Contact and Rolling

Rolling motion

A special case of rigid body motion is rolling without slipping on a stationary ground surface. This is defined by motion where the point of contact with the ground has zero velocity, so it matches the ground velocity and is not slipping.

Rolling without slipping on stationary ground surfaces: #rko-er

$$ \text{Contact point P has zero velocity}: \quad \vec{v}_P = 0 $$

Rolling without slipping means by definition that the contacting points have the same velocity. For a stationary ground the velocity is zero, so the contact point on the body must also have zero velocity.

It is helpful to think about the motion of the body in two ways:

The body rotates about the moving center $ C $.
The body rotates about the instantaneous center at the contact point $ P $.

These two ways of visualizing the motion can be seen on the figure below.

INSERT INTERACTIVE BLOCK HERE

Rolling on a flat surface

Insert Rolling on a 2D flat surface subsection here.

Rolling on curved surfaces

When a circular rigid body rolls without slipping on a surface which is itself curved, the radius of curvature of the surface affects the acceleration (but not velocity) of points on the rolling body.

INSERT INTERACTIVE BLOCK HERE

Geometric quantities for rolling on a curved surface. #rko-eg

$$ \begin{aligned} \left.\begin{aligned} R &= \rho - r \\ \vec{\omega} &= -\omega \,\hat{e}_b \\ \vec{\alpha} &= -\alpha \,\hat{e}_b \end{aligned}\right\} & \text{ when rolling on the inside of a curved surface} \\[1em] \left.\begin{aligned} R &= \rho + r \\ \vec{\omega} &= \omega \,\hat{e}_b \\ \vec{\alpha} &= \alpha \,\hat{e}_b \end{aligned}\right\} & \text{ when rolling on the outside of a curved surface} \end{aligned} $$

These formulas are all choices of sign conventions for $ \omega $ and $ \alpha $, and definitions of $ R $. Figure #rko-fc shows the appropriate geometry. Note that $ \omega $ and $ \alpha $ are defined with positive values corresponding to motion in the tangential direction.

Warning: Radii of curvature $ \rho $ and $ R $ may not be constant.

The radius of curvature $ \rho $ of the surface may be varying with position as the body rolls. If $ \rho $ changes then this will also cause $ R $ to change. These two variables will only be constant if the surface is in fact perfectly circular.

Applications

Bearings

This topic is in L27, slides 13-14. Pictures are from this book https://i-share-uiu.primo.exlibrisgroup.com/permalink/01CARLI_UIU/gpjosq/alma99955068260305899

Application for "Rolling on curved surfaces".