Question

A steel string guitar is strung so that there is negligible tension in the strings at a temperature of 24.9°C. The guitar is taken to an outdoor winter concert where the temperature of the strings decreases to –15.1°C. The cross-sectional area of a particular string is 5.5x10⁻⁶ m². The distance between the points where the string is attached does not change. For steel, Young’s modulus is 2.0x10¹¹ N/m²; and the coefficient of linear expansion is 1.2x10⁻⁵/C°. Use your knowledge of linear thermal expansion and stress to calculate the tension in the string at the concert.

Answer 1

The tension in the steel guitar string at the concert, due to a decrease in temperature to -15.1°C, is found to be 5.28 N after calculating the thermal stress and then using it to determine the force that results in this tension.

Step-by-step explanation:

To calculate the tension in the guitar string at –15.1°C, we need to understand that due to thermal contraction the string attempts to shorten its length, but since it is fixed at both ends, it cannot. This induces stress and tension within the string.

The equation for thermal stress without deformation is σ = α ΔT Y, where σ is stress, α is the coefficient of linear expansion, ΔT is the change in temperature and Y is Young's modulus.

The change in temperature (ΔT) is -15.1°C - 24.9°C = -40°C. Plugging the given values into the formula, we get σ = (1.2×10-5/C°)(-40C°)(2.0×1011 N/m²) = -960000 N/m² of stress. Since stress (σ) is also force (F) divided by area (A), we then calculate tension (T) using F = σA. The cross-sectional area (A) is 5.5×10-6 m².

Therefore, the tension in the string is T = (σ)(A) = (-960000 N/m²)(5.5×10-6 m²) = -5.28 N. The negative sign indicates that the calculated tension is compressive stress; however, since tension is a pulling force, we use the magnitude and say the tension in the string is 5.28 N.