Spinning top as spirit level
On a horizontal surface, a rotating gyroscope remains in one position. If a gyro doesn't run sideways on a smooth surface, you can conclude that the surface is horizontal. Thus you can use a gyro like a spirit level.
If a gyroscope rotates on a slightly inclined surface, then it moves perpendicular to the "downhill" direction on the surface. If the top rotates the other way round, it moves in the opposite direction.
Fig. 1: A spinning top on an inclined plane; one arrow indicates the direction of rotation of the spinning top, the other the direction of travel of the spinning top.
The reason is that the tip of the spinning top is never quite pointed, but slightly rounded. This means that the point on which the gyro stands is only exactly on the axis of rotation of the gyro if the surface is horizontal. With an inclined surface, the contact point of the gyro lies at a distance r from the axis of rotation. Thus the gyro rolls on a small circle around its axis of rotation and thus rolls a little bit further with each revolution.
The angle of inclination a of the surface can be calculated from the traveling speed v. The travel speed v depends on the inclination angle a of the surface, on the rotational speed n of the spinning top and on the radius RSpitze of the spinning tops' tip.
Fig. 2: The gyro rolls on a small circle around the top of the spinning top.
At each revolution of the spinning top, it rolls over the surface by a distance of circumference Ur of a circle with radius r:
The walking speed v of the spinning top is:
Thus the angle of inclination a of the surface is:
Example: A gyro with a tip radius of 1 mm rotates at 20 turns per second and travels at 10 mm/s over a surface. This surface has an angle of inclination of:
Since it is difficult to measure the speed of the spinning top and the radius of the tip of the gyro, it is better to use the spinning top only as a spirit level and not to measure the angle of inclination of the surface. Or you simply place a marble on the surface: if it does not roll, the surface is flat.