My system is a pendulum attached to moving horizontal mass m_1 and the pendulum m_2 that is shifted by X_o from origin. I have the lagrangian of my system what would be my equations of motions in terms of small angle approximation and what’s is their frequency? ### Transcription for Educational UseBelow is the transcription of a handwritten page containing physics equations related to mechanics, possibly dealing with motion and forces. Each equation is analyzed for educational purposes.---**Equations Transcribed:**1. **Lagrangian \( L \):** \[ L = \frac{\dot{x}^2}{2} \left( \frac{m_1 + m_2 + m_2 \cos \theta}{2} \right) \]2. **Potential Energy \( V \):** \[ V = -mg \, l \, \cos \theta \]3. **Total Lagrangian \( L \):** \[ L = T - V \]4. **Kinetic Energy \( T \):** \[ T = \frac{\dot{x}^2}{2} \left( m_1 + m_2 \right) + m_2 \, \dot{x} \, l \, \cos \theta \]---### Explanation of Variables:- \( L \): Lagrangian, a function that summarizes the dynamics of the system.- \( T \): Kinetic energy of the system.- \( V \): Potential energy of the system.- \( \dot{x} \): Derivative of \( x \) with respect to time, representing velocity.- \( m_1, m_2 \): Masses of the objects involved.- \( g \): Acceleration due to gravity.- \( l \): Length, possibly representing a pendulum or similar system component.- \( \theta \): Angle variable, typically used in problems involving pendulum motion or inclined planes.- \( \cos \theta \): Cosine of the angle theta, used for components of motion or forces along one axis.This example illustrates the formulation of a Lagrangian for a simple mechanical system, incorporating both kinetic and potential energy to analyze dynamics. It’s particularly useful in classical mechanics to describe motion. The image is a handwritten diagram with annotations on a lined notebook page. It appears to illustrate a physical system involving two masses. Here is a transcription and description of the diagram:### Transcription:- **Title**: Diagram Total Physical - **Labels and Notations**: - A horizontal arrow labeled \( X_0 \) points to the right with the notation "shifted distance from origin." - A vertical arrow pointing downwards from a rectangle labeled \( M_1 \). - A diagonal line starting from the bottom of \( M_1 \) going to the left, labeled \( l \), leads to a point labeled \( M_2 \). - An angle, marked \(\theta\), is indicated between the horizontal line and the diagonal line.### Diagram Description:The system depicted is a simple mechanical setup:- **\( M_1 \)**: This is a rectangular block or mass situated along the horizontal axis, marked by \( X_0 \).- **\( M_2 \)**: This is another mass located below and to the left of \( M_1 \). It is connected to \( M_1 \) by a line of length \( l \), which could represent a string or rod.- **\( X_0 \)**: Represents a baseline horizontal measurement indicating the initial position or distance from a defined origin.- **\(\theta\)**: The angle formed between the horizontal line (\( X_0 \)) and the line connecting \( M_1 \) and \( M_2 \).The diagram suggests the study of motion or forces in a physical system, possibly focusing on equilibrium or tension forces in a pendulum-like setup.

In this question we have to derive the Lagrangian equation of motion with respect to the possible…

Answered: . 2 xo (M₁ + m₂) + 2 Joizk • T = Xo V =…

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. 2 xo (M₁ + m₂) + 2 Joizk • T = Xo V = -mg I cos Ⓒ L= T-V ☺ ☺nic Mit M₂ + 1m₂ = 2 • 2 L = X₂ (M₁tM₂) 2