Oscillation of nonrelativistic Strings

[moriheru]

Homework Statement

Calculate the horizontal force dF_horizontal of a string .Show that for small oscillations this force is much smaller than the vertical force dF_vertical responsible for the transverse oscillations.
[I apolagize for the way the equations are written]

Homework Equations

dF_vertical=T₀dy/dx*-T₀dy/dx*¹
*for x+dx *¹ for x

T₀(d^2 y/d^2 x)dx=(μ₀dx)d^2 y/dt^2

3. The Attempt at a Solution
1.substituting y+dy and y for x+dx and x; substituting dx/dy for dy/dx for horizontal oscillation.
dF_horizontal=T₀dx/dy*-T₀dx/dy*¹
*for y+dy *¹ for y
2.exchanging d^2 y/ dx^2 with d^2 x/dy^2 and dx with dy...

T₀ (d^2 x/dy^2) dy= (μ₀dx)(d^2 x/dt^2)

This is as far as I get. I don't know how to show that the horizontal force is smaller than the vertical force for small oscillations.

Thanks for any clarifications and corrections.

 

[Orodruin]

I would instead express the forces (both horizontal and verical) in terms of y(x) and its derivatives wrt x. You can then compare the two with the assumption of small oscillations giving you dy/dx << 1.