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?


Both equations are wrong. It looks like you drew free body diagrams and equated the opposing forces. This would only be valid when the weight of the pendulum is not accelerating.

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 $$T - mg\cos\theta=mR\dot\theta^2$$ $$-mg\sin\theta=mR\ddot\theta.$$

The first of these is the equation you are after, and the second is the equation of motion.

  • $\begingroup$ How would the equations change if I decide to resolve tension and not weight. $\endgroup$ Jul 6 '20 at 11:16
  • 1
    $\begingroup$ 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. $\endgroup$
    – Puk
    Jul 6 '20 at 16:26

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