conditions even when dealing with simple quantities like weight.
