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Two masses and 3 springs. нотот Consider the longitudinal oscillations, i.e., along the axis, of a mechanical system composed of two particles of mass m con- nected to each other and to walls on either side by springs of constant k and rest length a, as seen in the figure. The distance between the two walls is 3a. I (a) Find the lagrangian and Lagrange's equations. (b) What are the normal-mode frequencies and eigenvectors? Describe the motions. (c) Construct explicitly the modal matrix A¡¡, the normal coordinates Çk, and the diagonal form L = Σk 1 (S² – w²5²) of the lagrangian (i.e. write L as an explicit sum with two terms and replace the frequencies by their values). (d) Suppose the mass on the left is initially displaced from equilibrium a distance a to the right, the mass on the right starts at its equilibrium position, and both masses start at rest. Compute the subsequent motion.

Two masses and 3 springs. нотот Consider the longitudinal oscillations, i.e., along the axis, of a mechanical system composed of two particles of mass m con- nected to each other and to walls on either side by springs of constant k and rest length a, as seen in the figure. The distance between the two walls is 3a. I (a) Find the lagrangian and Lagrange's equations. (b) What are the normal-mode frequencies and eigenvectors? Describe the motions. (c) Construct explicitly the modal matrix A¡¡, the normal coordinates Çk, and the diagonal form L = Σk 1 (S² – w²5²) of the lagrangian (i.e. write L as an explicit sum with two terms and replace the frequencies by their values). (d) Suppose the mass on the left is initially displaced from equilibrium a distance a to the right, the mass on the right starts at its equilibrium position, and both masses start at rest. Compute the subsequent motion.

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**1. Two Masses and Three Springs** Consider the longitudinal oscillations, i.e., along the axis, of a mechanical system composed of two particles of mass \( m \) connected to each other and to walls on either side by springs of constant \( k \) and rest length \( a \), as seen in the figure. The distance between the two walls is \( 3a \). ### Diagram Explanation: The diagram shows two masses connected by three springs. The two masses are between two walls, with one spring attached to each wall and one spring connecting the two masses. The springs are drawn in a horizontal line connecting a series of loops representing the masses. ### Questions: (a) **Find the Lagrangian and Lagrange’s Equations.** (b) **What are the Normal-Mode Frequencies and Eigenvectors?** Describe the motions. (c) **Construct Explicitly the Modal Matrix \( A_{ij} \),** the normal coordinates \(\zeta_k\), and the diagonal form \( L = \sum_k \frac{1}{2} \left(\dot{\zeta_k}^2 - \omega_k^2 \zeta_k^2 \right) \) of the Lagrangian (i.e., write \( L \) as an explicit sum with two terms and replace the frequencies by their values). (d) **Suppose the Mass on the Left is Initially Displaced** from equilibrium a distance \( a \) to the right, the mass on the right starts at its equilibrium position, and both masses start at rest. Compute the subsequent motion.
Transcribed Image Text:**1. Two Masses and Three Springs** Consider the longitudinal oscillations, i.e., along the axis, of a mechanical system composed of two particles of mass \( m \) connected to each other and to walls on either side by springs of constant \( k \) and rest length \( a \), as seen in the figure. The distance between the two walls is \( 3a \). ### Diagram Explanation: The diagram shows two masses connected by three springs. The two masses are between two walls, with one spring attached to each wall and one spring connecting the two masses. The springs are drawn in a horizontal line connecting a series of loops representing the masses. ### Questions: (a) **Find the Lagrangian and Lagrange’s Equations.** (b) **What are the Normal-Mode Frequencies and Eigenvectors?** Describe the motions. (c) **Construct Explicitly the Modal Matrix \( A_{ij} \),** the normal coordinates \(\zeta_k\), and the diagonal form \( L = \sum_k \frac{1}{2} \left(\dot{\zeta_k}^2 - \omega_k^2 \zeta_k^2 \right) \) of the Lagrangian (i.e., write \( L \) as an explicit sum with two terms and replace the frequencies by their values). (d) **Suppose the Mass on the Left is Initially Displaced** from equilibrium a distance \( a \) to the right, the mass on the right starts at its equilibrium position, and both masses start at rest. Compute the subsequent motion.
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