Boundry conditions on a string with a hoop at one end

[BareFootKing]

Homework Statement

problem 4.4

The question:
http://ocw.mit.edu/courses/physics/...ions-and-waves-fall-2004/assignments/ps4a.pdf

The solution:
http://ocw.mit.edu/courses/physics/...ons-and-waves-fall-2004/assignments/sol4a.pdf

Homework Equations

The Attempt at a Solution

For part a I am having trouble understanding why Tsinθ becomes -T∂y/∂x

If we were dealing with small angles would it not be the case that

sinθ ≈θ


and so we would have
Tsinθ≈Tθ

 

[bossman27]

$\frac{\partial y}{\partial x}$ is essentially the ratio of vertical tension to horizontal tension, much like $tan(\theta)$. It's a little confusing because obviously you're thinking about the hoop, which only moves vertically, but in this context you can think of it as representing the direction the hoop would go if it suddenly flew off of the rod, at which point the only force acting on the hoop would be the tension at the contact point (ignoring gravity), thus it would fly in that direction. Another way to think of it is simply as the instantaneous "slope" of the string.

The small angle approximation comes in because for small angles $tan(\theta) \simeq sin(\theta)$. (or equivalently, $T_{x} \simeq T_{tot}$.)

Thus we can put it all together: $sin(\theta) \simeq tan(\theta) = \frac{\partial y}{\partial x}$

 

[BareFootKing]

Thank you very much bossman27!