The transverse standing wave on a string fixed at both ends is vibrating at its fundamental frequency of 250 Hz. What would be the fundamental frequency on a piece of the same string that is twice as long and has four times the tension?

250

To find the fundamental frequency on a string that is twice as long and has four times the tension, we can use the formula for the fundamental frequency of a string:

f = (1/2L) * sqrt(T/μ)

Where:
- f is the fundamental frequency
- L is the length of the string
- T is the tension in the string
- μ is the linear mass density of the string

In this case, the original string vibrating at a fundamental frequency of 250 Hz with a certain length and tension. Let's call these values L1 and T1.

We want to calculate the fundamental frequency (f2) when the length (L2) is twice as long and the tension (T2) is four times greater.

Given that L2 = 2L1 and T2 = 4T1, we can substitute these values into the formula:

f2 = (1/2L2) * sqrt(T2/μ)

To find the value of f2, we need to know the linear mass density (μ) of the string. Linear mass density is calculated by dividing the mass of the string by its length:

μ = (m/L1)

However, we don't have the mass of the string. To proceed, we need to assume that the mass per unit length of the string remains constant, which is a common assumption for strings of the same material.

This assumption means that the linear mass density (μ) is the same for both strings (L1 and L2).

With this assumption, we can simplify the formula to:

f2 = (1/2L2) * sqrt(T2/μ)
= (1/2 * 2L1) * sqrt(4T1/μ)
= (1/L1) * sqrt(4T1/μ)
= 2 * sqrt(T1/μ)

Since T1/μ is a constant for a given string material, we can conclude that the fundamental frequency on a piece of the same string that is twice as long and has four times the tension will be twice the fundamental frequency of the original string.

Therefore, the fundamental frequency on the new string would be 2 * 250 Hz = 500 Hz.