Physics

Problem

A hiker with mass 70\,kg climbs from sea level to the summit of a mountain that is 1.20\,km above sea level. Assume the Earth is a perfect sphere of radius

R_{\!E}=6.37\times10^{6}\,\text{m}

and mass

M_{\!E}=5.97\times10^{24}\,\text{kg}

. Neglect air resistance and any motion of the Earth during the climb. (a) Calculate the magnitude of the local gravitational field

g_{\text{summit}}

at the mountain summit using universal gravitation. (b) Determine the hiker’s weight at the summit and the percent difference between this value and the hiker’s weight at sea level, where

g_{0}=9.80\,\text{N\,kg}^{-1}

. (c) The hiker’s equipment manual states that any weight measurement within 1.0\% of the true value is considered acceptable. Based on your calculation in part (b), evaluate whether taking

g=9.80\,\text{N\,kg}^{-1}

for all altitudes up to 1.20\,km satisfies this criterion, and justify your answer.

Intuitive explanation

This problem tests whether you understand that gravitational potential energy depends on distance from Earth's center, not just height above ground. The key insight is that $g$ isn't truly constant over 1.2 km—it decreases slightly as you move farther from the center—so you can't just use $\Delta PE = mgh$ . Instead, use the exact gravitational potential energy formula $PE = -\frac{GMm}{r}$ at both sea level and the summit, then find the difference. The small change reveals how the "constant $g$ " approximation breaks down at larger scales.

Intuitive explanation

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