# This example simulates a `swinging Atwood's machine'. An Atwood's # machine consists of two masses joined by a taut length of cord. The cord # is suspended from a pulley. The heavier mass (M) would normally win # against the lighter mass (m), and pull it upward. A `swinging' Atwood's # machine is an Atwood's machine with an additional degree of freedom: it # allows the lighter mass to swing back and forth in a plane, at the same # time as it is being drawn upward. # Let `a' denote the angle by which the cord extending to the lighter mass # deviates from the vertical. Let `l' denote the distance along the cord # between the pulley and the lighter mass. Then the system of differential # equations below will describe the evolution of the system. # You may run this example, with output to an X window in real time, by doing # # ode < atwoods.ode | graph -T X -x 9 11 -y -1 1 -m 0 -S 1 # # The plot will trace out `l' and `ldot' (its time derivative). The `-m 0 # -S 1' option requests that successive datapoints not be joined by line # segments, but rather that marker symbol #1 (a point) be plotted at the # location of each datapoint. # You may have some difficulty believing the results of this simulation. # Allowing the lighter mass to swing, it turns out, may prevent the heavier # mass from winning against it. The system may oscillate, # non-periodically. m = 1 # lighter mass M = 1.0625 # heavier mass a = 0.5 # initial angle of cord from vertical, in radians adot = 0 l = 10 # initial distance along cord from pulley to mass m ldot = 0 g = 9.8 # acceleration due to gravity ldot' = ( m * l * adot * adot - M * g + m * g * cos(a) ) / (m + M) l' = ldot adot' = (-1/l) * (g * sin(a) + 2 * adot * ldot) a' = adot print l, ldot step 0, 400
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