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  • Perplex

    Modelling (Lesson 4/7)

    Cubic Models

    1 / 2

    Discussion

    Amanda fixes magnets at positions ​x=−5cm​ and ​x=5cm. Then, she measures the force ​F​ applied to a third magnet at different positions. She plots her data on the graph below (negative positions and forces point toward Magnet ​1, positive positions and forces point toward Magnet ​2​).

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    Describe a curve that could be used to model force as a function of position.

    Solution:

    A good choice is a smooth, monotonically increasing S-shaped curve that goes through the points and has these visual features:

    • At the left (around ​x≈–3​ cm) it sits down near ​F≈–8​ units and is quite steep.

    • As you move toward the centre, the slope decreases, so by ​x≈–1​ cm it’s already flattening off.

    • Between ​x≈–1​ and ​x≈1​ cm the curve is almost flat, passing through ​(0,0)​ with a gentle incline.

    • Beyond ​x≈1​ cm the slope picks up again, becoming quite steep by ​x≈3​ cm where ​F≈8​ units.

    • The result is one continuous “S” shape, negative and getting steeper to the left, inflecting near ​x≈–1, nearly straight through the middle, then again near ​x≈1​ to become steeper up to the right.

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