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How does a digital scale work?

The sensor is called a load cell, and it uses a strain gauge. This can be some metal tracks (foil) bonded onto a printed circuit board, or on some other backing. You can buy them for industrial purposes. When the metal foil is "strained", as the backing flexes or stretches, there is a small change in resistance. To measure weight a lever and weights or springs requires a calibrated force for a given deflection, which is then applied to the strain gauge.

Strain gauges are used with 4 in a bridge connection to get a voltage proportional to strain. This is amplified, filtered then processed through an analogue to digital converter and microprocessor to get a digital value, some maths applied to make it stones or kg, and it is displayed. You can buy strain gauge amplifiers.

You can buy load cells to measure weights. These have inbuilt force transduce, strain gauge sensors, amplifiers etc and an output signal can be sent to some other location. They are used in factories for weighing just about anything.