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‘How can we describe mathematically the motion of a weighted cantilever when it is made to oscillate?’

Last Thursday the DP1 Maths HL & SL classes joined forces ready to use some serious heavy mathematics to describe a real life situation in Mr.Heyworth’s Physics lab.

The question was ‘How can we describe mathematically the motion of a weighted cantilever when it is made to oscillate?’ You may be asking “What’s a cantilever? ..well just think of twanging your ruler in school (be honest parents, you did it as well)  but on a much grander scale. The students were faced with a one metre long plywood cantilever with adjustable masses attached to its tip (0.5 kg to 3 kg). Part One With the aid of a distance sensor, a data logger and graphing software the students had to determine what the simple relationship was between the period T (time for one oscillation) and the mass m. This gave an idea to students about mathematical modelling, graphing the results and using data loggers and sensors.

Part Two was much more tricky as the students had to model the actual movement of the cantilever tip i.e. not only had they to use a suitably transformed  trigonometric function  but also contend with the fact the amplitude was decreasing with time (fading oscillations). The DP1s had to work fast and it was great to see how the majority of the students were actively striving for the mathematical model.  An example of a model equations for the one kg mass on the tip was

y = 4.9 x 2-t/4.5 x cos (12.87(t – 0.1)) + 0.34 where y is the distance of the tip from the sensor in centimetres.

This was an opportunity for students to understand the strong connection between real life situations with abstract looking mathematical equations. The students not only saw how to apply all these apparent abstract mathematical concepts but also gained further experience in using data loggers and graphing software.  Hopefully most appreciated how useful good mathematical models can be.

Typical cantilever with weights at the end

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