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Degrees of freedom of a massless rod, moving freely in space with a speck which is constrained to move on it?

It seems massless rod is ideal I am very confused how to regulate degrees of freedom. However, I know Degrees of freedom mean minimum number of coordinate to specify something. Any help will be appreciated. Thanks!

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  • $\begingroup$ Is the speck constrained just to move up and down the rod, i.e. along a line, or can it also move around the circumference of the rod? $\endgroup$
    – Not_Einstein
    Commented Sep 11, 2020 at 0:43

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The rod has six degrees of freedom: X,Y,Z location of its center of mass, two degrees for the direction the axis points, and one degree for rotation around the axis. The speck can move anywhere on the surface of the rod (a two dimensional surface), giving another two degrees of freedom. Presumably the speck is a dimensionless point, so does not have orientational degrees of freedom, which would otherwise give the speck an additional three degrees of freedom. So, there are a total of eight degrees of freedom, or eleven if the speck has a size and shape.

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  • $\begingroup$ Rod is massless nevertheless should we consider the center of mass? $\endgroup$
    – Unknown
    Commented Sep 11, 2020 at 1:39
  • $\begingroup$ Actually, any point rigidly attached to the rod serves the purpose as well as the center of mass. It's just a way to say where the rod is in space. $\endgroup$
    – S. McGrew
    Commented Sep 11, 2020 at 2:48
  • $\begingroup$ Here the speck works as a constraint. Right! $\endgroup$
    – Unknown
    Commented Sep 11, 2020 at 2:53
  • $\begingroup$ Then what is the difference between rod and massless rod as a prospective of degrees of freedom?? $\endgroup$
    – Unknown
    Commented Sep 11, 2020 at 3:12
  • $\begingroup$ There is no difference: an object has the ssme degrees of freedom whether it is massless or not. If the problem were to count the degrees of freedom of a point on a rod-shaped surface, that's different. In that case you need to decide if the surface is fixed in space or not, and how its orientation is constrained. That is, is the surface merely a constraint, or is it an object. $\endgroup$
    – S. McGrew
    Commented Sep 11, 2020 at 5:22
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Three coordinates $(x, y, z)$ are required to describe the pole's position. Two coordinates $(\theta, \phi)$ are required to describe the pole's orientation (think latitude and longitude). Finally, one coordinate is required to describe the position of the speck on the pole. That makes six total degrees of freedom.

Edit: To clarify, this answer assumes that the rod is infinitely thin. This appears not to be what the author intended, but I'll leave this answer here in case anyone finds it useful.

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