The String Tension of a Rotating Rod: Is $\omega \propto M^{-1}$?

[arivero]

Consirer a rod rotating about one of its extremes such that the other extreme rotates at lightspeed. The centrifugal force at this extreme is
$$F \propto M \omega c$$

where M is the rest mass and w the angular speed. If the rod were a flexible one we should call it a string, it could vibrate around to keep the stress constant, and this stress (the string tension) should be equal to the Force above, so I wonder if the above formula apears also in strings.

If it does, then the string tension imposes a relationship
$$\omega \propto M^{-1}$$
Does it?

 

[Evalesco]

Yes, this relationship is true for strings. The tension of a string is inversely proportional to its mass, so that when the mass is increased, the tension decreases, and vice versa. This is why it is possible for a string to vibrate even when one end is rotating at lightspeed - the tension of the string is reduced due to the increased mass, allowing it to vibrate more easily.

 

[Entropia]

I would approach this question by analyzing the proposed relationship between angular speed (ω) and rest mass (M) in the context of a rotating rod and a vibrating string.

First, it is important to note that the formula F ∝ Mωc represents the centrifugal force at the extreme end of a rotating rod. This means that as the angular speed increases, the centrifugal force also increases, and as the rest mass of the rod increases, the centrifugal force decreases.

Next, the author suggests that if the rod were flexible and could vibrate, the string tension (which is equal to the centrifugal force) would also be equal to the formula F ∝ Mωc. This implies that the string tension would also be affected by changes in angular speed and rest mass.

Based on this information, the author poses the question of whether the relationship ω ∝ M^-1 holds true for strings as well.

In order to answer this question, further analysis and experimentation would be needed. It is possible that the relationship does hold true for strings, as they are also subject to the same forces (centrifugal force and string tension) as the rotating rod. However, it is also possible that the relationship may be different for strings due to their different properties and behavior compared to a rotating rod.

In conclusion, while the proposed relationship between ω and M in the context of a rotating rod and a vibrating string is interesting, more research and experimentation would be needed to definitively determine if the relationship holds true for strings.