Question:

A string is fixed at both ends and set into oscillation with a frequency of 100 Hz. When the string is observed at a point P located 0.6 meters away from one end, three distinct nodes are formed. The tension in the string is known to be 50 N.

a) Calculate the wavelength (λ) of the wave on the string.

b) Determine the speed (v) of the wave on the string.

c) Find the amplitude (A) of the wave on the string.

d) Explain whether the interference at point P is constructive or destructive.

Answer:

a) The distance between two consecutive nodes in a standing wave is equal to half the wavelength of the wave. Therefore, the distance between the first and third nodes is equal to 1 wavelength. Given that the distance between these nodes is 0.6 meters, we can calculate the wavelength (λ) using the formula:

λ = 2d

Where d is the distance between consecutive nodes.

From the given information, we have: d = 0.6 meters

To calculate the wavelength: λ = 2 × 0.6 meters = 1.2 meters

Thus, the wavelength (λ) of the wave on the string is 1.2 meters.

b) The speed (v) of the wave on a string is given by the equation:

v = fλ

Where f is the frequency of the wave and λ is the wavelength.

From the given information, we have: f = 100 Hz λ = 1.2 meters

To calculate the speed: v = 100 Hz × 1.2 meters = 120 meters/second

Thus, the speed (v) of the wave on the string is 120 meters/second.

c) The amplitude (A) of a wave is the maximum displacement from equilibrium. However, since the amplitude is not explicitly given in the question, we cannot directly calculate it.

d) To determine whether the interference at point P is constructive or destructive, we need to analyze the relationship between the distance of point P from a node or antinode and the phase in the standing wave.

If point P is situated exactly at a node, the two waves interfere destructively at this point, resulting in zero displacement.

If point P is situated exactly at an antinode, the two waves interfere constructively at this point, resulting in maximum displacement.

Since it is mentioned that three distinct nodes are formed at point P, and nodes represent zero displacement, we can conclude that the interference at point P is destructive.

Therefore, the interference at point P is destructive.