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Chapter 14: Problem 31

A rope with \(280 \mathrm{g}\) of mass per meter is under \(550-\mathrm{N}\) tension. Find the average power carried by a wave with frequency \(3.3 \mathrm{Hz}\) and amplitude \(6.1 \mathrm{cm}\) propagating on the rope.

Short Answer

Expert verified
The average power carried by the wave is 44.8W

Step by step solution

01

Understanding and Writing Down the Given Details

The given details are: the mass per unit length (linear density) \(\mu = 280g/m = 0.28 kg/m\), the tension \(T=550-N\), frequency \(f = 3.3Hz\), and amplitude \(a=6.1cm = 0.061m\).
02

Deriving the Wave Speed

The speed (v) of a wave on a string under tension is found by \(\sqrt{T/\mu}\). Therefore, the speed \(v = \sqrt{550N / 0.28 kg/m} = 44.3 m/s\).
03

Applying the Power Formula

The average power \(P\) carried by a wave is given by \((2π)^2 \mu v f^2 a^2\). So, plug in the given parameters in the formula, we find \(P = (2π)^2 * 0.28 kg/m * 44.3 m/s * (3.3 s^-1)^2 * (0.061m)^2 = 44.8W\)

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Most popular questions from this chapter

A car horn emits \(380-\mathrm{Hz}\) sound. If the car moves at \(17 \mathrm{m} / \mathrm{s}\) with its horn blasting, what frequency will a person standing in front of the car hear?

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A rope is stretched between supports \(12 \mathrm{m}\) apart; its tension is \(35 \mathrm{N} .\) If one end of the rope is tweaked, the resulting disturbance reaches the other end 0.45 s later. Find the total mass of the rope.

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