Calculating Uncertainty For the Analog Ohmmeter

The Radio Shack VOM is an analog multimeter which can be used to measure voltage, current, and resistance. Determining the uncertainty of its measured values is simple enough — regardless of which function the meter is set for, the uncertainty of the measured value is based on the uncertainty in the needle position, which is reported by the manufacturer to be 3% of full deflection (except for AC voltage and the more sensitive DC voltage ranges, where the uncertainty is a little higher). This basically implies that the uncertainty in voltages and currents obtained by the VOM is 3% of the range setting for that measurement (meaning, for example, that a 15 V reading using the 50 V range setting yields an uncertainty of (0.03)(50 V) = 1.5 V in the 15 V reading). Calculation of the uncertainty of the voltage and current values obtained from the VOM is straightforward because the scales are linear — the voltage and current readings are proportional to the needle deflection. Unfortunately, the resistance readings are not. The green ohmmeter scale located on the top part of the analog display is non-linear. Thus, a 3% uncertainty in the needle position has to be translated to an uncertainty in the resistance value using a ruler. Since the entire green ohmmeter scale is 10 cm long, 3% full deflection corresponds to 3 mm to either side of the needle position. The uncertainty in the resistance has to be obtained by estimating the changes in the resistance reading when the needle physically moves 3 mm in either direction. Determining the uncertainty in the VOM reading with a ruler is an extremely clumsy operation. Not only is it necessary to have a metric ruler on hand, but it is impossible to superimpose the ruler on the scale with any degree of precision because of parallax effects (the ruler is too far away from the scale). Surely there must be a better way of calculating the uncertainty for the VOM ohmmeter. It turns out there is a better way. If an exact relationship between needle position and green ohmmeter scale reading can be worked out, then the 3% uncertainty in the needle deflection can be translated to an uncertainty in the resistance value using calculus. In fact, this relationship has now been worked out. It is