Gyroscopes in planes, specifically attitude gyros or artificial horizons, are designed to indicate the aircraft's attitude relative to the horizon. They remain horizontal to the Earth's horizon, regardless of the plane's travel distance, due to their inherent property known as rigidity in space.
The principle of rigidity in space states that once a gyroscope is set in motion, its spin axis remains fixed in space unless acted upon by an external force. This property allows gyroscopes to maintain their orientation with respect to the fixed stars in the background.
When a plane is flying, it moves through the Earth's atmosphere and experiences various forces and accelerations. However, the gyroscope's spinning rotor provides a reference that remains independent of the plane's motion and unaffected by external forces. Therefore, the gyroscope continues to indicate a horizontal position relative to the Earth's horizon.
While gyroscopes can maintain their orientation relative to the horizon, they do not compensate for the Earth's curvature. The curvature of the Earth is not a factor that affects the behavior of gyroscopes directly. Instead, it is the gyroscope's reference to the local horizon that remains fixed.
For navigation and maintaining a consistent attitude, pilots rely on other instruments such as altimeters, GPS systems, and attitude indicators that incorporate data from multiple sources to account for the Earth's curvature, altitude changes, and position relative to the ground. These instruments work together to provide accurate information about the aircraft's orientation and position during flight.