What is the equation used in computing the acceleration of the center of a rolling ball with a diameter of 8 inches?

The correct equation for computing the acceleration of the center of a rolling ball is indeed rw². In this context, 'r' represents the radius of the ball, and 'w' denotes the angular velocity.

When an object rolls without slipping, its linear acceleration at the center is directly related to its angular acceleration. This relationship is derived from the fact that the point of contact between the ball and the surface does not slide; therefore, the linear acceleration of the center of mass is equal to the radius multiplied by the square of the angular velocity.

In more detail, the acceleration of the center of the ball can be linked to the rolling motion through the equation a = r * α, where 'a' is the linear acceleration and 'α' is the angular acceleration. Since angular velocity (w) can be related to angular acceleration (α) by the equation α = w² (for uniform acceleration scenarios), we find that rw² is indicative of the acceleration imparted to the center of the rolling object.

This concept plays a significant role in mechanical systems where rolling elements are utilized, ensuring efficient energy transfer and minimizing energy loss through friction. Understanding the dynamics of rolling motion is essential for designing effective mechanical systems, such as in wheels, bearings