Question

a stretched string is 1.97 m long and has a mass of 19.3 g. when the string oscillates at 440 hz, which is the frequency of the standard a pitch, transverse waves with a wavelength of 15.3 cm travel along the string. calculate the tension t in the string. t = n

Explanation:

Step1: Calculate the wave speed

The wave speed formula is \(v = f\lambda\).
Given \(f = 440\space Hz\) and \(\lambda=15.3\space cm = 0.153\space m\).

$$v = 440\times0.153$$

$$v = 67.32\space m/s$$

Step2: Calculate the linear mass density

The linear mass density \(\mu=\frac{m}{L}\).
Given \(m = 19.3\space g=0.0193\space kg\) and \(L = 1.97\space m\).

$$\mu=\frac{0.0193}{1.97}$$

$$\mu\approx0.009797\space kg/m$$

Step3: Calculate the tension

The formula for the speed of a wave on a string is \(v=\sqrt{\frac{T}{\mu}}\), so \(T = v^{2}\mu\).
Substitute \(v = 67.32\space m/s\) and \(\mu\approx0.009797\space kg/m\) into the formula.

$$T=(67.32)^{2}\times0.009797$$

$$T = 4531.9824\times0.009797$$

$$T\approx44.4\space N$$

Answer:

\(44.4\)