A sonometer sir has length 0.1m under tension of 2kgwt when sounded together a tuning fork of 502hz,produces 2beats persecomd. When the fork is loaded with wax and sounded no beats are heard. Then linear density is

To determine the linear density of the sonometer string, we can use the information provided about the tension, frequency, and the behavior of the tuning fork. Let's break this down step by step.

Understanding the Problem

We have a sonometer string with a length of 0.1 m and a tension of 2 kgwt (kilogram-weight). The tuning fork has a frequency of 502 Hz and produces 2 beats per second when sounded together with the string. When the fork is loaded with wax, no beats are heard, indicating that the frequency of the fork matches the frequency of the vibrating string.

Key Concepts

• Frequency of a String: The frequency of a vibrating string can be calculated using the formula: f = (1/2L) * √(T/μ) where:
  • f = frequency of the string
  • L = length of the string
  • T = tension in the string
  • μ = linear density of the string
• Beats: The number of beats per second is the difference in frequencies of the two sources. Here, the frequency of the string when it produces beats with the fork is either 500 Hz or 504 Hz (502 Hz ± 2 Hz).

Calculating the Linear Density

First, we need to find the tension in Newtons. Since 1 kgwt is approximately equal to 9.81 N, the tension can be calculated as follows:

T = 2 kgwt × 9.81 N/kgwt = 19.62 N

Next, we can use the frequency formula. Since the fork produces 2 beats per second, we can assume that the frequency of the string is either 500 Hz or 504 Hz. Let's use 500 Hz for our calculations:

500 Hz = (1/2 × 0.1 m) * √(19.62 N/μ)

Rearranging this equation to solve for μ (linear density), we get:

μ = (19.62 N) / (4 × (500 Hz)² × (0.1 m)²)

Now, let's calculate it step by step:

μ = 19.62 N / (4 × 250000 Hz² × 0.01 m²)

μ = 19.62 N / (10000 N/m²)

μ = 0.001962 kg/m

Final Result

The linear density of the sonometer string is approximately 0.001962 kg/m. This value indicates how much mass is distributed along each meter of the string, which is crucial for understanding its vibrational properties.

In summary, by analyzing the tension, frequency, and the behavior of the tuning fork, we were able to calculate the linear density of the sonometer string effectively. If you have any further questions or need clarification on any of the steps, feel free to ask!