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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]],