I have a rigid bar of non-uniform density.
I know the total weight of the bar.
I would like to calculate the mass of individual sections of the bar, by suspending one point on a fulcrum, and using scales to measure the weight at another point (and of course repeating at different points)
I know the force (ie. weight) = mass * distance from fulcrum
What I can't figure out, is what formulae can I use to get the mass of sections A, B and C in the diagram below? (the blue and green triangles are the fulcrums and scales)
Is there a general formula I could use for a bar with many more than three sections?
I'm sure this shouldn't be as difficult as I'm finding it, and any help would be very much appreciated.
It discusses how to determine what weights are equal; I'm fairly sure you can set an alterable value as your known weight, then apply the formula and see how it all shakes out.
Thanks, but that's the same as what I already had: f = m * d
However, I've think I've just about got it figured out, with the help of Wolfram Alpha.
btw: Wolfram Alpha is great for klikers (I use it all the time) - you can figure out a formula for something in a way that's logical and easy to understand, and then put it into WA and it will give you back something ten times simpler and more efficient
It sounds like you've solved your problem, but I'd like to contribute a general solution as this is a good example of a physics question.
Let's say your bar is completely generally non-uniform in density. You'd need some function p(x) which defines its linear density in units of [mass / length]. This could be as simple as
p(x) = 0.3 between 0 and 0.1;
p(x) = 0.4 between 0.1 and 0.5;
etc.
or it could be much more general, say a bar with linearly increasing density
p(x) = 0.1 x
or even more complicated.
To determine the mass between points a and b, you would integrate p(x) dx from a to b. By multiplying p(x) by the differential bit of length dx, you're finding the mass in that little bit. By integrating over a range, you're finding the total mass in that range.
Now, when you say "force (ie. weight) = mass * distance" what you really mean is torque, which describes the sort of "rotational force" about that fulcrum. If the torques are balanced on each side of the fulcrum, the bar won't rotate. Furthermore, the weight is technically the mass times the acceleration due to gravity, which you can ignore in this case since the acceleration due to gravity is the same on both sides of your bar.
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Edited by Rick Shaw