Question

One end of a horizontal string is attached to a small-amplitude mechanical 60.0-hz oscillator. The string's mass per unit length is ____. The string passes over a pulley, a distance ____ away, and weights are hung from this end. What mass m must be hung from this end of the string to produce (a) one loop, (b) two loops, and (c) five loops of a standing wave? Assume the string at the oscillator is a node, which is nearly true?

Answer 1

This question involves the concepts of standing waves, tension, frequency, and mass on a string. The mass required for different modes of standing waves can be calculated using the relationship between tension, gravity, wave speed, and linear density of the string. Key formulas include those for tension in terms of mass and gravity, wave speed in terms of tension and linear density, and wavelength in terms of string length and mode number.

Step-by-step explanation:

The question involves the physical principles of standing waves on a string in relation to mass, tension, frequency, and wave speed. To determine the required mass to produce one, two, and five loops of a standing wave, the following formula can be used:

Tension (T) = mass (m) × gravity (g), where gravity (g) is typically 9.81 m/s2. The tension in the string is also related to the speed of the standing wave (v) and the string's linear density (μ) via v = √(T/μ). For a given mode of standing wave, represented by n, on a string of length L, the wavelength (λ) can be calculated using λ = 2L/n. With these relationships, you can derive the mass required to produce the standing wave patterns.

For instance, for one loop (first harmonic), n would be 1, and for two loops (second harmonic), n would be 2, and accordingly for five loops (fifth harmonic), n would be 5. The frequency of the oscillator connected to the string determines the standing wave pattern, with the frequency and wavelength related by f = v/λ. To create standing waves at certain harmonics, this educational physics problem requires a formula combination and an understanding of how tension, mass, frequency, and wavelength interact.