lew@ihuxr.UUCP (Lew Mammel, Jr.) (10/10/83)
On a real scale, the short duration force as the hourglass runs out can be treated as an impulse. It has the value m*l/t, where m is the mass of the sand, t is the running time, and l is the distance the sand is displaced. The scale can be regarded as a harmonic oscillator. It has a spring constant k, and an effective mass M. The spring constant is the weight per unit displacement of the pan. The effective mass can be calculated from the (angular) frequency of oscillation of the loaded scale: w = sqrt(k/M) , M = k/w^2 The scale's response to an impulse, I, is given by A*w = I/M. This just matches velocity due to the impulse with the velocity of oscillation. I tried this out on a home built balance I use for this sort of thing. (The balance is just a simple basswood frame, with plastic cups suspended in pivoted cradles for pans.) I jury-rigged one end to be like one of those sand box funnel toys, except I used salt. The parameters were: m = 23 pennies + 14 staples ~= 872 staples (1penny ~= 46 staples) l = 10 cm t = 22.7 sec (good to .1 sec over 7 trials, but this is overkill) k = 1 staple * g / .2 cm = 4900 staples/sec^2 w = 2pi/ 5 sec = 1.25 sec^-1 The intermediate values are: I = 466 staple cm/sec M = 3103 staples ... giving a predicted amplitude of A = .11 cm. On trying the experiment, I found that the scale balanced nicely while the salt was running, and received a barely perceptible impulse when it ran out - I measured its displacement at .05 cm. I didn't account for the reduced weight of the salt as it moved closer to the center of the earth :-) Lew Mammel, Jr. ihuxr!lew