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

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Start by recalling the equation for simple harmonic motion: 𝓍(t) = A cos(ωt + ϕ), where 𝓍(t) is the displacement, A is the amplitude, ω is the angular frequency, t is time, and ϕ is the phase constant.

At t = 0, substitute t = 0 into the equation: 𝓍(0) = A cos(ϕ). This simplifies to 𝓍(0) = A cos(ϕ).

The problem states that at t = 0, the displacement 𝓍 = -A. Substitute 𝓍(0) = -A into the equation: -A = A cos(ϕ).

Divide both sides of the equation by A (assuming A ≠ 0): -1 = cos(ϕ).

To solve for ϕ, use the inverse cosine function: ϕ = cos⁻¹(-1). The value of ϕ that satisfies this equation is π radians (or 180°).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. The motion can be described by a sine or cosine function, characterized by parameters such as amplitude, angular frequency, and phase constant. Understanding SHM is crucial for analyzing oscillatory systems, as it provides the foundational equations that govern the behavior of the mass-spring system.

Phase Constant (ϕ)

The phase constant, denoted as ϕ, determines the initial position of an oscillating object at time t = 0. It shifts the sine or cosine function along the time axis, allowing for the description of various starting conditions in oscillatory motion. In the context of the given problem, finding ϕ is essential to accurately represent the state of the mass when it is at its maximum negative displacement.

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. Knowing the amplitude is vital for determining the phase constant, as it sets the scale for the oscillation and helps in visualizing the motion of the mass.

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Textbook Question

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

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Textbook Question

Determine the phase constant ϕ in Eq. 14–4 if, at t = 0, the oscillating mass is at 𝓍 = ― 1/2 A.

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Textbook Question

A 0.25-kg mass at the end of a spring oscillates 3.2 times per second with an amplitude of 0.15 m. Determine the equation describing the motion of the mass, assuming that at t = 0, 𝓍 was a maximum.

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Textbook Question

An object with mass 2.7 kg is executing simple harmonic motion, attached to a spring with spring constant k = 310 N/m. When the object is 0.020 m from its equilibrium position, it is moving with a speed of 0.60 m/s. Calculate the maximum speed attained by the object.

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