Explanation:

The space centrode of a circular disc rolling on a straight path is a straight line.

Definition of Space Centrode:

The space centrode is the locus of the instantaneous centers of rotation of a rolling body. It is a curve that describes the path traced by the point of contact of two bodies in rolling contact. For a disc rolling on a straight path, the space centrode is a straight line.

Derivation:

Consider a circular disc of radius r rolling on a straight path. Let O be the center of the disc, and P be the point of contact between the disc and the path. Let C be the instantaneous center of rotation of the disc, and let Q be the point of contact between the disc and its image in a plane perpendicular to the path.

Now, as the disc rolls, the point P moves along the path, and the point Q moves along a circle of radius r centered at C. Therefore, the locus of C is a straight line passing through the point of contact P.

Hence, the space centrode of a circular disc rolling on a straight path is a straight line.

Applications:

The concept of space centrode is widely used in kinematics and dynamics of machinery. It is used to analyze the motion of rolling bodies, such as gears, cams, and rollers. The knowledge of space centrode helps in designing mechanisms with desired motion characteristics, such as constant velocity, acceleration, and deceleration. It also helps in minimizing wear and tear of the contacting surfaces, by ensuring that the instantaneous centers of rotation lie on the surfaces.