This broad question may have a number of answers. Which force is being measured? Magnetic? Electrostatic? Gravitational? Different forces are measured in different ways. Sometimes some clever experiments must be concocted to actually perform a measurement. As there are many answers, let's just look at a couple of instances. The phenomenon of piezoelectric effect is a simple concept, though it is a bit subtle in it's underlying nature. What this is all about is based on the fact that by compressing a crystal of something that exhibits piezoelectric properties, we can observe a tiny generated voltage (electromotive force or EMF). By setting up a tiny crystal of, say, lead zirconate titanate, a commonly selected material, we can observe the force created on the crystal by the weight (mass in a gravimetric field) of a small object. The mass of the tiny gem presses down on the crystal and a tiny voltage is created. We measure this voltage and convert it to a weight, and the weight of our gemstone is then discovered. We can weigh tiny gemstones with great accuracy using piezoelectric scales. The piezoelectric principle can be used to measure small forces in the same way it is used in this application. Electricity was a mystery for a long time. Then it was discovered that it "flowed" and, therefore, must be "liquid-like" tiny particles. The electron was discovered. And then we needed to know the charge on the electron. The electrostatic force an electron exerts on other electrons and between themselves and protons is a function of (among some other things) the field strength of the electron. How much force does that field generate? Robert Millikan, with considerable help from Harvey Fletcher, electrically charged tiny droplets of oil and let them fall through air in the presence of a vertically oriented electrostatic field. By varying the field voltage, they could actually suspend at drop in air. (The electrostatic force holding the drop offset the effects of gravity - the drop stood still in air.) By making computations based on the size of the drop (including some factors for the effect of the air) the number of carges suspending it could be calculated. Repeating the experiment over many trials allowed them to find a "fundamental unit charge" that represented the charge on the electron. It was good enough work to win a Nobel Prize in Physics for Millikan. A link is provided to the oil-drop experiment as posted by our friends at Wikipedia, where knowledge is free.
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