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# How to calculate tension in string

Updated July 20, 2017

Tension is the magnitude of pulling force exerted on a string. If you hold a length of string tight between your hands, any force that you exert on one end of the string is transmitted through the entire length of string to the other hand. The force transmitted by the string is called the tension in the string. This concept of tension is very important when you use a stringed instrument. The tune carried by a string is proportional to the speed that sound travels in the string, and that speed in turn depends on how tense and heavy the string is.

Write down the following formula for string tension T = 4 x L² x F² x r. In the formula, L equals string length, F equals frequency of a wave on the string and r equals mass per unit of volume or density of the string. In order to solve this, you'll need to measure the length of the string using a ruler, and you'll need to calculate the values for the frequency and the density.

Calculate the frequency F of a wave in the string using the following formula F = v ÷ W where v equals phase velocity of the wave in a string and W equals the wavelength. This can also be measured using an oscilloscope. A string attached at both ends, such as a string in a violin, will have a wavelength W equal to two times the length of the string. If, for example, you have a string that is not attached at both ends and you found that v = .33m/s and W = .002m, then F (the frequency) equals v ÷ W = .33m/s ÷ .002m, which equals 165 1/s or 165 H, Hertz.

Determine the linear density d of the string using a calculator through the following equation d = m ÷ L, where m equals the string mass and L equals the length of the string. For example, if the mass of the string is m = 3g and the length of the string equals 25cm, then the linear density is as follows: d = m ÷ L = 3g ÷ 25cm, which is equal to 0.12 gm/cm.

Solve the tension equation T = 4 x L² x F² x d by plugging in the string length you measured, your calculated frequency and your calculated linear density in to the formula. For example, the length of the string is 25cm, the frequency is 165 H and the linear density is 25cm. Plugging these into the original equation gives us T = 4 x L² x F² x d = 4 x (25cm)² x (165 H)² x 0.12 gm/cm, which equals 8,167,500 gm cm/s² or dynes.

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