Questions on Mechanics: Why is Tension at End of Whirling Rope Zero?

[benf.stokes]

Hi,

In one of the examples of "An Introduction to Mechanics" he argues that the tension at the end of a whirling rope must be zero as the end is free.

I just don't understand how this can be.
Below is the example

Thanks

 

[madhatter106]

is he talking about a length or rope being rotated as a straight line or where the rope is in a sine wave?

 

[madhatter106]

never mind, I see it now.
the end of the rope has no further mass beyond it or no more rope to provide the tension from the rotational acceleration, at any point r between the ends the tension will be based on the delta from r to the free end and the fixed end.

 

[benf.stokes]

I'm sorry but could you please explain better?

 

[CuriousGorge]

Sounds like you've reached the end of your rope...

Well if there's nothing beyond the end then how can there be any tension on it?

 

[Martin Rattigan]

The tension at a distance $x$ from the end of the rope acts on the segment of the rope between the end and distance x.

If the mass of this segment of the rope is $m_x$ and the tension is $T_x$, then $T_x=m_xa_x$, where $a_x$ is the acceleration of this segment of the rope.

Since $m_x\rightarrow 0$ as $x\rightarrow 0$, then unless the end of the rope is subject to infinite acceleration we must also have $T_x\rightarrow 0$ as $x\rightarrow 0$.