How Does Rsinθ Relate to Extra String Length in Physics?

[Dragster]

How do they get from the Rsinθ the value I have found the one I search: Rθ (extra string length)

I know that sinθ = θ when θ is small but there Rsinθ is the extra height the mass gain when we push it from initial position to a θ angle with vertical, I don't see how it could be the extra string length.

Thank you

 

[TSny]

Hi, Dragster. It would be very helpful if you could describe the physical setup. I think I see what some of the expressions in the figure represent. But I don't understand at all what the small circle with diameter $R\phi$ represents. Is that some physical object?

 

[Curious3141]

View attachment 54433

How do they get from the Rsinθ the value I have found the one I search: Rθ (extra string length)

I know that sinθ = θ when θ is small but there Rsinθ is the extra height the mass gain when we push it from initial position to a θ angle with vertical, I don't see how it could be the extra string length.

Thank you

Like TSny, I don't get the exact physical setup. But I can tell you that the arc length (length along the curve of the circumference) subtended by an angle $\phi$ along a circle of radius $R$ is $R\phi$. Does that help in relating the "extra string length" to that expression?

 

[Dragster]

I added the black circle with the ''?" on the top right.

Do you mind if I can get the rigorous explanation why the arc is Rθ? Thank you for the fast answers

Oh yea and for the problem, the string is attached on a cylinder. At theta = 0 with the vertical, the length is l0 and the asked result is the potential energy expression if the mass is pushed with an angle theta. I was only blocking on the part where I needed to know what was the extra string length.

 

[cepheid]

Do you mind if I can get the rigorous explanation why the arc is Rθ? Thank you for the fast answers

It's from the definition of how angles are measured:

http://en.wikipedia.org/wiki/Radian

 

[haruspex]

When the string is vertical, the 'extra' string is wrapped against the cylinder and subtends angle phi at the centre. Clearly the angle subtended by an arc of a circle at its centre is proportional to the length of the arc, and the radian is defined in such a way that the length is simply radius * angle.

 

[Dragster]

Oh well I am really stupid after all.
Thank you cepheid et harusphex :)
First university session makes me feel like a complete ignorant idiot.