. 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
. 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
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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 Use
Below 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b65ef36-cf51-4f81-80a6-74e205c9e9b1%2Fde6dfd52-211a-43c4-83a9-931182fe26d2%2Fnf5ozhj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Transcription for Educational Use
Below 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.
Transcribed Image Text: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.
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