Concept explainers
A string on a musical instrument is held under tension T and extends from the point x = 0 to the point x = L. The string is overwound with wire in such a way that its mass per unit length μ(x) increases uniformly from μo at x = 0 to μL at x = L. (a) Find an expression for μ(x) as a function of x over the range 0 ≤ x ≤ L. (b) Find an expression for the time interval required for a transverse pulse to travel the length of the string.
(a)
The expression for
Answer to Problem 69P
The expression for
Explanation of Solution
Given info: The tension in the string is
Assume the linear expression for the linear density
Here,
Substitute
Substitute
Substitute
Substitute
Conclusion:
Therefore, the expression for
(b)
The expression for the time interval required for transverse pulse to travel the length of the string.
Answer to Problem 69P
The expression for the time interval required for transverse pulse to travel the length of the string is
Explanation of Solution
Given info: The tension in the string is
Formula to calculate the speed of the wave is,
Here,
Rearrange the above equation.
Integrate the right hand side of the equation (3) from
Substitute
Assume,
When
Substitute
Integrate the left hand side of the equation (6) from
Substitute
This can be written as,
Conclusion:
Therefore, the expression for the time interval required for transverse pulse to travel the length of the string is
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Chapter 13 Solutions
Principles of Physics: A Calculus-Based Text
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- College PhysicsPhysicsISBN:9781285737027Author:Raymond A. Serway, Chris VuillePublisher:Cengage Learning