Hooke’s Law states that the force required to displace a spring \(x\) meters from equilibrium is given by \(F(x) = kx\), where \(k\) is a constant of proportionality. If \(60 J\) of work is required to stretch a spring \(3 m\) from its equilibrium position, find the amount of force required to stretch the spring \(2 m\) from its equilibrium position.
The force required is \(\answer {\frac {240}{9}} N\).
Note that the work, not the force, is given! Find the work in terms of \(k\) first. The
integral is:
\[ W = \int _{\answer {0}}^{\answer {3}} \answer { k x } \d x \]
Using \(W=60\) and evaluating the integral gives:
\[ 60 = \answer {\frac {9}{2}} k \]