Question

A \(2.00-\mathrm{m}\) -long string of mass \(10.0 \mathrm{~g}\) is clamped at both ends. The tension in the string is \(150 .\) N. a) What is the speed of a wave on this string? b) The string is plucked so that it oscillates. What is the wavelength and frequency of the resulting wave if it produces a standing wave with two antinodes?

Short answer

Answer: The speed of the wave is 170 m/s, the wavelength is 4.00 m, and the frequency is 42.5 Hz.

Step-by-step solution

Calculate the linear mass density of the string

First, we need to calculate the linear mass density (μ) of the string. This can be found using the formula: μ = m / L where m is the mass of the string and L is its length. Plugging in the given values: μ = 0.010 kg / 2.00 m = 0.005 kg/m

Calculate the wave speed on the string

Now we can calculate the wave speed (v) on the string using the formula: v = sqrt(T / μ) where T is the tension in the string and μ is the linear mass density. Substituting the given values: v = sqrt(150 N / 0.005 kg/m) = sqrt(30000 m^2/s^2) = 170 m/s So the speed of a wave on this string is 170 m/s.

Calculate the wavelength for a standing wave with two antinodes

The given problem states that the string is plucked so that it oscillates and produces a standing wave with two antinodes. In this case, the length of the string is equal to one-half of the wavelength (since there are two antinodes): L = λ / 2 To find the wavelength (λ), we can rearrange the equation: λ = 2 * L λ = 2 * 2.00 m = 4.00 m So the wavelength of the resulting wave is 4.00 m.

Calculate the frequency of the wave

Finally, we can calculate the frequency (f) of the wave using the wave speed (v) and the wavelength (λ) through the formula: v = f * λ To find the frequency (f), we can rearrange the equation: f = v / λ Substituting the calculated values: f = 170 m/s / 4.00 m = 42.5 Hz So the frequency of the resulting wave is 42.5 Hz. In summary, the speed of a wave on this string is 170 m/s, and the resulting standing wave with two antinodes has a wavelength of 4.00 m and a frequency of 42.5 Hz.