The number of independent constituent simple harmonic motions yielding a resultant displacement equation of the periodic motion as `y=8sin^2((t)/(2))sin(10t)` is

Options

A. 8
B. 6
C. 4
D. 3

Explanation

To solve the problem, we need to analyze the given displacement equation of the periodic motion, which is:

\[ y = 8 \sin^2\left(\frac{t}{2}\right) \sin(10t) \]

We will break down the equation step by step to identify the number of independent constituent simple harmonic motions.

### Step 1: Rewrite the Equation
We start with the equation:

\[ y = 8 \sin^2\left(\frac{t}{2}\right) \sin(10t) \]

We can factor out the constants:

\[ y = 4 \cdot 2 \sin^2\left(\frac{t}{2}\right) \sin(10t) \]

### Step 2: Use Trigonometric Identity
Next, we apply the trigonometric identity for \(\sin^2\left(\frac{t}{2}\right)\):

\[ \sin^2\left(\frac{t}{2}\right) = \frac{1 - \cos(t)}{2} \]

Substituting this identity into our equation gives:

\[ y = 4 \cdot 2 \cdot \frac{1 - \cos(t)}{2} \cdot \sin(10t) \]

This simplifies to:

\[ y = 4(1 - \cos(t)) \sin(10t) \]

### Step 3: Expand the Equation
Now we can expand the equation:

\[ y = 4 \sin(10t) - 4 \cos(t) \sin(10t) \]

### Step 4: Use Another Trigonometric Identity
We will now use the identity for the product of sine and cosine:

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

In our case, we can express \(-4 \cos(t) \sin(10t)\) as:

\[ -2 \cdot 2 \sin(10t) \cos(t) = -2 (\sin(10t + t) + \sin(10t - t)) \]

This gives us:

\[ y = 4 \sin(10t) - 2 \sin(11t) - 2 \sin(9t) \]

### Step 5: Identify Independent SHMs
From the final expression:

\[ y = 4 \sin(10t) - 2 \sin(11t) - 2 \sin(9t) \]

We can see that there are three distinct sine terms:

1. \(4 \sin(10t)\)
2. \(-2 \sin(11t)\)
3. \(-2 \sin(9t)\)

Each sine term represents an independent simple harmonic motion.

### Conclusion
Thus, the number of independent constituent simple harmonic motions is **3**.