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I was answering a question regarding two rotating particles $P$ and $Q$ with $P$ attached to a fixed point $O$ by a light inextensible string and $Q$ attached to $P$ by a light inextensible string with $P$ and $Q$ rotating about the vertical through $O$ in the same vertical plane and at constant angles to the vertical:

If I consider the whole system and consider the forces acting on the vertical plane I believe I get something like this:

Even if this is slightly wrong I know that I consider the tension $T_{1}$ acting in the direction $\vec{PO}$ but not in the direction $\vec{PQ}$ because it is cancelled out by the tension in the direction $\vec{QP}$ which is equal and opposite. My question is therefore why is there not an equal and opposite tension in the direction $\vec{OP}$ that cancels out $T_{1}$. Is it the fact that $O$ is a fixed point rather than a particle? If so why does this make a difference?

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If you are considering the whole system to be both particle $P$ and $Q$, then this makes sense. You are only drawing external forces in your second picture. The tension between $P$ and $Q$ is an internal force, so you have not drawn it. However, the tension between $P$ and $O$ is an external force, so you have drawn it.

If you wanted to "get rid of" the tension between $P$ and $O$ you would need to include point $O$ (whatever it is physically) in your system. However, you would then also need to consider all of the other forces acting on $O$ that keeps in fixed in space. So you would get rid of the tension between $P$ and $O$ from your diagram, but you would then include other forces acting on $O$.

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First of all, you must be precise with your system. I am assuming you are interested in knowing the forces acting on particle P, then forces on particle P are a)gravity downwards b)Both tensions T1 and T2 c)D'lemberts force axially outward on particle P. Don't confuse with tension T2. It pulls particle P towards right and particle Q towards left side. If you are selecting particle P as your system then there is no need to consider the tension with which particle P is pulling the string(which is just opposite to the tension with which string is pulling P). Just consider the external forces acting on the system, ignore all forces including which system may be applying to outside.

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