So,this question has been bothering me for quite sometime.
What exactly is an acceptable approximation?
Few days back,I did this-
Imagine there is a cylinder of sufficiently small diameter.A rope or string is wound around it as shown (curvature is a bit exaggerated in the figure).So now,I took a component of $x$ as $x\cos\theta$ as shown and did all my calculations to arrive at the exact answer that should be.
Now,some guys objected saying that I cannot use trigo like that as the surface is curved.Not at all an unreasonable claim.But my point is that if the triangle is small enough can't we assume the curved rope to be approximately straight?
For eg. in physics we often use that arc length of a sector ($s=r\theta$ for sufficiently small $\theta$) is approximately equal to the length of the straight line joining the endpoints of the sector.
So,isn't my argument sensible?
Thanks for any help!!
A good approximation is one that is good enough for the purpose. In the case of your cylinder, your $x \cos \theta$ is not an approximation, it is exact. You can unroll the cylinder to prove that. It works because the cylinder is flat. You need curvature in both directions to make a problem.
The arc length approximation you mention is basically using $\sin \theta \approx \theta$ This is the first term in the Taylor series and asserts that $\frac {\theta^3}6$ is negligible for the problem at hand. Whether this is true depends on how large $\theta$ is and how accurate you need to be.