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Physics for Scientists and Enginee...

10th Edition
Raymond A. Serway + 1 other
ISBN: 9781337553278

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BuyFindarrow_forward

Physics for Scientists and Enginee...

10th Edition
Raymond A. Serway + 1 other
ISBN: 9781337553278
Textbook Problem

A wire of density ρ is tapered so that its cross-sectional area varies with x according to

A = 1.00 × 10 5 x + 1.00 × 10 6

where A is in meters squared and x is in meters. The tension in the wire is T. (a) Derive a relationship for the speed of a wave as a function of position. (b) What If? Assume the wire is aluminum and is under a tension T = 24.0 N. Determine the waver speed at the origin and at x = 10.0 m.

(a)

To determine

The relation for the speed of wave as a function of position.

Explanation

The cross-sectional area of the wire is given as,

    A=1×105x+1×106

Formula to calculate the mass for small segment of wire is,

    dm=ρAdx

Here, dm is the mass for small segment of wire, ρ is the density of wire and dx is the length of small segment.

Formula to calculate the linear density for small segment is,

    μ=dmdx        (I)

Substitute ρAdx for dm.

    μ=ρAdxdx=ρA

Formula to calculate the speed of the wave is,

    v=Tμ        (II)

Here, v is the speed of the wave, T is the tension in the wire and μ is the linear mass density

(b)

To determine

The wave speed at the origin and x=10m.

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