4.1: Example linear system

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Suppose that we have three objects on a balanced beam. Also suppose we know that one has a mass of 2 kg, and we want to find the two unknown masses. Experimentation with a (assume weightless) meter stick produces these two balances. (diagram not to scale)

Figure \(\PageIndex{1}\): Image showing two balanced beams, each with three weights. In the top beam is unknown weight A is a distance of 40 to the left of the fulcrum, unknown weight B is a distance of 15 to the left of the fulcrum and a weight of 2 is 50 to the right of the fulcrum. In the bottom beam is the same unknown weights. Weight A is now a distance of 50 to the right of the fulcrum, weight B is a distance of 25 to the left of the fulcrum and the weight of 2 is a distance of 25 to the right of the fulcrum.

For the masses to balance we must have the sum of the moments on the left equal to the sum of the moments on the right, where the moment of an object is its mass times its distance from the balance point. That gives a system of two equations:

\[ 40A + 15B = 50 \times 2 \nonumber \]

\[ 25B = 25 \times 2 + 50A \nonumber \]

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Find a solution for the above systems of equations and place your solution in the following cell. Make sure you delete the instructional text in the cell first.

# Put your answer to the above question here

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Using Python as a calculator, verify that the solution you have found is correct.

# Put your answer to the above question here

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Now lets consider a system where we have three unknown masses instead of two. Experimentation with a meter stick produces the two balanced states shown below (diagram not to scale). Write the equations for this system.

Figure \(\PageIndex{2}\): Image showing two balanced beams, each with four weights. In the top beam is unknown weight A which is a distance of 35 to the left of the fulcrum, unknown weight B is a distance of 21 to the left of the fulcrum, unknown weight C is a distance of 11 to the right of the fulcrum and a weight of 2 is 50 to the right of the fulcrum. In the bottom beam is the same unknown weights. Weight A is now a distance of 10 to the right of the fulcrum, weight B is a distance of 24 to the right of the fulcrum, weight C is a distance of 25 to the left of the fulcrum and the weight of 2 is still at a distance of 50 to the right of the fulcrum.

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Find a solution to the second set of equations and report the mass for objects A, B and C.

# Put your answer to the above question here

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Using Python as a calculator, verify that the solution you have found is correct.

# Put your answer to the above question here