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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

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.

Movement:

flat

big flat

rock

inside 2-circle

inside 3-circle

inside rock

outside 1-circle

outside 2-circle

outside rock

Center \(C\)

none

\( \vec{v}_C \) (trans.)

\( \vec{\omega} \times \vec{r}_{CP} \) (rot.)

\( \vec{v}_P \) (total)

none

\( \vec{a}_C \) (trans.)

\( \vec{\alpha} \times \vec{r}_{CP} \) (ang.)

\( \vec{\omega} \times (\vec{\omega} \times \vec{r}_{CP}) \) (cent.)

\( \vec{a}_P \) (total)

Velocity and acceleration of points on a rigid body undergoing different rolling motions.

The most common and also the simplest form of rolling occurs on a flat surface.

Geometry and variables for rolling without slipping on a flat surface.

While rolling, the velocity and acceleration are directly connected to the angular velocity and angular acceleration, as shown by the next equations.

Center velocity and acceleration while rolling on a flat surface (Tangential-Normal Basis). #rko-ef

Center velocity and acceleration while rolling on a flat surface (Standard Basis). #rko-ef2

Another way to express the connection between angular and linear velocity and acceleration is via the distance \(s\) traveled by the rolling body:

Distance-angular relationships for rolling on a flat surface. #rko-eh

While rolling, the contact point will have zero velocity but will have a centripetal acceleration towards the rolling center:

Contact point \(P\) velocity and acceleration while rolling on a flat surface. #rko-eo

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.

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

Warning: Radii of curvature \(\rho\) and \(r\) may not be constant. #rko-wr

The center velocity for rolling on a curved surface is the same as for a flat surface, while the acceleration also depends on the radius of curvature of the surface:

Center velocity and acceleration while rolling on a curved surface (Tangential-Normal Basis). #rko-ec

Center velocity and acceleration while rolling on a curved surface (I-J Basis). #rko-ec2

The angular velocity of the rolling rigid body can be related to the derivative \( \dot{s} \) of the distance traveled on the surface as follows. In the special case of a perfectly circular surface, the angular acceleration also has a simple relationship with \( \ddot{s} \).

Distance-angular relationships for rolling on a curved surface. #rko-ew

Just as with rolling on a flat surface, when a rigid body rolls on a curved surface the contact point has zero velocity but nonzero acceleration, as shown below.

Contact point \(P\) velocity and acceleration while rolling on a curved surface. #rko-ep