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A rod of mass m and length I is suspended from two massless vertical springs with a spring constants k₁&k₂. What is the Lagrangian for the system, if x₁ & x₂ be the displacements from equilibrium position of the two ends of the rod.?
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A rod of mass m and length I is suspended from two massless vertical s...
Introduction
In this system, a rod of mass m and length l is suspended by two springs with spring constants k₁ and k₂. The displacements x₁ and x₂ represent the elongations from their respective equilibrium positions at the ends of the rod. The Lagrangian formalism will help us find the equations of motion for this mechanical system.
Kinetic Energy (T)
- The rod can move vertically as the springs stretch or compress.
- The center of mass of the rod moves due to the displacements x₁ and x₂.
- The vertical displacement of the center of mass (h) can be expressed as:
h = (x₁ + x₂) / 2
- The kinetic energy (T) of the system is given by:
T = (1/2) * m * (dh/dt)²
Potential Energy (V)
- Each spring contributes to the potential energy based on its displacement from equilibrium.
- The potential energy (V) due to the springs can be calculated as:
V = (1/2) * k₁ * x₁² + (1/2) * k₂ * x₂²
Lagrangian (L)
- The Lagrangian (L) is defined as the difference between kinetic energy (T) and potential energy (V):
L = T - V
- Substituting the expressions for T and V:
L = (1/2) * m * (dh/dt)² - [(1/2) * k₁ * x₁² + (1/2) * k₂ * x₂²]
Conclusion
- This Lagrangian comprehensively describes the dynamics of the rod-spring system.
- From here, one can derive the equations of motion using the Euler-Lagrange equation, leading to insights about the stability and oscillatory behavior of the system.
In this system, a rod of mass m and length l is suspended by two springs with spring constants k₁ and k₂. The displacements x₁ and x₂ represent the elongations from their respective equilibrium positions at the ends of the rod. The Lagrangian formalism will help us find the equations of motion for this mechanical system.
Kinetic Energy (T)
- The rod can move vertically as the springs stretch or compress.
- The center of mass of the rod moves due to the displacements x₁ and x₂.
- The vertical displacement of the center of mass (h) can be expressed as:
h = (x₁ + x₂) / 2
- The kinetic energy (T) of the system is given by:
T = (1/2) * m * (dh/dt)²
Potential Energy (V)
- Each spring contributes to the potential energy based on its displacement from equilibrium.
- The potential energy (V) due to the springs can be calculated as:
V = (1/2) * k₁ * x₁² + (1/2) * k₂ * x₂²
Lagrangian (L)
- The Lagrangian (L) is defined as the difference between kinetic energy (T) and potential energy (V):
L = T - V
- Substituting the expressions for T and V:
L = (1/2) * m * (dh/dt)² - [(1/2) * k₁ * x₁² + (1/2) * k₂ * x₂²]
Conclusion
- This Lagrangian comprehensively describes the dynamics of the rod-spring system.
- From here, one can derive the equations of motion using the Euler-Lagrange equation, leading to insights about the stability and oscillatory behavior of the system.
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A rod of mass m and length I is suspended from two massless vertical springs with a spring constants k₁&k₂. What is the Lagrangian for the system, if x₁ & x₂ be the displacements from equilibrium position of the two ends of the rod.?
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A rod of mass m and length I is suspended from two massless vertical springs with a spring constants k₁&k₂. What is the Lagrangian for the system, if x₁ & x₂ be the displacements from equilibrium position of the two ends of the rod.? for GATE Physics 2025 is part of GATE Physics preparation. The Question and answers have been prepared according to the GATE Physics exam syllabus. Information about A rod of mass m and length I is suspended from two massless vertical springs with a spring constants k₁&k₂. What is the Lagrangian for the system, if x₁ & x₂ be the displacements from equilibrium position of the two ends of the rod.? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A rod of mass m and length I is suspended from two massless vertical springs with a spring constants k₁&k₂. What is the Lagrangian for the system, if x₁ & x₂ be the displacements from equilibrium position of the two ends of the rod.?.
A rod of mass m and length I is suspended from two massless vertical springs with a spring constants k₁&k₂. What is the Lagrangian for the system, if x₁ & x₂ be the displacements from equilibrium position of the two ends of the rod.? for GATE Physics 2025 is part of GATE Physics preparation. The Question and answers have been prepared according to the GATE Physics exam syllabus. Information about A rod of mass m and length I is suspended from two massless vertical springs with a spring constants k₁&k₂. What is the Lagrangian for the system, if x₁ & x₂ be the displacements from equilibrium position of the two ends of the rod.? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A rod of mass m and length I is suspended from two massless vertical springs with a spring constants k₁&k₂. What is the Lagrangian for the system, if x₁ & x₂ be the displacements from equilibrium position of the two ends of the rod.?.
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