To determine The constant ‘c’ in the force expression. Explanation Write the expression for potential energy of the rope U = − ∫ x i x f F ( x ) d x (I) Here, U is the potential energy, F ( x ) is the force and d x is the distance element. Write the given expression for force F = − k x − c x 2 (II) Here, x is the position, k and c are the constant. Substitute − k x − c x 2 for F in equation (I) and integrate in the limit 0 to d . Here, d is the distance to which the rope is stretched. U = − ∫ 0 d ( − k x − c x 2 ) d x = ( k x 2 2 + c x 3 3 ) 0 d = k d 2 2 + c d

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Chapter 7, Problem 28E

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The constant ‘c’ in the force expression.

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