A Changing Ball Weight on Copper Wire: Impact on Wavelength

What is the change in the wavelength of the third harmonic caused by replacing the light ball with the heavy one?

Explanation:

Introduction to the Problem: A vertical, 1.10 m length of 18 gauge (diameter of 1.024 mm) copper wire has a 110.0 N ball hanging from it initially. The question is about the change in the wavelength of the third harmonic caused by replacing this light ball with a heavier one weighing 550.0 N.

Analysis of the Situation:

The situation involves a vertical copper wire with a hanging ball. The change in wavelength is determined by calculating different parameters related to the wire, tension, wave speed, and linear mass density.

Calculation:

According to the provided data, the wavelength is not affected by the change in ball weight. This is because the wave speed is determined by the tension in the wire and the linear mass density of the string, which are not influenced by the ball weight. The tension in the wire with the original ball hanging from it can be calculated using the formula F = mg + T, where m is the mass of the ball, g is the acceleration due to gravity, and T is the tension due to the wire itself. By substituting the values, the tension is found to be 108.82 N. Further calculations involve finding the wave speed and linear mass density of the string to determine the wavelength. The wave speed is calculated to be 296.48 m/s, and using the equation for the wavelength of a standing wave on a string, the new wavelength with the heavier ball is determined to be 0.733 m. Since the tension and wave speed are not affected by the change in ball weight, the wavelength remains the same as the original wavelength of 0.733 m. Conclusion: In conclusion, the change in the weight of the ball hanging from the copper wire does not impact the wavelength of the third harmonic. The wave speed and tension in the wire remain constant, leading to an unchanged wavelength in this scenario.
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