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Two particles move in one dimension at the junction of three springs as shown in the figure. The springs all have un-stretched lengths equal to a and the force constants and masses are shown. Find the eigenfrequencies and normal modes of the system.

Two particles move in one dimension at the junction of three springs as shown in the figure. The springs all have un-stretched lengths equal to a and the force constants and masses are shown. Find the eigenfrequencies and normal modes of the system.

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Two particles move in one dimension at the junction of three springs as
shown in the figure. The springs all have un-stretched lengths equal to a and
the force constants and masses are shown. Find the eigenfrequencies and
normal modes of the system.

**Diagram Description:** The diagram illustrates a mechanical system with three springs and two masses positioned between two fixed walls. Here's a detailed explanation: - **Left Wall:** The system begins with a fixed wall on the left. - **First Spring (k):** Connected to the left wall, this spring has a spring constant denoted by \( k \). - **First Mass (m):** A mass labeled \( m \) is connected to the first spring. - **Second Spring (3k):** Attached to the first mass, this spring has a higher spring constant, labeled \( 3k \). - **Second Mass (m):** Another mass labeled \( m \) follows the second spring. - **Third Spring (k):** Connected to the second mass, the spring is identical to the first, with a constant \( k \). - **Right Wall:** The system concludes with a fixed wall on the right side. Each spring is labeled with its spring constant, and the intervening masses are denoted by \( m \). The system may be used to study oscillations, resonant frequencies, or mechanical vibrations.
Transcribed Image Text:**Diagram Description:** The diagram illustrates a mechanical system with three springs and two masses positioned between two fixed walls. Here's a detailed explanation: - **Left Wall:** The system begins with a fixed wall on the left. - **First Spring (k):** Connected to the left wall, this spring has a spring constant denoted by \( k \). - **First Mass (m):** A mass labeled \( m \) is connected to the first spring. - **Second Spring (3k):** Attached to the first mass, this spring has a higher spring constant, labeled \( 3k \). - **Second Mass (m):** Another mass labeled \( m \) follows the second spring. - **Third Spring (k):** Connected to the second mass, the spring is identical to the first, with a constant \( k \). - **Right Wall:** The system concludes with a fixed wall on the right side. Each spring is labeled with its spring constant, and the intervening masses are denoted by \( m \). The system may be used to study oscillations, resonant frequencies, or mechanical vibrations.
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