The equation of the resultant motion of the number of simple harmonic motions is `E_(c)=(1+K sin omega_(2)t) sin omega_(1)t`. The number of simple harmonic components is/are.

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to analyze the given equation of the resultant motion of simple harmonic motions (SHM):

\[ E_c = (1 + K \sin(\omega_2 t)) \sin(\omega_1 t) \]

### Step 1: Expand the Equation
We start by expanding the equation:

\[ E_c = \sin(\omega_1 t) + K \sin(\omega_2 t) \sin(\omega_1 t) \]

### Step 2: Use the Product-to-Sum Formula
Next, we apply the product-to-sum identities to simplify the term \( K \sin(\omega_2 t) \sin(\omega_1 t) \). The product-to-sum formula states:

\[ 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \]

Using this, we can rewrite:

\[ K \sin(\omega_2 t) \sin(\omega_1 t) = \frac{K}{2} \left( \cos((\omega_1 - \omega_2)t) - \cos((\omega_1 + \omega_2)t) \right) \]

### Step 3: Combine Terms
Now, we can substitute this back into our equation:

\[ E_c = \sin(\omega_1 t) + \frac{K}{2} \left( \cos((\omega_1 - \omega_2)t) - \cos((\omega_1 + \omega_2)t) \right) \]

### Step 4: Identify the Components
From the equation, we can identify the components:

1. The term \( \sin(\omega_1 t) \) represents one simple harmonic motion.
2. The term \( \frac{K}{2} \cos((\omega_1 - \omega_2)t) \) represents another simple harmonic motion.
3. The term \( -\frac{K}{2} \cos((\omega_1 + \omega_2)t) \) represents a third simple harmonic motion.

### Conclusion
Thus, we can conclude that the total number of simple harmonic components in the equation is:

**3**