Question

The fundamental frequency of vibration of a string stretched between two rigid support is 50 Hz. The mass of the string is 18 g and its linear mass density is 20 g/m. The speed of the transverse waves so produced in the string is_______ ms^-1.

Answer 1

Correct Answer: 90

Understanding the Problem

We are given the mass and the linear mass density of the string, and we are asked to find the speed of transverse waves in the string. The formula for the speed \( V \) of waves on a string is:

\( V = \sqrt{\frac{T}{\mu}} \),

where \( T \) is the tension in the string, and \( \mu \) is the linear mass density of the string.

Finding the Tension

From the fundamental frequency formula for a vibrating string:

\( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \),

where \( f \) is the fundamental frequency, \( L \) is the length of the string, \( T \) is the tension, and \( \mu \) is the linear mass density. Rearranging for \( T \), we get:

\( T = (2L f)^2 \mu \).

We are not given \( L \), but the speed of the wave can also be determined directly from the tension and linear mass density using the formula \( V = \sqrt{\frac{T}{\mu}} \). To find the speed, we proceed to use the given information directly.

Substituting Known Values

Given:

\( \mu = 20 \, \text{g/m} = 20 \times 10^{-3} \, \text{kg/m}, \quad T = 18 \, \text{g} = 18 \times 10^{-3} \, \text{kg}. \)

Note: There is an error here. Tension should be in Newtons, not kg. The given value is mass, not tension. We will assume the tension is given in Newtons.

Assuming T = 18 N, then:

Using the formula for the speed of waves on a string:

\( V = \sqrt{\frac{T}{\mu}} \).

Substitute the known values:

\( V = \sqrt{\frac{18}{20 \times 10^{-3}}} = \sqrt{\frac{18000}{20}} = \sqrt{900} = 30 \, \text{m/s}. \)

Corrected Final Answer

Assuming the tension is 18 N, the speed of the wave is 30 m/s.

Note: The original solution incorrectly used the mass in grams as the tension in Newtons. Please verify the value of the tension.