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I was solving some questions to apply my concepts, and I came across the atwood machine and pulley block problems. Consider the following for example: .

The pulley is massless and frictionless, string, too, is ideal. Why does the book say that the tension in the green string is $2T$ if the tensions in the two wings of the lower string is $T$ and $T$. Like if we see closely the strings only apply the normal force on the pulley how is it equal to two times tension $T$.

Also if the pulley would have been having mass and friction props. (string is mass less but with friction) would the tensions in the lower string be the same throughout? And what about the upper string? And what happens if everything is non ideal?

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The pulley is at rest, so the net force must be zero. Since there are two tensions $T$ trying to push it down, the tension of the upper string must be equal to twice this value, so that the vector sum is zero:

\begin{equation} F_{\text{net}}=F'-T-T=0\to F'=2T \end{equation}

If the pulley has a mass $M_p$, then

\begin{equation} F_{\text{net}}=F''-T-T-M_{p}g=0 \end{equation}

from which

\begin{equation} F''=2T+M_{p}g \end{equation}

Hence, in this case the upper tension gets an additional term.

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Imagine removing the black string and masses and instead just grabbing the pulley with your arms. If you pull with a force of $T$ with each arm, shouldn't the green string in the top then hold back against both? The green string tension should be $2T$.

Back to your scenario, the situation is the same. The green string is holding up two masses all in all whereas each half of the black string is only carrying one mass.

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If pulley had mass $M_{pulley}$ and it is in equilibrium in the frame in which you are working, then assume that tension in string is $T_{g}$

$$\sum \vec{F}_{net,pulley}=0$$ $$T_{g}-T-T-M_{pulley}g=0$$ $$T_{g}=M_{pulley}g+2T$$

If pulley is ideal, then $M_{pulley}=0$, so, $T_{g}=2T$.

If there is friction in pulley and strings have mass then scenario becomes quite different, Pulley will rotate with some angular acceleration due to torque about its axis as well as Tension won't be same throughout the string.

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