# I am confused about resolving of forces in a certain situation. Where am I wrong?

I was solving questions related to circular motion of a pendulum hanging from a ceiling in an inertial frame having angular displacement $$θ$$ and if I resolve Tension into vertical and horizontal components I get

$$T\cosθ = mg$$

but if I resolve the weight $$mg$$ in the same scenario I get

$$T = mg\cosθ$$

and both possibly can't be right. Where am I wrong?

Let $$\dot\theta$$ denote the angular velocity, $$\ddot\theta$$ the angular acceleration and $$R$$ the length of the string. The centripetal acceleration along the tension force is $$a_{rad}=R\dot\theta^2$$ while the tangential acceleration perpendicular to the tension force and in the direction of increasing $$\theta$$ is $$a_{tan}=R\ddot\theta.$$
Now if you draw a free body diagram and write Newton's second law along each of these two directions, you get $$T - mg\cos\theta=mR\dot\theta^2$$ $$-mg\sin\theta=mR\ddot\theta.$$
• Ultimately you would arrive at the same equations, but with more algebra. Namely the acceleration in the upward direction is $a_{rad}\cos\theta+a_{tan}\sin\theta$, and in the horizontal direction $-a_{rad}\sin\theta+a_{tan}\cos\theta$. You can then write Newton's second law along the vertical and horizontal.