Question

Consider a pendulum making an angle theta with the vertical. The two forces on it are mg due to the earth and T , the tension in the string.
We can
1) decompose T as T cos(theta) equal and opposite to mg and Tsin(theta) back to equilibrium position
2) decompose mg as mg cos(theta) equal and opposite to T and mg sin(theta) back to equilibrium position.
Both are valid vector operations.
But the first case is clearly not equivalent to the second.how is this resolved?
My question concerns basic vector operation(s) the has , for some reason, really confused me.

Answer 1

Both decompositions are correct, however in case 1, Tcos(theta) is not equal to mg becuase the pendulum is in fact moving and accelerating in the horizontal direction. Case 2 is correct because the rod connected to the object is rigid and therefore the object does not move in the direction of the rod.

Answer 2

Makes sense, thanks a lot.