Instantaneous power delivered in a string

Question

When the supports of a string are moving, why is the instantaneous power delivered by the string $0$?
So we say that at all moving contacts of the string, $$sum vec{T_i}cdot vec{v_i}=0$$ where $T$ and $v$ are the tensions and velocities at a point.
We often use this method to solve the problems of constrained motion in mechanics.
I want to know the proof of this method.

Agnius Vasiliauskas · Accepted Answer

Internal forces doesn't accomplishes work (such as tension in constrained-dynamics systems,- pulleys, etc.). Thus :
$$ W=vec{T_1} cdot vec{s_1} + vec{T_2} cdot vec{s_2} + ldots + vec{T_n} cdot vec{s_n} = 0 $$
Where $vec{s}$ are displacement vectors.  Differentiating with respect to time, we get :
$$ vec{T_1} frac{d vec{s_1}}{dt} + vec{T_2} frac{d vec{s_2}}{dt}  + ldots + vec{T_n} frac{d vec{s_n}}{dt} = 0 $$
Which is :
$$ vec{T_1} ~vec{v_1} + vec{T_2} ~vec{v_2}  + ldots + vec{T_n} ~vec{v_n} = 0 $$
Or simply in short notation :
$$ sum_i vec{T_i} ~ vec{v_i} = 0 $$

Instantaneous power delivered in a string

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