FIGURE EX15.7 is the position-versus-time graph of a particle in simple harmonic motion. What is the phase constant?

Verified step by step guidance

1. Step 1: Recall the equation for simple harmonic motion: x(t) = A * cos(ωt + φ), where A is the amplitude, ω is the angular frequency, t is time, and φ is the phase constant. The phase constant determines the initial position of the particle at t = 0.
2. Step 2: Analyze the graph to determine the initial position of the particle at t = 0. From the graph, at t = 0, the position x is at its maximum value, which is 20 cm.
3. Step 3: Substitute the initial condition into the equation x(0) = A * cos(φ). Since x(0) = 20 cm and A = 20 cm (the amplitude), the equation becomes 20 = 20 * cos(φ). Simplify to find cos(φ) = 1.
4. Step 4: Use the property of the cosine function to determine φ. If cos(φ) = 1, then φ = 0 radians (or 0 degrees). This indicates that the particle starts at its maximum displacement.
5. Step 5: Conclude that the phase constant φ is 0 radians, meaning the particle starts at its maximum position in the positive direction.

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 is characterized by a restoring force proportional to the displacement from the equilibrium, leading to sinusoidal position, velocity, and acceleration graphs. In SHM, the position of the particle can be described by a sine or cosine function, which reflects the repetitive nature of the motion.

Phase Constant

The phase constant is a parameter in the mathematical description of oscillatory motion that determines the initial position of the particle at time t=0. It shifts the sine or cosine function along the time axis, allowing for the representation of different starting points in the oscillation. The phase constant is crucial for accurately describing the motion of the particle in relation to its equilibrium position.

Position-Time Graph

A position-time graph visually represents the position of an object as a function of time. In the context of simple harmonic motion, this graph typically shows a sinusoidal pattern, indicating the periodic nature of the motion. The peaks and troughs of the graph correspond to the maximum and minimum displacements from the equilibrium position, while the x-axis represents time, allowing for the analysis of the motion's frequency and amplitude.