Two shms are given by y1 and y2 . ThePhase difference btw these after 1 sec.

To determine the phase difference between two simple harmonic motions (SHMs) represented by \( y_1 \) and \( y_2 \) after 1 second, we first need to understand the general form of SHM equations. Typically, SHMs can be expressed as:

Understanding Simple Harmonic Motion

For two SHMs, we can represent them as follows:

• \( y_1 = A_1 \sin(\omega_1 t + \phi_1) \)
• \( y_2 = A_2 \sin(\omega_2 t + \phi_2) \)

Here, \( A \) represents the amplitude, \( \omega \) is the angular frequency, \( t \) is time, and \( \phi \) is the phase constant. The phase difference between the two SHMs at any time \( t \) can be calculated using the phase terms of the sine functions.

Calculating Phase Difference

The phase difference \( \Delta \phi \) at any time \( t \) is given by:

\( \Delta \phi(t) = (\omega_2 t + \phi_2) - (\omega_1 t + \phi_1) \)

After 1 second, this becomes:

\( \Delta \phi(1) = (\omega_2 \cdot 1 + \phi_2) - (\omega_1 \cdot 1 + \phi_1) \)

Which simplifies to:

\( \Delta \phi(1) = (\omega_2 - \omega_1) + (\phi_2 - \phi_1) \)

Example Calculation

Let’s say we have the following parameters:

• \( \omega_1 = 2 \, \text{rad/s} \), \( \phi_1 = 0 \, \text{rad} \)
• \( \omega_2 = 3 \, \text{rad/s} \), \( \phi_2 = \frac{\pi}{4} \, \text{rad} \)

Plugging these values into our phase difference formula gives:

\( \Delta \phi(1) = (3 - 2) + \left(\frac{\pi}{4} - 0\right) \)

Thus:

\( \Delta \phi(1) = 1 + \frac{\pi}{4} \)

This means that after 1 second, the phase difference between the two SHMs is \( 1 + \frac{\pi}{4} \) radians.

Final Thoughts

Understanding how to calculate the phase difference in SHMs is crucial for analyzing wave phenomena in various fields, including physics and engineering. By knowing the angular frequencies and phase constants, you can easily determine how two oscillating systems relate to each other over time.