Combined Translation & Rotation

Applet shows a disc rolling to the right on a horizontal surface. You can change the linear and angular velocities of the disc. You can use the check boxes in the display tab at the bottom right end to display the velocity vector of a point on the disc and/or the path traced by a point on the disc or see the motion without the clutter by turning them off. The velocity vector of the point due to rotation is displayed in red and the velocity vector due to translation is displayed in yellow. The green vector is the velocity vector of the point due to combined translation and rotation. When the disc reaches the right end of the applet area it stops. You can use the reset button to shift it back to the left end of the applet area. ( The start/stop button turns blue in to a reset button ).

The orange colored line on the floor shows the linear displacement of the center of the disc. The angle subtended by the orange colored arc on the edge of the disc shows the angular displacement of the disc. If the length of the orange arc on the edge of the disc is equal to the length of the orange line on the floor, the disc is rolling without slipping on the surface. The velocity of the bottom most point is zero in this case. You can observe that the red velocity vector and yellow velocity vector are equal in length in this case and since at the bottom most point they are opposite they add up to zero. Also note that the disc travels a distance equal to its circumference before it stops. This is the length of the orange line on the floor.

Increase or decrease the linear/angular velocities and observe what happens. Try different combinations, finding the combination that causes the body to roll without slipping. Also observe the velocity vectors. The red and yellow vectors are parallel at the top most point and anti parallel at the bottom most point.

Can you make out the ratio of the angular and linear speeds by observing the path? By observing the lengths of the velocity vectors? (Radius of the disc is assumed to be one unit. )