MOTION OF DEVELOPABLE CYLINDERS. 87 



follows : — Suppose that originally we had a thin lamina 

 resting on a flat plane ; now the amount of work necessary 

 to raise a particle of weight w to the height y is wy ; and 

 to raise an aggregate of particles the work will be %ivy or 

 #My, where y denotes the vertical height of the centre of 

 gravity and M the whole mass. In the rolled-up cylinder 

 this is stored up as potential energy ; during the motion it 

 assumes the kinetic form, and would of itself be sufficient 

 to keep up the motion on a smooth plane. In what precedes 

 I have supposed the centre of gravity to lie in the normal 

 to the plane drawn through the point of contact of the 

 cylinder with the plane. This would not be exactly true ; 

 on account of the cylinder not being perfectly circular, there 

 will be an extremely small couple due to gravity tending to 

 produce rotation. 



If in equation (2) we suppose the length of the tape to 

 be infinite, for the time of motion during any finite length 



we shall have t = ~ . 



v 



In the above problem I have supposed the external edge 

 of the tape to be fixed. We may, however, have the inter- 

 nal edge fixed and the external in motion, as in the fol- 

 lowing problem : — An indefinitely thin lamina is wound 

 round a fixed horizontal cylinder of indefinitely small cross 

 section; to the external edge of the lamina a weight is fixed: 

 to determine the motion of this edge. Suppose we take 

 moments about the fixed axis. Then the expression 



^ / d z y d z x\ 



for the whole mass may be divided into two parts. For the 

 portion that is still coiled up the angular velocity will be 

 the same for each particle, and equal to the angular velo- 



