Here's the reasoning: Newton's third law tells us that every force has an equal but opposite reaction force. So besides my weight pushing down on the scale from above, there's also an equal force pushing up on the scale from below, so the spring or pressure-sensor-thingy inside the scale is being squeezed from two directions by double the force! That's tipping the scales, and therefore everybody is actually just half as heavy as it says on the readout.
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3$\begingroup$ The equal but opposite force paired with the downwards force of your weight is the upwards force exerted by the spring on you. In other words, the reaction force does not act on the scale itself. $\endgroup$– RationsCommented May 14, 2015 at 22:10
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4$\begingroup$ Apart from the whole misconception of the action-reaction thing, calibration of the scale (when it displaces x mm, we call that y kg because the standard kg mass causes a displacement of x/y) means that it doesn't matter. We define your weight as "what the calibrated scale says". $\endgroup$– FlorisCommented May 14, 2015 at 23:35
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$\begingroup$ More on forces and factors of two: physics.stackexchange.com/q/41291/2451 and links therein. $\endgroup$– Qmechanic ♦Commented May 14, 2015 at 23:58
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1 Answer
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You've inserted the scale between yourself and the planet Earth. The scale's spring is compressed by the force exerted on you by Earth's gravitation.
The compressed spring tries to uncompress itself and pushes back on you with linear restoring force. Linear restoring force is exactly equal to the force exerted on you by the Earth's gravitation.
By recording its linear restoring force on a scale, the spring measures the force exerted on you by Earth's gravitation. The distance the spring is compressed will depend on its stiffness, which is measured by Hooke's Law, which says that distance compressed is directly proportional to the stiffness of the spring.
Force applied to the spring = + spring stiffness constant * distance of compression
Force applied back to you = - spring stiffness constant * distance of compression
The spring stiffness constant is a dimensionless number calculated according to Hooke's Law.
The net result is that the spring measures BOTH the force you apply to it, and the force it applies to you. But as the measure is exactly the same and measures the same quantity, there is no need to halve it.
Here is a graphic explanation, which also discusses the potential energy stored in a compressed spring: http://theory.uwinnipeg.ca/physics/work/node5.html
