Consider a spring of stiffness \(k\) stretched by a distance \(x_{0}\) from its free length. One end of the spring is fixed at point \(O\) and the other end is connected to a roller as shown in Fig. 1.90. The

roller is constrained to move in the horizontal direction with no friction. Find the force \((F)\) displacement \((x)\) relationship of the spring when the roller is moved by a horizontal distance \(x\) to position \(B\). Discuss the resulting force-displacement relation and identify the stiffness constant \(\widetilde{k}\) along the direction of \(x\).

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0 * 00000000000 AO B FIGURE 1.90 One end of spring with lateral movement.