We want to know what the velocity is at these points on this rolling wheel moving forward with velocity v. The instant center is the spot about which all points on a body rotate for a particular instant in time. When the instant center is determined we can use vector geometry to determine the velocity at any point on the wheel. What is the instant center for this wheel? (Click Here!) | |
Given the forward velocity of the wheel, shown in green, and the location of the instant center, the velocity at all other points on the wheel can be determined. | |
A velocity triangle, in white, can be drawn connecting the zero velocity point (the instant center) and the endpoints of the known velocity vector (at the wheel axis). The velocities of the points of interest on the circle which intersect the white line can be determined immediately. | |
Geometrically determining the velocity of the other points (orange, yellow, and blue) is a little trickier. The direction and magnitude of the velocity at the orange point will be determined first. First, draw a circle with center at the instant center and intersecting the point of interest, and the line tangent to the circle at the point (thin green lines). The magnitude is determined based on the width of the velocity triangle at that radius. The vector is rotated to meet the orange point, and the process is complete. | |
Repeat the process described above to determine the velocity at the yellow and blue points. Notice that the points to the right of the wheel axis are actually travelling toward the ground and the points to the left of the axis are travelling up--rather than in a circular fashion about the axis!
By definition, the magnitude of the velocity at any point is equal to the product of the known velocity with the ratio of the distance from the instant center to the point of interest and the distance from the instant center to the point of known velocity. But it is more fun to draw, eh? |