Problem 4: The time-dependent equations of motion for the mass-spring system illustrated below are: \begin{array}{l} \ddot{x}_{1}+\frac{\left(k_{1}+k_{2}\right)}{m_{1}} x_{1}-\frac{k_{2}}{m_{1}} x_{2}=0 \\ \ddot{x}_{2}-\frac{k_{2}}{m_{2}} x_{1}+\frac{\left(k_{2}+k_{3}\right)}{m_{2}} x_{2}-\frac{k_{3}}{m_{2}} x_{3}=0 \\ \ddot{x}_{3}-\frac{k_{3}}{m_{3}} x_{2}+\frac{\left(k_{3}+k_{4}\right)}{m_{3}} x_{3}=0 \end{array} Assuming solutions

of the form x; = X; sin(@t), transform the above system into an eigenvalue problem; then, find the three natural frequencies (w1, w2, and w3) and the three eigenvectors of the system using the values \begin{array}{l} \mathbf{k}_{\mathbf{1}}=\mathbf{k}_{\mathbf{4}}=15 \mathbf{N} \mathbf{m} \\ \mathbf{k}_{\mathbf{2}}=\mathbf{k}_{\mathbf{3}}=\mathbf{3 5} \mathbf{N}_{\mathbf{m}} \\ \mathbf{m}_{\mathbf{1}}=\mathbf{m}_{\mathbf{2}}=\mathbf{m}_{\mathbf{3}}=\mathbf{m}_{\mathbf{4}}=\mathbf{2} \mathbf{k g} \end{array} You may use the "eig" function in MATLAB. Explain what the eigenvectors tell you about the directions the three masses will move when oscillating at the three different natural frequencies.