Steel Wire Tension Calculator

This calculator computes the tension required in a steel wire to produce a certain frequency in the wire when it vibrates. 
CalculationInputs﻿﻿Diameter, d
:5.00mm​
﻿
﻿﻿Length, L
:0.80m​
﻿
﻿﻿Density, ρ
:1,000kg/m^3​
﻿
﻿﻿Frequency, f
:220Hz​
﻿
Output﻿﻿Mass per unit length, μ
:19,635kg/m​
﻿
﻿﻿μ=ρ 4πd2​
﻿
﻿﻿Tension, T
:2,433N​
﻿
﻿﻿T=4 L2 f2 μ
﻿
Where:
Diameter, ﻿d
 is the diameter of the steel wire in millimeters ﻿(mm)
﻿
Length, ﻿L
 is the length of the steel wire in meters ﻿(m)
﻿
Density, ﻿ρ
 is the density of the steel wire material in kilograms per cubic meter ﻿(kg/m3)
﻿
Frequency, ﻿f
 is the fundamental frequency of the wire, in Hertz ﻿(Hz)
﻿
Mass per unit length, ﻿μ
 is in kilograms per meter ﻿(kg/m)
﻿
Tension, ﻿T
 is the tension in the steel wire, in Newtons ﻿(N)
﻿
ExplanationThe formula for tension incorporates the wire's physical dimensions, density, and vibrational frequency. This relationship is vital for assessing the wire's ability to endure different types of forces.
﻿
Steel wire in tension
ApplicationsExample 1: calculating tension for a desired frequency
In musical instrument manufacturing, such as crafting a guitar or piano, achieving the correct pitch (frequency) for each string requires adjusting the tension. Here, desired frequency and the physical characteristics of the string is known, and is required to find the tension to apply.
The tension, ﻿T
 required in a string to produce the correct pitch (frequency) is given by the equation:
﻿
T=4 L2 f2 μT=4\space L^2\space f^2\space \muT=4 L2 f2 μ
💡This calculator uses the equation above. 
Example 2: calculating frequency for a desired tension 
In civil engineering, particularly in the maintenance of bridge cables or suspension structures, knowing the current tension in a cable (through on-site measurements) can help calculate the fundamental frequency. This information is crucial for assessing the structure's resonance characteristics.
The equation in example 1 can be re-arranged such that the fundamental frequency, ﻿f
 becomes the subject:
﻿
f=12LTμ​​f=\dfrac{1}{2L}\sqrt{\dfrac{T}{μ}}​ ​f=2L1​μT​​​​
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ReferencesTimoshenko, S., & Young, D. H. (1962). Vibration Problems in Engineering. John Wiley & Sons.
Gere, J. M., & Timoshenko, S. P. (1997). Mechanics of Materials. PWS Publishing Company.