A string under tension has a fundamental frequency of 220 Hz. What is the fundamental frequency if the tension is doubled?

Verified step by step guidance

1. Identify the relationship between the tension in the string and the frequency of vibration. The frequency of a vibrating string is proportional to the square root of the tension. Mathematically, this is represented as f \propto \sqrt{T}, where f is the frequency and T is the tension.
2. Set up the proportionality equation for the initial condition. Let f_1 = 220 Hz be the initial frequency and T_1 be the initial tension. The relationship can be written as f_1 = k \sqrt{T_1}, where k is the proportionality constant.
3. Express the new frequency f_2 in terms of the new tension T_2, which is double the initial tension (T_2 = 2T_1). Using the same proportionality constant, the equation becomes f_2 = k \sqrt{T_2}.
4. Substitute T_2 = 2T_1 into the equation for f_2 to find f_2 = k \sqrt{2T_1}.
5. Using the relationship between f_1 and f_2, and knowing that f_1 = k \sqrt{T_1}, solve for f_2 in terms of f_1. This gives f_2 = f_1 \sqrt{2}. Calculate f_2 to find the new fundamental frequency when the tension is doubled.

Key Concepts

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system oscillates. In the context of a string, it is determined by the string's length, tension, and mass per unit length. This frequency is crucial for understanding how musical notes are produced and perceived.

Tension in a String

Tension refers to the force applied along the length of a string, which affects its vibration characteristics. Increasing the tension in a string raises its fundamental frequency, as a tighter string vibrates faster. This relationship is essential for analyzing how changes in tension influence sound production.

Relationship Between Tension and Frequency

The relationship between tension and frequency in a vibrating string is described by the formula f = (1/2L)√(T/μ), where f is the frequency, L is the length of the string, T is the tension, and μ is the mass per unit length. Doubling the tension results in an increase in frequency, specifically by a factor of √2, illustrating how tension directly impacts the pitch of the sound produced.