A string can have a "free" end if that end is attached to a...

Name: Problem 85, Chapter 15: Physics for Scientists and Engineers with Modern Physics
Uploaded: 2020-11-11T20:43:27-08:00
Duration: 2 min 5 s
Channel: Sachin Rao
Description: A string can have a "free" end if that end is attached to a ring that can slide without friction on a vertical pole (Fig. 40). Determine the wavelengths of the resonant vibrations of such a string with one end fixed and the other free.

A string can have a "free" end if that end is attached to a ring that can slide without friction on a vertical pole (Fig. 40). Determine the wavelengths of the resonant vibrations of such a string with one end fixed and the other free.

Key Concepts

Standing Waves

Standing waves occur when waves of the same frequency and amplitude traveling in opposite directions interfere, creating nodes and antinodes that appear stationary. In the context of vibrating strings, the interference results from the superposition of the incident and reflected waves, forming distinct patterns that depend on the length of the string and its boundary conditions.

Boundary Conditions in Vibrations

Boundary conditions determine how a string is constrained at its ends, which in turn affects the formation of standing waves. A fixed end enforces zero displacement (a node), whereas a free end allows maximum displacement (an antinode). These conditions critically influence the positions of nodes and antinodes, and therefore dictate the possible wavelengths and frequencies of the resonant vibrations.

Resonant Frequencies and Mode Shapes

Resonant frequencies are the specific frequencies at which a system naturally oscillates with the largest amplitude due to constructive interference of waves. For a string with one fixed end and one free end, only certain vibrational modes are allowed, resulting in quantized wavelengths that meet the prescribed boundary conditions. This quantization leads to a discrete set of resonant frequencies.

Wavelength Quantization in Heterogeneous Boundary Systems

In systems with different boundary conditions at each end, such as a fixed end paired with a free end, the allowed wavelengths are determined by the necessity to have a node at the fixed end and an antinode at the free end. This requirement typically yields a relationship where the resonant wavelengths are a function of the string’s length divided by odd fractions, often expressed in the form ? = 4L/(2n+1), where n is an integer starting from zero.

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