Proof for double tension in a string

Question

I want to know if there is a proof for double tension to be present in the string for 6kg mass. Like some mathematical proof , not just saying.

Answer 1

You seem to be asking why the force that the string exerts on the $6$ kg block is twice the force it exerts on the $4$ kg block. A conservation of energy argument shows that this must be the case.

If the $4$ kg block moves a distance $d$ against a tension $T$ in the string then it does work $Td$ on the string. The string itself cannot store any energy because it is light and inextensible. So by conservation of energy the string must do work $Td$ on the $6$ kg block. But the $6$ kg block only moves a distance $\frac d 2$. So the force $F$ exerted by the string on the $6$ kg block must satisfy

$F \frac d 2 = T d \\ \Rightarrow F = 2T$

Answer 2

Double tension isn’t present in the string. Twice the tension is the net force on the upper block because there is a force from the upper leg of the string and also one from the lower leg.

If the pulley confuses you, imagine the upper block is glued down. Then the pulley will not rotate. In that case, you can remove the pulley and just tie both the upper and lower legs of the string to the block. Each has tension $T$, so the net force of the string on the block is $2T$.

Answer 3

It is supposed that the pulleys have no bearing friction or moment of inertia. So, the tension in the string due to the weight on the right has to be fully supported by the vertical wall where it is attached.

And that is valid in a static or dinamic situation. Once accepting that, it is clear that the total force at the left pulley is $2T$.

Maybe the reason for the doubt is that for a real situation, the pulley itself requires some torque to rotate. In that case, the torque should come from a difference between the tension in the upper leg and in the bottom leg.