A Hand Saw's Cutting Rate as a Function of Its Filing State: Rate Decay Over a Working Day and the Price of Filing Against the Rate It Buys Back

A hand saw does not cut at one rate. It cuts fastest just after filing and slower with every board, and the person using it must decide, repeatedly, whether to stop and file or to keep cutting with a dulling saw. This paper models that decision for a 660 mm crosscut saw with 8 teeth per inch cutting 45 mm seasoned pine boards. A freshly filed saw needs 14 strokes per board, or 205.7 boards per hour at the stated pace; a fully dull saw needs 40.0 strokes per board, or 72.0 boards per hour. With sharpness decaying exponentially over 90 boards, the filing interval that maximises the net rate is 112 boards for a full 14-minute filing, giving 110.9 boards per hour sustained, and 72 boards for a 6-minute touch-up that restores sharpness only to 0.85, giving 123.1 boards per hour. The touch-up, although it restores less sharpness, beats the full filing on sustained rate because it costs less time. A saw that is never filed completes 671 boards in an eight-hour day, against 887 with filing repeated at the optimal interval. The marginal rule for stopping is explicit: file when sharpness falls to 0.291 of fresh, which occurs after 111.1 boards, the point at which the instantaneous rate has fallen to the best sustained average rate. Two defects are priced as well. Teeth set too narrow reduce the sustained rate to 83.3 boards per hour, and a damaged section of teeth reduces it to 92.1 boards per hour, even with optimal filing. One common habit does not pay: filing the moment the saw feels dull is not always correct, because at 120 boards since filing, filing before a final batch of only 30 boards takes 23.7 minutes against 17.6 minutes to finish unfiled; filing at that point only breaks even if at least 54 further boards remain to be cut.