How to identify component vectors correctly?

Question

Firstly I'm sorry for asking this very basic (& seemingly repeated) question but I want some more insight on it. The main question is this only: Velocity of a ring tied to an inextensible rope

Now I HAVE read the accepted answer and Yes MATHEMATICALLY its CORRECT. But............

• It doesn't answers why questioner's following logic is incorrect:

The rope is pulling the ring at an angle so only the component of the the velocity of the rope in the direction of movement of ring should act make it move, which ultimately makes the velocity of ring as ${v}_{ring}={v}_{rope}\cos\left(\theta \right)$

• Also it seems counterintuitive to say that velocity of ring has a component along rope whereas it sounds more reasonable to say that velocity of rope has a component along the ring because ORIGINALLY the force is applied on rope only. So at at intuitive level, how to make sense of this?
• Even if ${v}_{rope}={v}_{ring}\cos\left(\theta \right)$ is correct, is there any shorter way to find out the CORRECT components of a vector in such confusing situations? I don't think while solving mechanics problems one would go through such long process just to identify the component vectors?

Answer 1

The reason that

$$v_{ring}=v_{rope}\cos (\theta)$$

is incorrect is that this assumes $\theta$ is constant, which it is not. Once you realise this, the correct derivation is quite straightforward. If the distance of the ring from the pulley is $y$ and the horizontal distance (along the bar) is $x$ then:

$y^2=x^2+\text{ constant} \\ \displaystyle \Rightarrow 2y \frac {dy}{dt}=2x\frac {dx}{dt} \\ \displaystyle \Rightarrow v_{rope}=\frac x y v_{ring} = v_{ring}\cos (\theta)$

Although this result looks superficially like “taking components” it is better not to think of it like this.