Final answer:
To find the distance between two points on a vibrating string with half the maximum amplitude in the fundamental mode, calculate a quarter of the wavelength based on the total length of the string and then double it.
Step-by-step explanation:
The question is about calculating the distance between two points on a string vibrating in its fundamental mode with varying amplitudes. In the fundamental mode, the shape of the string is a standing wave that looks like a sine curve, and the points along the string will have amplitudes that vary from zero at the ends to a maximum at the center.
Given that the amplitude at the center is 4 mm, and we're looking for the distance between points where the amplitude is 2 mm, we can visualize this as finding the points on the sine curve that are at half the maximum amplitude. Since the string is symmetric, we are effectively looking for the points where the sine function has half its maximum value, which will be at quarter wavelengths from the center.
As such, the wavelength λ of the fundamental mode is twice the length of the string, so λ = 1.5 m × 2 = 3.0 m. Then, a quarter wavelength (λ/4) is 3.0 m / 4 = 0.75 m. Therefore, the distance between the two points with half the amplitude is twice the quarter wavelength, which is 0.75 m × 2 = 1.5 m.