91Ó°ÊÓ

In simple harmonic motion, the concept of phase difference is crucial to understanding the relative positions or motions of oscillating particles at a given time. It measures how a wave is shifted in relation to another wave. Simply put, phase difference can indicate which wave is ahead or lagging.
In this exercise, we focus on the phase difference between the velocity equations of two particles. Velocity, being the derivative of displacement, means that its phase can differ from the displacement itself. Examining velocity allows us to understand the timing of these motions more precisely.
To calculate the phase difference here, we concentrate on the arguments inside the trigonometric functions that represent velocities. We find this by subtracting the phase angle of one particle's velocity from the phase angle of the other. The result shows us how far one velocity's wave is shifted from the other. In our problem, after simplification, the phase difference turns out to be \(\frac{\pi}{6}\). This means that one particle reaches the same point in its cycle \(\frac{\pi}{6}\) radians later or earlier than the other.

When dealing with simple harmonic motion, velocity equations are derived by taking the time derivative of the displacement equations. This process is central because velocity represents the rate of change of position. The derivatives often contain cosines or sines, just like the original displacement equations, but shifted by half a period.
For particle 1, we started with the equation \(y_1 = 0.1 \sin(100\pi t + \pi/3)\). Taking the derivative gives us the velocity: \(y_1'(t) = 0.1(100\pi)\cos(100\pi t + \pi/3) \). Note the transition from sine to cosine indicates a shift in phase of \(\pi/2\) radians.
For particle 2, from the equation \(y_2 = 0.1\cos(\pi t)\), the derived velocity is \(y_2'(t) = -0.1\pi\sin(\pi t)\). Notice the introduction of a negative sign and the conversion to sine form, illustrating the change in phase perception. In both cases, these transformations highlight the intricacy involved in moving from position to the speed of the oscillating particles.

Trigonometric identities are mathematical tools that allow us to simplify expressions and solve equations involving angles. In this exercise, identities are pivotal in transforming sine to cosine or vice versa to draw parallels between velocity equations.
One key identity utilized here is \( \sin(x) = \cos(\frac{\pi}{2} - x)\), which allows us to express a sine function in terms of cosine. This is particularly helpful in aligning two equations so their phase difference can be calculated. For particle 2, \(y_2'(t) = -0.1\pi\sin(\pi t)\) is converted to cosine format as \(-0.1\pi\cos(\pi t + \pi/2)\), making phase comparisons feasible.
Understanding these identities deepens our ability to interpret the functions in different forms and is key in solving problems related to wave interaction, such as this one. Having a strong grasp of how these identities work prepares students to tackle more complex oscillation and wave phenomena.