Obtain differential equations of motion using Lagrange equations. k 24k m k m
Obtain differential equations of motion using Lagrange equations. k 24k m k m
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![**Title: Differential Equations of Motion Using Lagrange Equations**
**Objective:**
To derive the differential equations of motion for the given mechanical system using Lagrange equations.
**Diagram Explanation:**
The provided diagram depicts a mechanical system consisting of two masses \( m \) each, connected by springs. The setup is as follows:
1. There are two masses, designated as \( m \), which are part of the system.
2. The first mass \( m \) is positioned on the left and is connected to a wall via a spring with a spring constant \( k \).
3. The second mass \( m \) is present above the first mass and is connected to a vertical wall on the right by a spring with a spring constant of \( 24k \).
4. Additionally, there’s another spring with the spring constant \( k \) connecting the two masses horizontally.
5. Both masses appear to have wheels, indicating they can slide horizontally without friction.
**Objective:**
To find the differential equations of motion for this system by applying Lagrange equations, which is a fundamental approach in classical mechanics.
**Procedure:**
To obtain the differential equations of motion using the Lagrange method, follow these steps:
1. **Determine the Kinetic Energy (T):**
- Calculate the kinetic energy contributions from both masses.
2. **Determine the Potential Energy (V):**
- Compute the potential energy stored in each spring.
3. **Construct the Lagrangian (L):**
- The Lagrangian \( L \) is given by the difference between the kinetic and potential energy: \( L = T - V \).
4. **Apply Lagrange's Equations:**
- Use the Lagrange equation to derive the equations of motion:
\[
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0
\]
where \( q_i \) represents the generalized coordinates, and \( \dot{q}_i \) their corresponding velocities.
The steps outlined above, when applied correctly, will yield the set of differential equations describing the motion of the masses in the system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe25a2d5b-a922-4cc2-9728-89f28b4232be%2F25500baa-eb6c-41f5-9944-cb8e7d5bcf0c%2F154ndqj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Differential Equations of Motion Using Lagrange Equations**
**Objective:**
To derive the differential equations of motion for the given mechanical system using Lagrange equations.
**Diagram Explanation:**
The provided diagram depicts a mechanical system consisting of two masses \( m \) each, connected by springs. The setup is as follows:
1. There are two masses, designated as \( m \), which are part of the system.
2. The first mass \( m \) is positioned on the left and is connected to a wall via a spring with a spring constant \( k \).
3. The second mass \( m \) is present above the first mass and is connected to a vertical wall on the right by a spring with a spring constant of \( 24k \).
4. Additionally, there’s another spring with the spring constant \( k \) connecting the two masses horizontally.
5. Both masses appear to have wheels, indicating they can slide horizontally without friction.
**Objective:**
To find the differential equations of motion for this system by applying Lagrange equations, which is a fundamental approach in classical mechanics.
**Procedure:**
To obtain the differential equations of motion using the Lagrange method, follow these steps:
1. **Determine the Kinetic Energy (T):**
- Calculate the kinetic energy contributions from both masses.
2. **Determine the Potential Energy (V):**
- Compute the potential energy stored in each spring.
3. **Construct the Lagrangian (L):**
- The Lagrangian \( L \) is given by the difference between the kinetic and potential energy: \( L = T - V \).
4. **Apply Lagrange's Equations:**
- Use the Lagrange equation to derive the equations of motion:
\[
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0
\]
where \( q_i \) represents the generalized coordinates, and \( \dot{q}_i \) their corresponding velocities.
The steps outlined above, when applied correctly, will yield the set of differential equations describing the motion of the masses in the system.
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