If a plank is moving horizontally on a solid cylinder with a spring attached to it on a wall, why is its velocity twice the velocity of the cylinder? Pretty much the title, why is the velocity of the plank on the cylinder twice the velocity of the cylinder?
 A: I assume the speed of the cylinder (v) is defined as the speed of its centre of mass, which should be at the axle. At any point in time while the cylinder is rolling, the current speed of its top point is 2v, the speed of its lowest point touching the table is 0. Average of all points of the cylinder is v.
A: Is the cylinder is rolling on a table with a plank on top of it?
Suppose you are an ant sitting on the axle of the cylinder. You see the periphery of the cylinder has speed v. You see the cylinder pushes the table backward and the plank forward with speed v.
Suppose you are an ant on the table. You see the table sitting still, and the ant on the axle moving forward with speed v. You see the plank moving forward with speed 2v.
A: At an instant, paint a vertical line across the side of the cylinder, from the ground, through its axis, and to the top (where the plank sits).
Then, observe the line as the cylinder rolls (let's say it's rolling right from your vantage point.) The line will rotate clockwise.
The center of the line is moving with the axis of the cylinder. But, in addition to that, the top of the line is moving forward (right) and the bottom is moving back (left). If you watch carefully, the bottom of the line will lift straight up off the ground -- at that instant, it's not moving horizontally at all.
Thus, we have a line segment where the middle is moving with the cylinder, the bottom is not moving at all (horizontally) and the top is moving faster than the cylinder.
Because it's symmetric, the amount by which the top is faster than the middle is the same as the amount by which the bottom is slower. If the middle is moving at 1m/s, and we know the bottom is moving at 0m/s, then the top is moving at 2m/s.
