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Can somebody explain how to calculate the restoring force in a uniform circular rod with known Young's modulus and diameter. I would need the restoring force in a specified distance from the origin with also a specified deviance.

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The force of a stress uniformly distributed over an area A is $$ F = \sigma \times A $$

Stress = Young's modulus * strain $$ \sigma = E \times \epsilon $$

Strain = displacement / Length $$ \epsilon = {u \over L} $$

so putting it together, along with the area of a circle: $$ F = E \times {u \over L} \times \pi \times d^2/4 $$

That gives positive force for a positive displacement. If you want the restoring force applied to the thing that causes the displacement, negate it.

The force is the same at any distance along the bar.

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