What relationship is described between angular velocity and linear velocity?

The relationship between angular velocity and linear velocity is defined by the formula ( v = r \times \omega ). In this equation, ( v ) represents the linear velocity, ( r ) is the radius of the circular path, and ( \omega ) is the angular velocity in radians per second.

This relationship shows that the linear velocity of an object moving in a circular path is directly proportional to both the radius of the circle and the angular velocity. When the radius increases, the linear velocity increases for a given angular velocity. Similarly, if the angular velocity increases while keeping the radius constant, the linear velocity also increases.

This fundamental concept is essential in mechanics, particularly in analyzing motion in circular paths. It helps in understanding how different physical quantities relate to each other in rotational motion and is widely applicable in various engineering and physics problems involving rotational dynamics.