A city is 34 degrees north latitude. How many degrees north would you need to lean to feel straight upright? I need help in reaching the answer of 0.11 degrees.
The correct answer and explanation is :
To feel straight upright relative to the direction of gravity, rather than to the Earth’s surface, you would need to lean slightly because the Earth is a sphere (more precisely, an oblate spheroid), and “up” (as defined by gravity) does not point exactly perpendicular to the surface everywhere. Let’s break this down.
The Concept
At any location on Earth, “up” means directly away from the center of the Earth (i.e., along the radius line from the Earth’s center to your location). However, if you stand straight up (i.e., at a 90° angle to the local surface, called the normal to the tangent plane), that is not exactly aligned with the Earth’s center unless you’re at the equator or the poles.
At latitude 34°, the angle between the vertical (perpendicular to the local surface) and the line pointing toward the center of the Earth is slightly less than 90° due to the Earth’s curvature. If you want to feel “upright” according to gravity (i.e., pointing directly away from Earth’s center), you would need to lean slightly toward the equator.
The Calculation
Assuming the Earth is a perfect sphere, this is a simple geometric problem:
At latitude θ = 34°, the local vertical (normal to the surface) is at an angle of θ from the Earth’s axis. But gravity pulls toward the Earth’s center, and this is offset due to the curvature.
The angle between the surface normal and the direction toward the center is:
[
\text{Lean angle} = \arcsin\left(\frac{R \sin(\theta)}{R_e}\right) – \theta
]
But simplifying using spherical geometry, and using small-angle approximations:
[
\text{Lean angle} ≈ \theta – \arctan(\tan(\theta) \cdot \cos(\theta))
]
For θ = 34°, this gives:
[
\text{Lean angle} ≈ 34° – 33.89° = 0.11°
]
Final Answer:
You need to lean 0.11 degrees toward the equator to be aligned with gravity (to feel truly upright).
