Find the displacement equation of the simple harmonic motion obtained by combining the motion.
`x_(1) = 2sin omega t`, `x_(2) = 4sin (omega t + (pi)/(6))` and `x_(3) = 6sin (omega t + (pi)/(3))`

Explanation

To find the displacement equation of the simple harmonic motion obtained by combining the motions given by the equations:

1. \( x_1 = 2 \sin(\omega t) \)
2. \( x_2 = 4 \sin\left(\omega t + \frac{\pi}{6}\right) \)
3. \( x_3 = 6 \sin\left(\omega t + \frac{\pi}{3}\right) \)

we can follow these steps:

### Step 1: Identify the Amplitudes and Phases
- For \( x_1 \): Amplitude \( A_1 = 2 \), Phase \( \phi_1 = 0 \)
- For \( x_2 \): Amplitude \( A_2 = 4 \), Phase \( \phi_2 = \frac{\pi}{6} \)
- For \( x_3 \): Amplitude \( A_3 = 6 \), Phase \( \phi_3 = \frac{\pi}{3} \)

### Step 2: Convert to Phasor Representation
We can represent each motion as a phasor in the complex plane:
- \( x_1 \) corresponds to \( 2 \) at \( 0 \) degrees.
- \( x_2 \) corresponds to \( 4 \) at \( 30 \) degrees (since \( \frac{\pi}{6} = 30^\circ \)).
- \( x_3 \) corresponds to \( 6 \) at \( 60 \) degrees (since \( \frac{\pi}{3} = 60^\circ \)).

### Step 3: Calculate the X and Y Components
Using the cosine and sine components:
- For \( x_1 \):
- \( A_{x1} = 2 \cos(0) = 2 \)
- \( A_{y1} = 2 \sin(0) = 0 \)

- For \( x_2 \):
- \( A_{x2} = 4 \cos\left(\frac{\pi}{6}\right) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} \)
- \( A_{y2} = 4 \sin\left(\frac{\pi}{6}\right) = 4 \cdot \frac{1}{2} = 2 \)

- For \( x_3 \):
- \( A_{x3} = 6 \cos\left(\frac{\pi}{3}\right) = 6 \cdot \frac{1}{2} = 3 \)
- \( A_{y3} = 6 \sin\left(\frac{\pi}{3}\right) = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3} \)

### Step 4: Sum the Components
Now, we sum the x and y components:
- Total \( A_x = A_{x1} + A_{x2} + A_{x3} = 2 + 2\sqrt{3} + 3 \)
- Total \( A_y = A_{y1} + A_{y2} + A_{y3} = 0 + 2 + 3\sqrt{3} \)

### Step 5: Calculate the Resultant Amplitude
Using the Pythagorean theorem:
\[ A = \sqrt{A_x^2 + A_y^2} \]

### Step 6: Calculate the Phase Angle
The phase angle \( \phi \) can be found using:
\[ \tan(\phi) = \frac{A_y}{A_x} \]

### Step 7: Write the Final Displacement Equation
The displacement equation can then be expressed as:
\[ x(t) = A \sin(\omega t + \phi) \]

### Final Calculation
1. Calculate \( A_x \) and \( A_y \) numerically.
2. Calculate \( A \) and \( \phi \).
3. Substitute these values into the displacement equation.