Rolling Ball inside a Cylinder

A ball rolling along the inside of a cylinder without slipping performs a surprising return to the starting height. This is related to the phenomenon of a golf ball rising out of the cup after it starts to fall in. This Demonstration uses the equations of motion to show the rolling ball together with the axis of rotation and an energy plot. A key point is the proper coordinate system for studying the axis of rotation. The rotation axis is given by , where refers to the vertical axis through the center of the ball, refers to the horizontal axis through the point of contact and normal to the cylinder, and refers to the horizontal axis perpendicular to that of . The third component of the angular velocity, , turns out to remain constant. The graph on the right shows the breakdown of two types of energy. Total energy is constant, but the energy can be divided into a conventional part (kinetic energy plus angular energy due to rotation of and plus potential energy) and a surprising part, the spinning around the axis, . The -spinning is due to a coriolis torque and it is this that causes the sign of to reverse; because , the reversal of -spin causes the reversal of the -motion.

The controls:

The ratio control gives where is the ball radius and the cylinder radius. is always set to the golf ball cup radius of 5.1 cm. The golf ball setting for is 2.1 cm.

is the acceleration due to gravity in .

is the entry angle, in radians measured from the horizontal.

is the initial speed of the ball.

is the initial rate of spin, in radians per second, about the horizontal axis normal to the cylinder—the axis. For example, a golf ball will have such spin as it enters the cup. The default ties this to via , but there is a checkbox to decouple these.

The number of revolutions sets the number of horizontal revolutions of the ball around the cylinder.

Time is given in seconds.