A 2.0-m-long string vibrates at its second-harmonic frequency with a maximum amplitude of 2.0 cm. One end of the string is at x = 0 cm. Find the oscillation amplitude at x = 10, 20, 30, 40, and 50 cm.

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Step 1: Understand the problem. The string is vibrating in its second harmonic, which means it has two antinodes and one node (excluding the fixed ends). The wave equation for the standing wave can be expressed as y(x, t) = A * sin(kx) * cos(ωt), where A is the maximum amplitude, k is the wave number, and ω is the angular frequency. The amplitude at any point x is determined by the term A * sin(kx).

Step 2: Determine the wave number k. The wave number k is related to the wavelength λ by the equation k = 2π/λ. For the second harmonic, the wavelength is equal to the length of the string (L) divided by 2, so λ = L/2. Substituting L = 2.0 m, we find λ = 1.0 m. Therefore, k = 2π/λ = 2π/1.0 = 2π rad/m.

Step 3: Write the expression for the amplitude at any position x. The amplitude at position x is given by A(x) = A_max * sin(kx), where A_max is the maximum amplitude of the wave (2.0 cm) and k = 2π rad/m. Convert A_max to meters: A_max = 2.0 cm = 0.02 m.

Step 4: Calculate the amplitude at each specified position. Substitute x = 10 cm, 20 cm, 30 cm, 40 cm, and 50 cm into the equation A(x) = A_max * sin(kx). Convert x to meters before substituting: x = 10 cm = 0.10 m, x = 20 cm = 0.20 m, and so on. For each x, calculate sin(kx) using k = 2π rad/m.

Step 5: Interpret the results. The amplitude at each position will vary depending on the sine function. Note that the amplitude will be zero at nodes (where sin(kx) = 0) and maximum at antinodes (where sin(kx) = ±1). Use this understanding to verify the calculated amplitudes at each position.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Harmonic Frequencies

Harmonic frequencies refer to the specific frequencies at which a system, such as a vibrating string, can oscillate. The second harmonic, for instance, is the first overtone and occurs when the string vibrates in two segments, producing a frequency that is twice that of the fundamental frequency. Understanding these harmonics is essential for analyzing the behavior of the string at various points along its length.

Amplitude of Oscillation

Amplitude is the maximum extent of a vibration or oscillation, measured from the position of equilibrium. In this context, the maximum amplitude of 2.0 cm indicates how far the string moves from its rest position during oscillation. The amplitude can vary along the length of the string, especially in different harmonic modes, affecting the displacement at specific points.

Wave Equation and Node/Antinode Locations

The wave equation describes how waves propagate through a medium, and in the case of a vibrating string, it helps determine the positions of nodes (points of no displacement) and antinodes (points of maximum displacement). For the second harmonic, there are nodes at both ends of the string and an antinode at the center. This understanding is crucial for calculating the oscillation amplitude at various points along the string.