Lab Prep - Speed of a wave on a stretched string

[Hurly]

Homework Statement

1) Show that the expression for the mass per unit length, μ, of a wire in terms of its density and diameter is

μ = [itex]\frac{πρd^2}{4}[/itex]

2) Using equations (3) and (4) show that the expression for the error in frequency in terms of the error in length and the error in diameter of the wire is given by equation 5. Assume the error in the mass is negligible.

Δf = f(ΔL/L + Δd/d)

Homework Equations

(3) f = [itex]\frac{n}{2L}[/itex] [itex]\sqrt{\frac{T}{μ}}[/itex]

(4) μ = [itex]\frac{πρd^2}{4}[/itex]

The Attempt at a Solution

Mass of Wire = Vol x Density

= πV^2 ρ

or of density = 2V V^2 = [itex]\frac{d}{2}[/itex]^2

∴Mass = [itex]\frac{πd^2ρ}{4}[/itex]

 

[jbsteele]

2) Substituting equation (4) into equation (3), we getf = \frac{n}{2L} \sqrt{\frac{Tπρd^2}{4μ}}Differentiating w.r.t. L and d, we getΔf = f(ΔL/L + Δd/d)