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

Verified step by step guidance

1. Identify the general form of the equation for simple harmonic motion, which is typically given by x(t) = A cos(ωt + ϕ), where A is the amplitude, ω is the angular frequency, t is the time, and ϕ is the phase constant.
2. Substitute the given values into the equation. Here, at t = 0, x = -1/2 A. Plugging these values in, we get -1/2 A = A cos(ϕ).
3. Simplify the equation to isolate the cosine term: cos(ϕ) = -1/2.
4. Determine the values of ϕ that satisfy the equation cos(ϕ) = -1/2. Recall that cosine of an angle is -1/2 at specific standard angles in the unit circle.
5. Identify the principal value of ϕ and consider the periodic nature of the cosine function to list possible values of ϕ that satisfy the condition in the context of the problem.

Key Concepts

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 motion.

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 SHM, knowing the phase constant is essential for accurately predicting the position and velocity of the mass at any given time.

Amplitude (A)

Amplitude is the maximum displacement of an oscillating object from its equilibrium position. It represents the extent of the oscillation and is a key parameter in the equations of SHM. In this scenario, the relationship between the phase constant and the amplitude is critical, as it helps determine the specific position of the mass at t = 0, which is given as 𝓍 = -1/2 A.