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Chapter 15: Problem 1

The wave speed on a string under tension is 200 m/s. What is the speed if the tension is doubled?

Short Answer

Expert verified
The speed of the wave when the tension is doubled is approximately 282.84 m/s.

Step by step solution

01

Analyze the Given Information

The given wave speed(v) which is 200 m/s, represents the speed under the initial tension, let's denote this as \(T_1\). Now the tension is doubled, denote this as \(T_2 = 2*T_1\). We are to find out the wave speed under this new tension, let's denote this as \(v_2\)
02

Apply the Wave Speed Formula

The formula relating wave speed, tension, and mass per unit length is \(v = \sqrt{T/碌}\). While we don't know the values of \(T_1\) or 碌, we do know that \(T_2 = 2*T_1\) and the relation between the initial wave speed and the new wave speed is expressed as a function of the tension: \(v_1/v_2 = \sqrt{T_1/T_2}\). We know that \(v_1 = 200 m/s\) and \(T_2 = 2*T_1\). Substitute these values into the relation.
03

Calculate the New Wave Speed

Substituting the values from the last step into the relation, we get \(v_2 = v_1 * \sqrt {T_2 / T_1} = 200 m/s * \sqrt {2} \approx 282.84 m/s\). Round as appropriate for the level of precision required.

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