How can we prove that the forces acting on a body can form a closed-up

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Forces acting on a body can form a closed polygon, indicating equilibrium, as a resultant force exists only if they do not. A zero resultant force occurs when the forces create a closed polygon, satisfying the condition for equilibrium. In scenarios with no forces or only two forces, the object can still be in equilibrium, but the vectors won't form a polygon. The two-force case can be visualized as moving along a line and returning along the same line. Ultimately, the sum of the force vectors must equal a zero vector for the object to be in equilibrium.
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How can we prove that the forces acting on a body can form a closed-up polygon?
 
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If they didn't, then there would be a resultant force. The magnitude of the resultant is zero only if the other forces form a closed polygon. The condition for equilibrium is, of course, no net force
 


If there are no forces or only two forces on the object then that object could be in equilibrium but the force vectors can't be connected to form a polygon.
 


Well you could think of the two forces case as going along a line and back along the same line.

The idea to remember is that the sum of the force vectors is a zero vector for the object in equilibrium.
 
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