Viewing page 191 of 212

This transcription has been completed. Contact us with corrections.

73.

[[underlined]] Appendix B [[/underlined]].

[[underlined]] Theory of the Displacements for a Simple Harmonic Motion [[/underlined]].

In addition to the notation given under Appendix [[underlined]] A [[/underlined]], the following additional notation must be employed:

Let m [[subscript]] s [[/subscript]] = the mass of the spring,

F [[subscript]] 1 [[/subscript]] = the force in dynes which produces unit extension of the spring,

m[[subscript]] 1 [[/subscript]] = the mass in dynes which produces unit extension of the spring,

and s = the upward displacement of [[underlined]] M [[/underlined]], resulting from the firing, that would be had if there were no friction, 

Then, allowing for the mass of the spring, we have, from the theory of simple harmonic motion:

^[[Fx = (M + Ms/3) (2[[pi]]/P)[[superscript]] 2 [[/superscript]] x,]]

where [[underlined]] x [[/underlined]] is any displacement, and [[underlined]] P [[/underlined]] is the period of the motion.

But V is the maximum velocity during the motion and hence 

V = w.s where [[underlined]] s [[/underlined]] is the maximum displacement, and [[underlined]] w [[/underlined]] is a constant, having the usual significance; 

also P = 2[[pi]]/w

Hence m [[subscript]] 1 [[/subscript]]g = (M + Ms/3) ∇[[superscript]] 2 [[/superscript]]/s[[superscript]] 2 [[/superscript]].

But by the conservation of linear momentum,

^[[(M + Ms/3)∇ = mov.]]

Hence 

^[[m[[subscript]] 1 [[/subscript]]g = (M + Ms/3) m[[superscript]] 2 [[/superscript]] v[[superscript]] 2 [[/superscript]]/(M + ms/3)[[superscript]] 2 [[/superscript]] 1/s[[superscript]] 2 [[/superscript]],