Lagrangian for a rolling disk on horizontal plane

In summary, when a disk is rolling on a horizontal plane, the total kinetic energy can be calculated by adding the kinetic energy of the center of mass with respect to the origin and the rotational kinetic energy about the center of mass. This can be represented by the formula T= 1/2 (M V^2) +1/2(I ω^2), where M is the mass of the disk, V is the velocity of the center of mass, I is the moment of inertia, and ω is the angular velocity about the center of mass. However, some books only consider the rotational kinetic energy and ignore the term for translational motion. It would be helpful to know which books you are referring to so that I can
  • #1
sayf alawneh
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for a disck rolling on a horizontal plane the kinetic energy should be the kinetic energy of the CM of the disk with respect to the origin plus the kinetic energy due to the rotation of the disc about his CM
so T= 1/2 (M V^2) +1/2(I ω^2)
where M is the mass of the disk and V is the velocity of the CM and I is the moment of inertia of the disk and W is the angular velocity about the CM

am i wrong ?
why some books sole this problem for rotational kinetic energy only and ignore the term of kinetic energy that depends on the velocity of the center of mass with respect to the origin
in ohter words they consider T = 1/2 I ω^2 only :(
 
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  • #2
You would normally have to consider the translational motion also. In which books did you find this so that I can give a look at the problem.
 
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1. What is the Lagrangian for a rolling disk on a horizontal plane?

The Lagrangian for a rolling disk on a horizontal plane is a mathematical expression that describes the total kinetic and potential energy of the disk as it rolls on a flat surface. It is derived from the principle of least action and is used in classical mechanics to analyze the motion of the disk.

2. How is the Lagrangian for a rolling disk on a horizontal plane calculated?

The Lagrangian for a rolling disk on a horizontal plane is calculated by considering the kinetic and potential energy of the disk. The kinetic energy is determined by the motion of the disk's center of mass, while the potential energy is a result of the disk's position in a gravitational field. The total Lagrangian is then the sum of these two energies.

3. What is the significance of the Lagrangian for a rolling disk on a horizontal plane?

The Lagrangian for a rolling disk on a horizontal plane is significant because it allows us to analyze the motion of the disk using the principle of least action. This principle states that the actual path taken by the disk is the one that minimizes the difference between the kinetic and potential energy. The Lagrangian provides a concise and efficient way to determine this path.

4. Can the Lagrangian for a rolling disk on a horizontal plane be used to analyze other rolling objects?

Yes, the Lagrangian for a rolling disk on a horizontal plane can be extended to analyze the motion of other rolling objects, such as cylinders or spheres. The principles and equations used to calculate the Lagrangian for a rolling disk can be applied to these objects as well.

5. How does the Lagrangian for a rolling disk on a horizontal plane differ from that of a disk on an incline?

The Lagrangian for a rolling disk on a horizontal plane differs from that of a disk on an incline because the potential energy of the disk is affected by the incline's angle. In the case of a horizontal plane, the potential energy is constant, but on an incline, it varies depending on the angle of the incline. This difference affects the overall Lagrangian and therefore, the motion of the disk.

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