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Consider the mass attached to four identical springs, as
Chapter 5, Problem 5.19(choose chapter or problem)
Consider the mass attached to four identical springs, as shown in Figure 5.7(b). Each spring has force constant k and unstretched length \(l_{\mathrm{o}}\), and the length of each spring when the mass is at its equilibrium at the origin is a (not necessarily the same as \(l_{\mathrm{o}}\)). When the mass is displaced a small distance to the point (x, y), show that its potential energy has the form \(\frac{1}{2} k^{\prime} r^{2}\) appropriate to an isotropic harmonic oscillator. What is the constant \(k^{\prime}\) in terms of k? Give an expression for the corresponding force.
Questions & Answers
QUESTION:
Consider the mass attached to four identical springs, as shown in Figure 5.7(b). Each spring has force constant k and unstretched length \(l_{\mathrm{o}}\), and the length of each spring when the mass is at its equilibrium at the origin is a (not necessarily the same as \(l_{\mathrm{o}}\)). When the mass is displaced a small distance to the point (x, y), show that its potential energy has the form \(\frac{1}{2} k^{\prime} r^{2}\) appropriate to an isotropic harmonic oscillator. What is the constant \(k^{\prime}\) in terms of k? Give an expression for the corresponding force.
ANSWER:Step 1 of 3
The two springs are attached along the x-axis and two along the y-axis.
The potential energy of the two springs along x-axis is,
\(U_{x}=k\left[x^{2}+\left(1-\frac{I_{0}}{a}\right) y^{2}\right]\)
The potential energy of the two springs along y-axis is,
\(U_{y}=k\left[y^{2}+\left(1-\frac{I_{0}}{a}\right) x^{2}\right]\)