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Identify the system
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Determine all of the forces that are external
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Draw a free body diagram
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Sum the magnitude of all of the forces in every direction that is important in the problem, using the free body diagram.
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Insert negative signs to match your choice of axis. Eg, if the component of a force is pointing to the left, and right is positive that component is negative.
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Set the sum of the forces, for each direction, equal to the mass of the system times the acceleration in that direction.
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Rearrange equations to solve for the required quantity. This can involve some ugly algebra, including solving simultaneous equations.
Lecture video highlighting general steps to solving mechanics problems;
Other Algorithms
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Problem Solving strategies chapter in OpenStax;
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A pdf to a 6 step-by-step approach.
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The University of Wisconsin here provides a general algorithm for force problems and links multiple different problem examples as well.