A guitar string has a tension of 100 N and is supposed to have a frequency of 110 Hz. When a standard tone of that value is sounded while the string is plucked, a beat frequency of 2 Hz is heard. The peg holding the string is loosened (decreasing the tension) and the beat frequency increases. What should the tension in the string be in order to achieve perfect pitch?

F = 110 - 2 = 108 Hz.

Tension = (110Hz/108Hz) * 100N=101.85 N.

To determine the tension in the string needed to achieve perfect pitch, we need to understand the relationship between tension, frequency, and beat frequency.

In this scenario, the guitar string is initially under a tension of 100 N and is supposed to have a frequency of 110 Hz. When plucked, a beat frequency of 2 Hz is heard.

First, let's understand what is happening when we hear a beat frequency. In music, a beat occurs when two slightly different frequencies are played simultaneously. When these frequencies are close to each other, they interfere and create a sound with a pulsating or "beating" quality. The beat frequency is the difference between the two frequencies. In this case, the beat frequency is 2 Hz.

Now, let's consider the effect of changing the tension on the beat frequency. When the peg holding the string is loosened, the tension decreases. This decrease in tension will result in a decrease in the frequency of the string because tension affects the speed at which waves propagate along the string. As a result, the beat frequency increases.

To achieve perfect pitch, we need to adjust the tension in the string so that the beat frequency is zero. A beat frequency of zero means that there is no interference between frequencies and the string's frequency matches the desired frequency (110 Hz).

To find the required tension for perfect pitch, we need to analyze the relationship between tension and frequency. The frequency of a string under tension can be calculated using the fundamental frequency equation:

f = (1/2L) * sqrt(T/μ)

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

Since the length and linear mass density remain constant, we can assume that they are not changing in this scenario. Therefore, we can rewrite the equation as:

f ∝ sqrt(T)

Now let's analyze the relationship between the beat frequency and the difference in tension. We know that a change in tension causes a change in beat frequency:

Δf beat = k * ΔT

Where:
- Δf beat is the change in beat frequency
- k is a constant

From the given information, we know that the initial beat frequency is 2 Hz. To achieve perfect pitch, we need to find the tension at which the beat frequency becomes zero.

0 = k * ΔT

Since zero is equal to any constant multiplied by ΔT, we can conclude that ΔT (the change in tension) is zero when the beat frequency is zero, indicating perfect pitch.

Therefore, the required tension in the string to achieve perfect pitch is 100 N.