Apparent Weight and the Earth's Rotation

At the equator, objects move in a circle due to Earth's rotation, requiring centripetal acceleration toward the rotational axis. The spring scale reading (apparent weight) equals true weight minus the centripetal force needed for this circular motion. At the poles, centripetal force is zero, so apparent weight equals true weight.

Centripetal force is proportional to Earth's radius and the square of its angular speed. Since Earth's angular speed is relatively small, the centripetal force is also small. At the equator, this force accounts for only a 0.34% difference between true and apparent weight, making it a minor correction.

Three forces act on the object: gravitational force directed toward Earth's center, tension force in the spring directed away from the center, and centripetal force directed toward the rotation axis. The tension force (apparent weight) balances the gravitational force minus the centripetal force requirement.

For objects at the equator to feel weightless, the centripetal force would need to equal the gravitational force. Calculations show Earth's orbital period would need to be only 84 minutes instead of 24 hours. This demonstrates how slowly Earth actually rotates relative to what would be required for weightlessness.

At the poles, objects do not move in a circle around Earth's rotational axis; they remain stationary relative to the axis. Since there is no circular motion, no centripetal acceleration is needed. Therefore, apparent weight equals true weight at the poles.

The tension force in the spring scale equals the apparent weight of the object. This reading represents the normal force preventing the object from falling through the scale. At the equator, this tension is less than the gravitational force because some gravitational force provides the necessary centripetal acceleration.

Earth's rotation causes variation in acceleration due to gravity near the earth s surface by reducing the effective gravitational acceleration at the equator. The centripetal acceleration required for circular motion at the equator reduces the net downward acceleration, making objects appear lighter on a scale compared to the poles.