. American engineer and railroad journal . indenberger announces this problem as follows : Lft it be supposed that the stroke of the pistons of a loco-motive is 2 ft., the diameter of the driving-wheels 8 ft. and thevelocity 60 miles per hour : what is the maximum and mini-mum velocity of the piston relatively to the earth and not withregard to the locomotive, and when does each occur : The author has chosen the usual analytical method for find-ing maxima aud minima— , by writing that the differentialof the velocity — 0 for a maximum or minimum ; this he hasthe right to do, although this m
. American engineer and railroad journal . indenberger announces this problem as follows : Lft it be supposed that the stroke of the pistons of a loco-motive is 2 ft., the diameter of the driving-wheels 8 ft. and thevelocity 60 miles per hour : what is the maximum and mini-mum velocity of the piston relatively to the earth and not withregard to the locomotive, and when does each occur : The author has chosen the usual analytical method for find-ing maxima aud minima— , by writing that the differentialof the velocity — 0 for a maximum or minimum ; this he hasthe right to do, although this method is much more compli-cated in this case than the geometrical solution ; but the mis-take begins in solving the equation of third degree resultingfrom this analytical method. The detail of this last calcula-tion is not published in the article above mentioned ; and myopinion is that the formula adopted to solve the equation ofthird degree must have been only a formula of approximation,for Mr. Lindenberger concludes by saying : X;S. So that the maximum and minimum velocity is not forthese parts [the crank-pin radius and the connecting-rod] atright angles, as some solutions assume, but is very near thatplace. Now, I would ask the author whether he has examined tisolutions that assume, or. rather, whether he thinks that amathematical demonstration proves beyond tiny doubt orjust assumes. To decide this question. I will take the lib-erty of recalling here in a few words the usual geometricalsolution of this problem : Let A T> be the connecting-rod. Let 0 ]! In- the crank pin radius. Let / = the velocity of the piston. Let u =z the angular velocity of the crank-pin radius, or theangle described by this radius in the unit of time. Thisangular velocity is constant, since the velocity of the engine issupposed to be constant. Vol. LXVIII, No. i.] AND RAILROAD JOURNAL First, it is evident that the maximum and minimum veloci-ties of the piston relatively to the earth occur at t
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