   Chapter 14, Problem 81AP

Chapter
Section
Textbook Problem

By proper excitation, it is possible to produce both longitudinal and transverse waves in a long metal rod. In a particular case, the rod is 1.50 m long and 0.200 cm in radius and has a mass of 5.09 g. Young’s modulus for the material is 6.80 × 1010 Pa. Determine the required tension in the rod so that the ratio of the speed of longitudinal waves to the speed of transverse waves is 8.

To determine
The required tension in the rod.

Explanation

Given Info: Longitudinal sound wave are produced in a long metal rod

Formula to find the speed of the longitudinal wave produced in the rod is,

vlong=Yρ

• vlong is the speed of the longitudinal sound wave.
• Y is the young’s modulus for the material.
• ρ is the density of the material.

Use m/V for ρ and πr2L for V in above relation.

vlong=YVm=Y(πr2L)m (1)

• r is the radius of the rod.
• L is the length of the rod.
• m is the mass of the rod.

Formula to find the speed of the transverse wave produced in the rod is,

vtrans=Fμ

• vtrans is the speed of the transverse sound wave.
• F is the tension force.
• μ is the linear density of the material.

Use m/L for μ in above relation.

vtrans=FLm (2)

The ratio of the speed of longitudinal wave and transverse wave is,

vlongvtrans=8

Rewrite the relation to find the tension force.

vlong=8vtrans

Compare the equation (1) and (2)

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