Rolling without Slipping
Rolling Without Slipping is a form of General Plane Motion
Rolling without slipping is a common type of general plane motion. Rolling without slipping is also an elucidating example of how:
General Plane Motion = Translation + Fixed Axis Rotation
The figure below is meant to depict a wheel that is rolling without slipping from left to right. Slipping would mean that sliding occurs between the wheel and the ground. Rolling without slipping means that the bottom point of the wheel--the point in contact with the ground--does not move relative to the ground (i.e. ). Point is not a point on the wheel, it is the point where the wheel and the ground meet.Use the Figure Below to Understand how General Plane Motion is Translation Plus Fixed Axis Rotation
Click the "Just Translate" button to see translation 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 depicted on four points on the circle. All points on the circle translate at the same rate when we consider translation in isolation.
Click the "Just Rotate" button see fixed axis rotation about the center axis in isolation. The vector represents the velocity of the four points due to rotation about the center axis.
Now click the "Roll W/out Slip" button. Rolling without slipping includes both translation and rotation. To find the total velocity of any point on the wheel, you can use:
This is analogous to:
In the Figure above, Click "Roll W/out Slip" and Check "Show Vector Sum"
If the wheel is rolling without slipping with a speed of , what is the speed of the bottom point of the wheel? (select one)
In the Figure above, Click "Roll W/out Slip" and Check "Show Vector Sum"
If the wheel is rolling without slipping with a speed of , what is the speed of the top point of the wheel? (select one)
In the Figure above, Click "Roll W/out Slip" and Check "Show Vector Sum"
If the wheel is rolling without slipping with a speed of , what is the speed of a point on the edge of the wheel at mid-height of the wheel? (select one)
Let's call the center axis of the wheel . And let's say that the wheel is rolling with a velocity of .
The translational velocity describes the speed at which the wheel translates left to right. This means that:
The velocity from the fixed axis rotation is related to the angular velocity of the wheel--the rate at which the wheel is spinning. The velocity from the fixed axis rotation can be found from:
where is the radial distance from the fixed axis.
Since rolling without slipping means that there is no motion of the bottom point of the wheel relative to the ground, we can say that:
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:
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 .
For rolling without slipping, we know that:
and
where is the radius of the wheel that is rolling without slipping.
Center of Rotation
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:
Since the point where the wheel meets the ground has zero velocity (i.e. ), the velocity of any other point on the body can be found from , where is the position vector from the bottom point of the wheel to point .
Since point is constantly changing, rolling without slipping cannot be classified as rotation about a fixed axis. But because we can define a point that is momentarily stationary, we can treat it like fixed-axis rotation for an instant--only an instant because point changes every instant. Point is called an Instantaneous Center of Rotation (ICR).