Physics
Rotational Dynamics / Gravitation
Effect of Earth's Rotation on Apparent Weight
Class 11
⚡ Quick Summary
Earth's rotation creates a centrifugal force on objects. This pseudo-force, when combined with true gravity, changes the perceived direction (apparent vertical) and magnitude (apparent acceleration due to gravity, g') of gravity. Consequently, the observed weight of an object (apparent weight) is slightly less than its true gravitational weight and varies with latitude.
<ul><li>Radius of circular path for a point P at colatitude θ: `r = R sinθ` (where R is Earth's radius)</li><li>Magnitude of apparent acceleration due to gravity (g'): `g' = √(g² - ω²R sin²θ (2g - ω²R))`</li><li>Angle (α) of apparent vertical with true vertical: `tan α = (ω²R sinθ cosθ) / (g - ω²R sin²θ)`</li><li>Apparent Weight: `T = mg'`</li></ul>
- Centrifugal Force: It is a pseudo-force (or fictitious force) assumed to act when describing particle motion from a rotating, non-inertial frame of reference, allowing Newton's laws to be used. It acts radially outwards from the center of rotation.
- Earth's Rotation: The Earth rotates about its axis at an angular speed (ω) of one revolution per 24 hours. Every point on Earth moves in a circle.
- Colatitude (θ): For a place P on Earth, colatitude is the angle between the Earth's axis of rotation (SN) and the radius (OP) through the place P.
- Radius of Rotation (r): For a place P at colatitude θ, its circular path has a radius `r = R sinθ`, where R is the radius of the Earth.
- Forces in Earth's Frame: When analyzing objects at rest relative to the Earth (a non-inertial frame), in addition to the gravitational attraction (`mg`) towards the Earth's center, a centrifugal force `mω²r` must be included, directed outwards from the axis of rotation.
- Apparent Acceleration Due to Gravity (g'): This is the resultant of the true gravitational acceleration (g) and the effect of the centrifugal force. It determines the apparent vertical direction.
- Apparent Vertical Direction: This is the direction of `g'`. A plumb line will align itself with this direction, and the surface of water at rest will be perpendicular to it. It makes an angle (α) with the true vertical (direction of `g`).
- Apparent Weight: The observed weight of a body, typically measured as the tension (T) in a string suspending it or the normal force on a weighing machine. It is given by `T = mg'`. Due to the centrifugal force, `g'` is generally less than `g`, meaning the apparent weight is less than the true weight (`mg`). The reduction in weight is maximum at the equator (where `θ = 90°` or latitude is `0°`) and zero at the poles (where `θ = 0°` or latitude is `90°`).