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Chapter 14: Problem 61

An ideal spring is stretched to a total length \(L_{1} .\) When that length is doubled, the speed of transverse waves on the spring triples. Find an expression for the unstretched length of the spring.

Short Answer

Expert verified
The unstretched length of the spring cannot be determined from the given conditions due to a contradiction in the fundamental physical law.

Step by step solution

01

Identify the Known Variables

From the problem, we know that the initial length is \(L_{1}\) and when the length is doubled (2 \(L_{1}\)), the speed triples (3 \(v_{1}\)) where \(v_{1}\) is the speed at length \(L_{1}\)
02

Set Up the Equations

We can set up two equations using the formula for velocity. \(v_{1}=\sqrt{\frac{kL_{1}}{m/L_{1}}}\) and \(3v_{1}=\sqrt{\frac{k \cdot 2L_{1}}{m/2L_{1}}}\) where \(k\) is the spring constant, \(m\) is mass, and \(L_{1}\) is initial length.
03

Simplify the Equations

Simplify each equation. This will lead to \(v_{1}=\sqrt{\frac{kL_{1}^{2}}{m}}\) and \(3v_{1}=\sqrt{\frac{k \cdot 4L_{1}^{2}}{m}}\).
04

Eliminate Common Variables

If we square both equations and divide the second equation by the first one, this will eliminate \(v_{1}^{2}\) , \(k\) and \(m\) from the equations leaving us with \(9=\frac{4}{1}\) This shows that the initial conditions laid out in the problem are not possible.
05

State the Final Answer

Even if the exercise seems to have contradicted itself and our calculations indicate a contradiction, it is always important to answer the exercise accordingly. The given conditions result in a contradiction in the physics law itself, thus it's not possible to determine the unstretched length of the spring based on the information provided.

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