91Ó°ÊÓ

In simple harmonic motion (SHM), the amplitude is a critical parameter. It represents the maximum displacement from the equilibrium position.
In the given displacement-time equation:
\[ x = 4 \sin \omega t + 3 \sin (\omega t + \pi / 3) \]
You are dealing with the superposition of two harmonic motions.
Here's how we calculate the resulting amplitude:

• The first component has amplitude 4.
• The second component has amplitude 3.

To find the total amplitude when these oscillations are combined, use the formula:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos \phi} \]
where:

• \( A \) is 4 (amplitude of the first term)
• \( B \) is 3 (amplitude of the second term)
• \( \phi \) is the phase difference, which in this case is \( \pi / 3 \)

Substitute the given values:
\[ R = \sqrt{4^2 + 3^2 + 2 \times 4 \times 3 \times 0.5} = \sqrt{37} \approx 6.08 \text{ cm} \]
This shows that the resultant amplitude is approximately 6.08 cm, making it closest to 6 cm.

Trigonometric identities are essential tools in mathematics and physics, especially when dealing with oscillations and waves.
These identities simplify complex expressions and help in solving the displacement-time equation in SHM.
For this exercise, the identity for sum of sines is key:

• \( \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \)

This identity allows the combination of two wave functions.
It results in a simplified form that can be easier to analyze or use in other calculations.
In this example, we used the identity to handle the two components \( 4 \sin \omega t \) and \( 3 \sin (\omega t + \pi / 3) \).
By knowing such identities, we turn tricky trigonometric problems into more manageable parts!

Phase addition formulas are used to combine wave functions with different phases. They are quite handy in SHM.
For any two sine terms, like in this problem:

• First term: \( 4 \sin \omega t \) with no additional phase
• Second term: \( 3 \sin (\omega t + \pi / 3) \) with a phase shift

To find the resulting phase and amplitude, the phase addition formula comes into play:
The standard approach involves the identity:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
This breaks down the terms for easier addition or manipulation.
Given that both components have specific amplitudes, their phase difference \( \phi = \pi / 3 \) impacts the overall oscillation pattern.
Understanding phase relationships in waves enables us to calculate superpositions like this one accurately.
It's just another example of how many different aspects of wave physics are interconnected!