Topic 8 - Measurement
From KstructIB
[edit] 8.1 Standards of measurement
[edit] 8.1.1
The Sine and Cosine rules are both in the data book.
The sine rule is A/sin a = B/sin b = C/sin c. The lower case letters are lengths of the sides of a triangle, and the upper case are the angles opposite these sides. This should allow you to find unknown sizes and angles given enough information about the triangle.
The Cosine rule is a2 = b2 + c2 - 2bc cos A. The values represented by a, b, c and A are as above. Obviously, the triangle can be relabelled as required (or the letters can be switched around). The easiest way to remember this is as an extended version of Pythagorus' theorem, with a correction for when the angle is not 90 degrees. Then again, it's in the data book, so I guess we don't have to remember it.
[edit] 8.1.2
Being able to use logarithms is something to learn in maths really, but here's a brief recap.
log means log in base 10. ln or loge = log in base e (e is about 2.71...).
Taking the log of both sides of an equation is the equivalent of reversing the effect of putting both sides to the power of the base (i.e. log (109) = 9). This needs to be done with some of the decay equations to see their exponential nature. Anyway, if you don't know how to handle this any decent maths book should explain it.
[edit] 8.1.3
The tricky thing here is identifying what the functions look like (often it's possible to graph it on a graphic calculator, but sometimes there are two variables, making things rather tricky). The amplitude of the resulting function will be the difference between the maximum and minimum values, which can be found by knowing where the sin and cos curves peak (and this can always be graphed).
Usually, the easiest way to work anything else out is to graph each separate segment, then apply the transformations mentally (i.e. multiply the two curves together or whatever), and then think about the resulting graph. This is tricky (and I've never seen a question on it either).
For graphs where a linear transformation involves adding constants to a function, the technique is rather simple. For instance, in a graph of f(x) = [(x^2) - 3] + 3, we know to shift the graph of a common parabola y = x^2 to the "right" of the Cartesian plane by 3 units and "up" 3 units so that the origin is at (3,3).
However, when we have graphs in which there more advanced functions, it becomes somewhat trickier. The trick is to find some fundamental points such as x and y intercepts and fundamental loci such as vertical and horizontal asymptotes. For instance, for the graph of f(x) = y = (2x+3)/(x-1), we must find horizontal asymptote, or lim y(x approaches positive infinity) = lim (2x / x)(x approaches positive infinity) = 2. The vertical asymptote, of course, is the solution of x-1 = 0, which is 1. Then, we test points above and below the axes to determine where our curves are relative to their asymptotes.
Also, for another example 0 = (y^2) - x + y - 6, it often helps to take the x and y intercepts (by setting y to 0 and x to 0 resp.) and simply knowing the general nature of such a graph. Then you can test points and perhaps approximate the graph. Moreover, more approximation can be made with calculus by using second derivatives and such to find out points of inflection, maximums, and minimums, which are also useful for any "normal" graph.
[edit] 8.2 Graphical techniques
[edit] 8.2.1
Using log scales is, once again, rather tricky to describe without proper diagrams (though I seriously doubt there would be any questions on it). Does anyone have a neat explanation ? Personally, I find it easier to do the transformations into a log scale (which is what the IB suggests you shouldn't do), but that's just me.
[edit] 8.2.2
If a scale has been logged, then the unit must be put to the power of the base to compensate when used in any expression derived from the graph.
[edit] 8.2.3
This is similar to the SL section, except now we can use powers and reciprocals on log graphs as well as with 'normal' graphs.
[edit] 8.3 Uncertainties and errors
[edit] 8.3.1
Uncertainties can be expressed in two different ways. Absolute uncertainty refers to the actual uncertainty in a quantity measured. For example, with an average of 1.4538 ± 0.0001 g for three trial measurements, the absolute uncertainty is ± 0.0001 g.
Relative uncertainties express the uncertainty as a specific percentage of the quantity in question. For example, an object weighing 1.4538 ± 0.0001 g, the relative uncertainty is 0.0001 g/1.4538 g which equals ± 6.9 x 10-5. This relative uncertainty is expressed as 6.9 x 10^-5 multiplied by 100 to get the precentage of ± 0.0069 %. Relative uncertainty is often used to grasp a qualitative idea of the precision and accuracy of your measuring device.
[edit] 8.3.2
Carrying uncertainties through a series of calculations is fairly simple. Just remember to calculate the uncertainty at the end of each calculation, then carry this value and uncertainty on to the next. Follow these rules based on the type of calculations you are carrying out:
Addition and Subtraction - Add absolute uncertainties
Multiplication and Division - Add relative uncertainties
[edit] 8.3.3
The following problems can be caused by digital equipment :
Quantization : This is caused by converting continuous analogue data into individual digital numbers.
Sampling frequency : The digital systems can only 'grab' a piece of data every x sec (let's say 0.01). If the data is changing significantly within this amount of time, then the sampled results will be almost random compared to the actual signal.
