Dynamic Calculus

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Hooke's Law

A spring has its left end fixed and its right end (point \(P\)) free to move along the \(x\)-axis. When the spring is at its 'natural length' (ie neither compressed nor stretched), the point \(P\) is at the origin. According to 'Hooke's Law' for elastic springs, the force \(F(x)\) that must be exerted on the spring to hold its right end at the point \(P\) is proportional to the displacement \(x\) of the right end from its rest position. This can be expressed mathematically by the equation \(F(x) = kx\), where \(k\) is a constant known as the 'spring constant'.

 

 

1. The natural length of this particular spring is 5 cm and a force of 1.25 Newtons (N) is required to compress it to a length of 2.5 cm. By substituting into the formula \(F(x) = kx\), confirm that the spring constant \(k = 0.5{\rm{ N/cm}}\).
   
2. Move the point \(P\) so that the total length of the spring is 15 cm. The work done in stretching the spring is given by the area of the shaded region under the curve. Without using calculus, calculate the work done.
   
3. How would we calculate the work done using calculus? Confirm your answer to question (2) using calculus then check your answer by clicking the 'checkbox'.

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