Final answer:
The tension required for the D-string on a guitar to be properly tuned at a fundamental frequency of 146.8 Hz is approximately 63.97 newtons.
Step-by-step explanation:
To determine the tension a D-string on a guitar must be under for it to be properly tuned at a fundamental frequency of 146.8 Hz with a length of 0.65 m and a mass of 1.4x10-3 kg, the following equation for the fundamental frequency of a string can be used:
f = (1/2L) * √(T/μ)
Where f is the frequency (146.8 Hz), L is the length of the string (0.65 m), T is the tension, and μ is the mass per unit length (mass/length).
To solve for T, we rearrange the equation to get:
T = (μ * (2Lf)²)
Assuming the given string mass and length corresponds to the linear density μ:
- Mass (m) = 1.4x10-3 kg
- Length (L) = 0.65 m
Thus, μ = m/L = 1.4x10-3 kg / 0.65 m = 0.002153846 kg/m
Substituting the values into the tension formula:
T = (0.002153846 kg/m * (2 * 0.65 m * 146.8 Hz)²)
After calculation:
Tension (T) ≈ 63.97 N
The D string must be stretched under a tension of approximately 63.97 newtons to reach the desired pitch.