Determine the phase constant ϕ in Eq. 14–4 if, at t = 0, the oscillating mass is at 𝓍 = A .

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Understand the context of the problem: The equation for simple harmonic motion is given as 𝓍(t) = A cos(ωt + ϕ), where 𝓍(t) is the displacement at time t, A is the amplitude, ω is the angular frequency, and ϕ is the phase constant. We are tasked with finding the phase constant ϕ when the displacement 𝓍 = A at t = 0.

Substitute the given conditions into the equation: At t = 0, the displacement 𝓍 = A. Substituting these values into the equation 𝓍(t) = A cos(ωt + ϕ), we get A = A cos(ϕ).

Simplify the equation: Divide both sides of the equation by A (assuming A ≠ 0), which gives 1 = cos(ϕ).

Interpret the result: The cosine of the phase constant ϕ is equal to 1. From trigonometry, we know that cos(ϕ) = 1 when ϕ = 0 radians (or 0 degrees).

Conclude the solution: Therefore, the phase constant ϕ is 0 radians in this case, as it satisfies the condition that 𝓍 = A at t = 0.

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주요 개념

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

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. The motion is characterized by a restoring force proportional to the displacement from the equilibrium, leading to sinusoidal motion. In SHM, parameters such as amplitude, frequency, and phase constant are crucial for describing the motion's characteristics.

Amplitude (A)

Amplitude is the maximum displacement of an oscillating object from its equilibrium position. In the context of SHM, it represents the peak value of the oscillation, indicating how far the object moves from the center point during its motion. The amplitude is a key factor in determining the energy of the oscillating system, as greater amplitudes correspond to higher energy levels.

Phase Constant (ϕ)

The phase constant, denoted as ϕ, is a parameter in the equation of motion for SHM that determines the initial position of the oscillating object at time t = 0. It effectively shifts the sine or cosine function used to describe the motion, allowing for the accurate representation of the system's state at the start of the observation. The value of ϕ is crucial for solving problems related to the timing and position of oscillations.