Question:

A 2.0 m long rope is fixed at both ends and has a tension of 100 N. A sinusoidal wave with a frequency of 10 Hz is generated at one end of the rope. At a certain point along the rope, the wave interferes with its reflection to create a standing wave. What is the wavelength of the standing wave?

(A) 0.2 m (B) 0.4 m (C) 0.6 m (D) 0.8 m

Answer:

The wavelength (λ) of a standing wave on a rope is related to the length of the rope (L) and the harmonic number (n) by the equation:

λ = 2L / n

Where n is the harmonic number (the number of loops or segments that form the standing wave).

To find the wavelength of the standing wave, we need to determine the harmonic number at which the standing wave is formed. For a wave reflected at a fixed end, only odd harmonics (n = 1, 3, 5, etc.) can form standing waves.

Given that the rope length (L) is 2.0 m, and the standing wave is created by interference with its reflection, we can find the wavelength as follows:

For the first harmonic (n = 1):

λ = 2L / 1 = 2 * 2.0 / 1 = 4.0 m

For the third harmonic (n = 3):

λ = 2L / 3 = 2 * 2.0 / 3 = 1.3 m

For the fifth harmonic (n = 5):

λ = 2L / 5 = 2 * 2.0 / 5 = 0.8 m

For the seventh harmonic (n = 7):

λ = 2L / 7 = 2 * 2.0 / 7 ≈ 0.6 m

The wavelength of the standing wave is 0.6 m, so the correct answer is (C) 0.6 m.