Setting Up KE for 2 Point Masses and a Rod

  • Thread starter Reshma
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In summary: If the rod is free to rotate about its centre, then another angle is need. This could be, for exampe, the angle that the rod makes with respect to horizontal, or the angle that the rod makes with respect to the line (radius) that joins the centre of the rod to the centre of the circle.
  • #1
Reshma
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Two point masses are joined by a rigid weightless rod of length "l", the centre of which is constrained to move on a circle of radius 'a'. Set up the kinetic energy in generalised coordinates.

So in this case, since the 2 mass m are joined by a rod, I consider the reduced mass of the system [itex]\mu[/itex]:
[tex]{1\over {\mu}} = {1\over m} + {1\over m}[/tex]
[tex]\mu = {m\over 2}[/tex]

Am I going right? How do I set up the KE in this case?
 
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  • #2
Reshma said:
Two point masses are joined by a rigid weightless rod of length "l", the centre of which is constrained to move on a circle of radius 'a'. Set up the kinetic energy in generalised coordinates.

So in this case, since the 2 mass m are joined by a rod, I consider the reduced mass of the system [itex]\mu[/itex]:
[tex]{1\over {\mu}} = {1\over m} + {1\over m}[/tex]
[tex]\mu = {m\over 2}[/tex]

Am I going right? How do I set up the KE in this case?
It's not clear to me that the reduced mass will be helpful. I think you need to pick your generalized coordinates. It looks to me like a pair of angles would be a natural choice.
 
  • #3
Pair of angles? I don't get it :frown:. I can consider one angle i.e. the polar angle on the circle but what is the other angle?
 
  • #4
Reshma said:
Pair of angles? I don't get it :frown:. I can consider one angle i.e. the polar angle on the circle but what is the other angle?

If the rod is free to rotate about its centre, then another angle is need. This could be, for exampe, the angle that the rod makes with respect to horizontal, or the angle that the rod makes with respect to the line (radius) that joins the centre of the rod to the centre of the circle.

Some choices make the analysis easier than other choices.
 

Related to Setting Up KE for 2 Point Masses and a Rod

1. What is the purpose of setting up KE for 2 point masses and a rod?

The purpose of setting up KE (kinetic energy) for 2 point masses and a rod is to analyze and understand the energy distribution and movements of these objects in a system. This can help in predicting their future positions and velocities, as well as determining the overall stability of the system.

2. How do you calculate the KE for 2 point masses and a rod?

The KE of each point mass can be calculated using the formula KE = 1/2 * m * v^2, where m is the mass of the point mass and v is its velocity. For the rod, the KE can be calculated using the formula KE = 1/2 * I * ω^2, where I is the moment of inertia of the rod and ω is its angular velocity.

3. What are the assumptions made when setting up KE for 2 point masses and a rod?

The main assumptions made in this setup are that the point masses are moving in a straight line and the rod is rotating about its center of mass. It is also assumed that there are no external forces acting on the system, and that there is no friction or air resistance present.

4. How does the distribution of KE differ between the point masses and the rod?

The point masses will have a linear distribution of KE, meaning their KE will be directly proportional to their mass and velocity. The rod, on the other hand, will have both linear and rotational KE, with the rotational KE being dependent on its moment of inertia and angular velocity.

5. Can the KE of a system with 2 point masses and a rod ever be completely conserved?

No, the law of conservation of energy states that energy cannot be created or destroyed, only transferred between different forms. In this system, there will always be some energy lost due to external forces such as friction or air resistance. However, the total KE of the system can remain constant as long as no external forces are present.

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