- #1
jwxie
- 281
- 0
Homework Statement
Suppose an element of a string, called [tex]\[\triangle x\][/tex] with T being the tension.
The net force acting on the element in the vertical direction is
[tex]\[\sum F_{y} = Tsin(\theta _{B}) - Tsin(\theta _{A}) = T(sin\theta _{B} - sin\theta _{A})\]
[/tex]
1. Since the angels are small, we can use the small-angle approximation [tex]\[sin\theta \approx tan\theta \][/tex], and rewrite
[tex]\[\sum F_{y} = T(tan\theta _{B} - tan\theta _{A})\][/tex]
I know what small-approximation is, but I suspect there is a definitive reason to why we choose sin ~= tan and not sin ~= delta. But y/x is arctan.. if we are talking about that.. So what is it?
If we extended the displacement outward and gives infinitesimal x and y components, then the tangent of the angle with respect to the x-axis for this displacement is [tex]\[\frac{d_{y}}{d_{x}}\][/tex]
2. Because we evaluate this tangent at a particular instant of time, we must express it in partial form as [tex]\[\frac{\partial y }{\partial x}\][/tex]
To be more clear, the reason we use partial is because the function contains two variables, x and t, right?
Any help is appreciated! Thank you!