A string with a length of 1.3 m is fixed at both ends. What is the longest possible wavelength for a standing wave on this string?
To find the longest possible wavelength for a standing wave on a string fixed at both ends, you need to consider the fundamental frequency, which corresponds to a standing wave with the longest wavelength.
The fundamental frequency, f, of a standing wave on a string can be calculated using the formula:
f = v / (2L)
where v is the velocity of the wave and L is the length of the string.
In this case, the length of the string is given as 1.3 m. The velocity of a wave traveling along a string depends on the tension in the string and the mass per unit length of the string. However, since the tension and mass per unit length are not provided, we cannot calculate the exact velocity. Therefore, we need to make some assumptions.
For simplicity, let's assume the string is a light string under moderate tension. In such a case, we can consider the wave velocity to be approximately equal to the speed of a transverse wave in a string, which is about 100 m/s.
Substituting the values into the formula, we have:
f = 100 m/s / (2 * 1.3 m)
f ≈ 100 m/s / 2.6 m
f ≈ 38.46 Hz
The frequency, f, is inversely proportional to the wavelength, λ, of the standing wave:
λ = v / f
Substituting the values we have:
λ ≈ 100 m/s / 38.46 Hz
λ ≈ 2.6 m
Therefore, the longest possible wavelength for a standing wave on this string is approximately 2.6 meters.
To find the longest possible wavelength for a standing wave on a string, we need to consider the boundary conditions. In this case, the string is fixed at both ends, which means that the ends of the string must be nodes for the standing wave.
In a standing wave, the distance between two consecutive nodes (or antinodes) corresponds to half a wavelength. Since the string is fixed at both ends, the longest possible wavelength occurs when the distance between the two ends is exactly half a wavelength.
Given that the string has a length of 1.3 m, we can consider half of this length as the distance between the two ends. So, the longest possible wavelength (λ) for a standing wave can be calculated as:
λ = 2 * (Distance between ends)
λ = 2 * (1.3 m / 2)
λ = 1.3 m
Thus, the longest possible wavelength for a standing wave on this string is 1.3 meters.