86 DR. JAMES BOTTOMLEY ON THE 



If we suppose the motion to take place on a horizontal 

 plane, the equation of motion changes; the differential 

 equation now becomes 



d z x c fd.r\ z 



= o. 



dt z 27ry z \dtJ 

 A first integral of this equation will be 



( 



dx\ z _ A 



dt ) 7rR z — cx 



(I) 



To determine A, suppose the mass set in motion by a 

 blow parallel to the plane so that the initial velocity is V, 

 thus A=ttR 2 V\ 



Substituting, integrating again, and determining the 

 constant we have 



-jH*K- ■)'}■ ■■ ■ 



(2) 



To unwind the whole length of the tape, this equation 

 would make the time to vary as the length directly and as 

 the initial velocity inversely. 



Equation (i) may also be written 



. /dx\ z _ 

 l \di) ~ 



M,[^l = MV\ 



Hence, as the mass diminishes the velocity increases, but 

 the kinetic energy in the direction of motion is constant 

 and equal to its initial value. Hence it would seem that 

 none of the energy of the blow is consumed during the 

 rotation of the variable cylinder ; once started it would 

 continue of itself. In the rolled-up cylinder there is an 

 amount of potential energy which may be estimated as 



