Rolling without slipping is a common type of [i]general plane motion.[/i] Rolling without slipping is also an elucidating example of how:[br][center]General Plane Motion = Translation + Fixed Axis Rotation[/center]The figure below is meant to depict a wheel that is rolling without slipping from left to right. [i]Slipping[/i] would mean that sliding occurs between the wheel and the ground. Rolling [i]without slipping[/i] means that the bottom point of the wheel--the point in contact with the ground--does not move relative to the ground (i.e. [math]\vec{v}_B = 0 [/math]). Point [math]B[/math] is not a point on the wheel, it is the point where the wheel and the ground meet.[br][br]

Click the "Just Translate" button to see [i]translation[/i] in isolation. (You can also use the slider next to any button to control the motion.) You can choose to "Show TR Vectors" to see translational velocity [math] \vec{v}_{TR} [/math] depicted on four points on the circle. All points on the circle translate at the same rate when we consider translation in isolation. [br][br]Click the "Just Rotate" button see [i]fixed axis rotation[/i] about the center axis in isolation. The vector [math]\vec{v}_{FAR} [/math] represents the velocity of the four points due to rotation about the center axis. [br][br]Now click the "Roll W/out Slip" button. Rolling without slipping includes both translation and rotation. To find the total velocity [math] \vec{v}_{TOT}[/math] of any point on the wheel, you can use:[br][center][math]\large \vec{v}_{TOT} = \vec{v}_{TR} + \vec{v}_{FAR}[/math][/center]This is analogous to:[br][br][center][math]\large \vec{v}_A = \vec{v}_B + \vec{\omega} \times \vec{r}_{BA} [/math][/center]

Let's call the center axis of the wheel [math] G [/math]. And let's say that the wheel is rolling with a velocity of [math]\vec{v}_G [/math].[br][br]The translational velocity [math] \vec{v}_{TR} [/math] describes the speed at which the wheel translates left to right. This means that: [br][br][center][math] \large \vec{v}_{TR}=\vec{v}_G [/math][/center]The velocity from the fixed axis rotation [math]\vec{v}_{FAR} [/math] is related to the angular velocity [math] \omega [/math] of the wheel--the rate at which the wheel is spinning. The velocity from the fixed axis rotation can be found from: [br][br][center][math]\large \vec{v}_{FAR} = \omega r [/math][/center]where [math]r [/math] is the radial distance from the fixed axis. [br][br]Since rolling [i]without slipping[/i] means that there is no motion of the bottom point of the wheel relative to the ground, we can say that:[br][br][center][math]\large \vec{v}_B = 0[/math][/center]and since the total velocity at any point is the vector sum of the translational velocity and the velocity from fixed axis rotation, we can add that:[br][br][center][math]\large \vec{v}_B = 0 = \vec{v}_{TR} + \vec{v}_{FAR} [/math][/center]You can see this in the figure above. For a point at the bottom of the circle, the rightward translational velocity added to the leftward velocity from fixed axis rotation result in [math] \vec{v}_{TOT}=0 [/math].[br][br]For rolling without slipping, we know that:[br][br][center][math] \large |\vec{v}_{TR}| = |\vec{v}_{FAR}| [/math][/center]and [br][br][center][math] \large \vec{v}_G = \omega r_w [/math][/center]where [math] r_w [/math] is the radius of the wheel that is rolling without slipping.

When a wheel is rolling without slipping, the point where the wheel meets the ground is effectively the axis about which the wheel is rotating. We know this from:[br][br][center][math]\large \vec{v}_A = \vec{v}_B + \vec{\omega} \times \vec{r}_{BA} [/math][/center]Since the point where the wheel meets the ground has zero velocity (i.e. [math]\vec{v}_B=0[/math]), the velocity of any other point [math]A[/math] on the body can be found from [math]\vec{\omega} \times \vec{r}_{BA} [/math], where [math]\vec{r}_{BA} [/math] is the position vector from the bottom point of the wheel to point [math]A[/math].[br][br]Since point [math]B[/math] is constantly changing, rolling without slipping cannot be classified as rotation about a fixed axis. But because we can define a point [math]B[/math] that is momentarily stationary, we can treat it like fixed-axis rotation for an instant--only an instant because point [math]B[/math] changes every instant. Point [math]B[/math] is called an [i]Instantaneous Center of Rotation (ICR)[/i].