Calculate Phase Angle Differences Between Position Displacement and Velocity

What is Phase Angle?

The phase angle represents the relative timing between two periodic functions, such as displacement and velocity in harmonic motion. It measures the phase difference in radians or degrees between the two quantities.

In simple harmonic motion, the phase angle (φ) relates the displacement (x) and velocity (v) of an object. When an object moves in simple harmonic motion, its displacement and velocity are 90 degrees out of phase with each other. This means that when displacement is at its maximum, velocity is zero, and vice versa.

Calculating Phase Angle Differences

To calculate the phase angle difference between position displacement and velocity, you need to know the amplitude and angular frequency of the system. The phase angle is determined by the relationship between these quantities.

The phase angle difference between displacement and velocity in simple harmonic motion is always π/2 radians (90 degrees). This is because velocity is the derivative of displacement with respect to time, introducing a 90-degree phase shift.

The Formula

The phase angle difference (φ) between displacement (x) and velocity (v) in simple harmonic motion is given by:

Where:

• φ is the phase angle difference in radians
• v is the velocity
• ω is the angular frequency (2πf, where f is frequency)
• x is the displacement

For simple harmonic motion, this simplifies to φ = π/2 radians (90 degrees).

Worked Example

Let's calculate the phase angle difference for a system with:

• Displacement (x) = 2 meters
• Velocity (v) = 3 m/s
• Angular frequency (ω) = 5 rad/s

Using the formula:

Converting to degrees: 0.2915 × (180/π) ≈ 16.75 degrees

This shows the phase angle difference is approximately 16.75 degrees, which is close to the expected 90 degrees for simple harmonic motion.