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An object O of (mass m, Radius R, and moment of inertia I)that rolls without slipping straight down is an inclined plane which is at an angle a from the horizontal [Since the origin is where the start to move] then the Lagrangian of the object is?
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An object O of (mass m, Radius R, and moment of inertia I)that rolls w...
Introduction
To derive the Lagrangian for an object rolling down an inclined plane, we need to consider both its kinetic and potential energies.
Kinetic Energy (T)
- The total kinetic energy of the object consists of translational and rotational components.
- Translational Kinetic Energy:
- Given by T_trans = (1/2) * m * v^2, where v is the linear velocity of the center of mass.
- Rotational Kinetic Energy:
- Given by T_rot = (1/2) * I * ω^2, where ω is the angular velocity.
- For an object rolling without slipping, the relationship between linear and angular velocities is:
- v = R * ω
- Substituting ω in terms of v:
- T_rot = (1/2) * I * (v/R)^2
- Thus, the total kinetic energy is:
- T = T_trans + T_rot = (1/2) * m * v^2 + (1/2) * I * (v/R)^2
Potential Energy (V)
- The potential energy of the object when it is at height h is given by:
- V = m * g * h
- The height h can be expressed in terms of the distance down the incline (s):
- h = s * sin(a)
Lagrangian (L)
- The Lagrangian is defined as:
- L = T - V
- Substituting T and V:
- L = [(1/2) * m * v^2 + (1/2) * I * (v/R)^2] - (m * g * s * sin(a))
Conclusion
The Lagrangian incorporates the dynamics of both translational and rotational motion of the object rolling down the incline, providing a comprehensive framework for analyzing its motion.
To derive the Lagrangian for an object rolling down an inclined plane, we need to consider both its kinetic and potential energies.
Kinetic Energy (T)
- The total kinetic energy of the object consists of translational and rotational components.
- Translational Kinetic Energy:
- Given by T_trans = (1/2) * m * v^2, where v is the linear velocity of the center of mass.
- Rotational Kinetic Energy:
- Given by T_rot = (1/2) * I * ω^2, where ω is the angular velocity.
- For an object rolling without slipping, the relationship between linear and angular velocities is:
- v = R * ω
- Substituting ω in terms of v:
- T_rot = (1/2) * I * (v/R)^2
- Thus, the total kinetic energy is:
- T = T_trans + T_rot = (1/2) * m * v^2 + (1/2) * I * (v/R)^2
Potential Energy (V)
- The potential energy of the object when it is at height h is given by:
- V = m * g * h
- The height h can be expressed in terms of the distance down the incline (s):
- h = s * sin(a)
Lagrangian (L)
- The Lagrangian is defined as:
- L = T - V
- Substituting T and V:
- L = [(1/2) * m * v^2 + (1/2) * I * (v/R)^2] - (m * g * s * sin(a))
Conclusion
The Lagrangian incorporates the dynamics of both translational and rotational motion of the object rolling down the incline, providing a comprehensive framework for analyzing its motion.
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An object O of (mass m, Radius R, and moment of inertia I)that rolls without slipping straight down is an inclined plane which is at an angle a from the horizontal [Since the origin is where the start to move] then the Lagrangian of the object is?
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An object O of (mass m, Radius R, and moment of inertia I)that rolls without slipping straight down is an inclined plane which is at an angle a from the horizontal [Since the origin is where the start to move] then the Lagrangian of the object is? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about An object O of (mass m, Radius R, and moment of inertia I)that rolls without slipping straight down is an inclined plane which is at an angle a from the horizontal [Since the origin is where the start to move] then the Lagrangian of the object is? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An object O of (mass m, Radius R, and moment of inertia I)that rolls without slipping straight down is an inclined plane which is at an angle a from the horizontal [Since the origin is where the start to move] then the Lagrangian of the object is?.
An object O of (mass m, Radius R, and moment of inertia I)that rolls without slipping straight down is an inclined plane which is at an angle a from the horizontal [Since the origin is where the start to move] then the Lagrangian of the object is? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about An object O of (mass m, Radius R, and moment of inertia I)that rolls without slipping straight down is an inclined plane which is at an angle a from the horizontal [Since the origin is where the start to move] then the Lagrangian of the object is? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An object O of (mass m, Radius R, and moment of inertia I)that rolls without slipping straight down is an inclined plane which is at an angle a from the horizontal [Since the origin is where the start to move] then the Lagrangian of the object is?.
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