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Newtonian Physics: Balance Beam Problem Structuring

Need to obtain a working mathematical model of the balance beam problem, preferably with a cart, not a ball. i.e. Newtons laws leading through to equations of motion, aswell as transfer functions for use in matlab, any help will be much appreciated, thank you...

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I am attaching to this note one file called Cart_pole_dynamic.mdl that is a Simulink ...

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