Tension in a Massive Rotating Rope with an Object

[AzimD]

Looking back through the thread, unless I misunderstand how the variables are defined, the equation of mine you quoted in post #31 is wrong. It has a sign error.
See if you can correct it.
(I believe T(0) should be the tension at the pole if we ignore the point mass.)

Also, what do you have for T(0)?

For T(0) I have $T(0)=-m_1\omega^2l$
Should this be a positive value?

 

[jbriggs444]

For T(0) I have $T(0)=-m_1\omega^2l$
Should this be a positive value?

As I understand it, we are considering "tension" $T(r)$ in the sense of "the force from the anchored direction on the remainder of the cable from point $r$ on out"

As I understand it, we are adopting a sign convention in which outward is positive and inward is negative.

If so, then $T(0)$ would properly be negative. The rope is being pulled inward.

However, $-m_1 \omega^2 l$ would be correct for a mass rotating in a circular trajectory at radius $l$. What is the relevant rotational radius for mass $m_1$ here?

 

[haruspex]

As I understand it, we are considering "tension" $T(r)$ in the sense of "the force from the anchored direction on the remainder of the cable from point $r$ on out"

As I understand it, we are adopting a sign convention in which outward is positive and inward is negative.

If so, then $T(0)$ would properly be negative. The rope is being pulled inward.

That works, but then I would not call it tension (nor label it T). To me, tension is a state, not a force. It is more like a pair of equal and opposite forces at a point, or a distribution of such pairs along a body. A suitable sign convention for that would be positive for a tension and negative for a compression - or the converse.

 

[jbriggs444]

That works, but then I would not call it tension (nor label it T).

Yes. I'd drafted a paragraph pontificating on how tension is more properly a scalar or a portion of a tensor rather than a vector-valued force, but then discarded it as more confusing than helpful.

 

[AzimD]

As I understand it, we are considering "tension" $T(r)$ in the sense of "the force from the anchored direction on the remainder of the cable from point $r$ on out"

As I understand it, we are adopting a sign convention in which outward is positive and inward is negative.

If so, then $T(0)$ would properly be negative. The rope is being pulled inward.

However, $-m_1 \omega^2 l$ would be correct for a mass rotating in a circular trajectory at radius $l$. What is the relevant rotational radius for mass $m_1$ here?

$T(0)$ for $m_1$ would be at length $l$ no? Of course, the force caused by $m_2$ isn't added in because I come back to add that in later.