Question

M With particular experimental methods, it is possible to produce and observe in a long, thin rod both a transverse wave whose speed depends primarily on tension in the rod and a longitudinal wave whose speed is determined by Young's modulus and the density of the material according to the expression v = √(Y/rho) . The transverse wave can be modeled as a wave in a stretched string. A particular metal rod is 150cm long and has a radius of 0.200cm and a mass of 50.9 g . Young's modulus for the material is 6.80× 10¹⁰N/m² . What must the tension in the rod be if the ratio of the speed of longitudinal waves to the speed of transverse waves is 8.00 ?

Answer 1

The tension in the rod when the ratio of speed is 8 is 1.34 × 10⁴ N.

The longitudinal waves speed is determined using the density of the material and the Youngs modulus with the expression:

v = √(Y/rho)

A wave in a stretched string can be used to represent the transverse wave.

The length of the metal rod is 150 cm and the radius of the rod is  cm.

The mass of the rod is 50.9 g.

Now, the Youngs modulus for the material is 6.80× 10¹⁰ N/m².

For the longitudinal wave

v(l) = (Y/ρ)^{1/2}

For the transverse wave :

v(T) = (Y/μ)^{3/2}

Now, we need

v(L) / v(T) = 8

And we know,

T = (μY) / (64ρ) where μ= m /L

And the density, ρ = mass / volume = (m) / (πr²L)

Therefore,

T = (πr²Y) / 64

T =  (π)(2 × 10⁻³)² × (6.8 × 10¹⁰) / 64

T = 1.34 × 10⁴ N

Learn more about longitudinal waves here:

brainly.com/question/25827374

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