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A measurement doesn't have to be direct to be a measurement. In other words, the aspect ratio of a desk is still obtained by measurement and thus is a measured
by glastra 11y ago
A measurement doesn't have to be direct to be a measurement. In other words, the aspect ratio of a desk is still obtained by measurement and thus is a measured quantity, subject to error and so on.
Of course, both the quantity and the associated error come from previous measurements.
- danharaj 11y agoi'm just wondering if there's an example of a direct measurement of a ratio, rather than one computed from the measurement of dimensional quantities.
- JoeAltmaier 11y agoWell, you've got to measure some quantity. How about an angle? Measure any angle in a right triangle and you know something about the ratio of the sides.
- danharaj 11y agoAt first i thought "aha! yes, how obvious.", but then i thought about how one might measure an angle. i would use a protractor. but how is a protractor calibrated? surely the angles determine a relationship between commensurable quantities, and a protractor is calibrated by performing measurements on lengths held at angles and measuring the observed relationship. can one measure an angle directly without performing an experiment on commensurate lengths? a protractor only works because a measurement of an angle can be transported and remain the same (to an approximation). that's a nontrivial fact that i took for granted, but that's really what a protractor does: it is the stored result of a bunch of experiments to determine angles. those experiments must have been done using observations of dimensional quantities. i would like an example of an experiment that directly measures a dimensionless quantity without inferring it from direct measurements of dimensional quantities. it's just a curiosity.
- kazinator 11y agoHowever, we can use angles in such a way that we never have any estimate that correlates with the size of the rectangle whose aspect ratio we are measuring. Suppose some rectangle is visible at a distance. We can use parallax angles from our vantage point to estimate its aspect ratio. We have no idea how long or wide that rectangle is because we don't know how far away it is. (Indeed, we don't even know whether it is a rectangle, or whether it is some quadrilateral that appears as a rectangle under perspective.)
- danharaj 11y agoI think you're saying something a little too strong. I think such a measurement does provide some information about the dimensions of the quadrilateral being measured. I think it would be possible with a bounded number of observations to measure the lengths purely by measuring the angles. You would have to shift your perspective each time, but it shouldn't take too many. So, each such measurement must contribute a non-zero amount of information about the dimensions. Either that, or i'm misinterpreting your first sentence.
- gaze 11y agoImagine a balance with an arm that can be somehow folded in half again and again (think a snap bracelet. Those are rigid in one direction and stiff bent in the other). Fold the beam in half and mark the center. Use this point as the fulcrum to balance two objects. Pick the side that comes up. Divide this side in half and use this new half way point as the new fulcrum. starting with 0. if the beam reads on the heavy side, write down a 1. If reads on the light side, write a 0. Eventually you'll have 0.101000101110101 or some such. This is the ratio of the weights of the two objects in base 2. You can binary search the ratio of the weights.
- gaze 11y agoHere's another. Imagine you want the ratio of two clock sources. Put them both through frequency dividers, with one divider output put through a phase shifter. compare the resultant divided frequencies by xoring the signals together. The ratio of the frequencies is given by the ratio of the dividers when you can find a phase shift such that the xor output doesn't change. The time you wait to see if the xor output doesn't change represents the certainty you have in your ratio. This is called the time frequency uncertainty relation, also known as the Heisenberg uncertainty principle.
- gaze 11y agoOkay we have mass and time nailed down. I think the subdivision method should work for length as well. Let's try voltage. Consider two voltage sources that you want to measure the ratio of. Place one source on one arm of a resistor, and the other through a pulse width modulator, (ideal) low pass filter with a cutoff frequency much lower than the pulse width modulator frequency, and then attach this to the other lead of the resistor. Turn on the voltage sources and adjust the pulse width modulator until the resistor doesn't heat up. You could place it in a cup of water and monitor the temperature. Remember we're free to push up the current in this thought experiment
- gaze 11y agoResistance should be easy as well. I think the wikipedia page for a Wheatstone bridge explains this best. If this sort of thing interests you, I'd encourage you to look at the quantum metrology triangle. Frequency, resistance, and current are all currently related to a josephson voltage standard, the quantum hall resistance, and the current generated by a single electron transistor. This allows us to standardize all these back to a frequency. Length can be related to frequency by wavelength of light. Everything has been related back to frequency. Basically the only thing not referred back to something absolute is mass, which I think is still referred back to a mass standard, which is a physical thing stored in a vault. Okay well more than you ever wanted to know.
- danharaj 11y agoAll your posts are great! These are all really illuminating experiments.