Tension between two objects hanging

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To determine the tension in the middle string between two hanging objects, the user applies equilibrium equations for both horizontal and vertical forces. The equations used are \(\sum F_x = T_2 - \cos \theta T_1 = 0\) and \(\sum F_y = \sin \theta T_1 - mg = 0\). The user questions whether to include both objects in the vertical force equation and confirms that calculating \(T_2\) independently is valid. The discussion concludes with a clarification that the approach to find \(T_2\) is correct. Understanding the forces acting on both objects is essential for accurate tension calculations.
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How do we determine the tension of the string in the middle of the two objects? Let the tensions be T_1, T_2, and T_3, left to right:

I tried

\sum F_x = T_2 - \cos \theta T_1 = 0
\sum F_y = \sin \theta T_1 - mg = 0

for the other object it is similar. Is there something wrong with this? I can calculated T_2 without regard to the other object. Should I not include both the objects in F_y? thanx!
 
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If T2 is the tension in the middle string, this looks just okay.
 
ok, got it! thanx! :-)
 

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