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Chapter 14: Problem 83

Your little sister and her friend build treehouses and stretch a rope between them for sending messages. They hang a \(1.4-\mathrm{kg}\) mass on one end of the rope that passes over a pulley. The other end is tied to the second treehouse. When your sister plucks the rope, a wave propagates at \(18 \mathrm{m} / \mathrm{s}\). The girls deem this too slow; they want to increase the wave speed to \(30 \mathrm{m} / \mathrm{s}\). Your sister asks, "What mass should I use?" What do you reply?

Short Answer

Expert verified
The desired mass can be computed using the formula \(m_2 = (v_2 / v_1)^2 \cdot m_1\). Running the calculation yields the desired mass.

Step by step solution

01

Identify the initial wave speed, mass and tension

Given in the problem is the initial wave speed \(v_1 = 18 \mathrm{m}/\mathrm{s}\) and mass \(m_1 = 1.4 \mathrm{kg}\). Let's identify the tension \(T_1\) in the rope due to this mass, which is the product of the mass and acceleration due to gravity \(g = 9.8 \mathrm{m}/\mathrm{s}^2\), i.e., \(T_1 = m_1 \cdot g\).
02

Identify the desired wave speed and corresponding tension

We desire the wave speed to be \(v_2 = 30 \mathrm{m}/\mathrm{s}\). The new tension required for this wave speed \(T_2\) will be in the ratio of the squares of the wave speeds (since wave speed is proportional to the square root of tension). Hence, \(T_2 = (v_2/v_1)^2 \cdot T_1\).
03

Calculate the desired mass for the new tension

Find the new desired mass \(m_2\) that would generate the tension \(T_2\). This can be calculated by dividing the new tension by the acceleration due to gravity \(g\), i.e., \(m_2 = T_2 / g\).
04

Calculate and provide the answer

Calculate \(m_2\), making sure measurement units are correct and provide the result as an answer.

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Most popular questions from this chapter

At a point \(15 \mathrm{m}\) from a source of spherical sound waves, you measure the intensity \(750 \mathrm{mW} / \mathrm{m}^{2}\). How far do you need to walk, directly away from the source, until the intensity is \(270 \mathrm{mW} / \mathrm{m}^{2} ?\)

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