The displacement `y` of a particle executing periodic motion is given by `y = 4 cos^(2) ((1)/(2)t) sin(1000t)`
This expression may be considereed to be a result of the superposition of

Options

A. two
B. three
C. four
D. five

Explanation

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

\[ y = 4 \cos^2\left(\frac{1}{2} t\right) \sin(1000t) \]

We will simplify this expression using trigonometric identities to determine how many simple harmonic motions (SHMs) are superimposed in this equation.

### Step-by-step Solution:

1. **Rewrite the Equation**:
Start with the given equation:
\[ y = 4 \cos^2\left(\frac{1}{2} t\right) \sin(1000t) \]

2. **Use the Trigonometric Identity**:
We can use the trigonometric identity:
\[ \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \]
Here, let \(\theta = \frac{1}{2} t\). Thus:
\[ \cos^2\left(\frac{1}{2} t\right) = \frac{1 + \cos(t)}{2} \]

3. **Substitute the Identity**:
Substitute the identity back into the equation:
\[ y = 4 \left(\frac{1 + \cos(t)}{2}\right) \sin(1000t) \]
Simplifying this gives:
\[ y = 2(1 + \cos(t)) \sin(1000t) \]

4. **Distribute the Terms**:
Distributing the terms:
\[ y = 2 \sin(1000t) + 2 \cos(t) \sin(1000t) \]

5. **Use Another Trigonometric Identity**:
Now, we can use the identity:
\[ 2 \cos(a) \sin(b) = \sin(a + b) - \sin(a - b) \]
Here, let \(a = t\) and \(b = 1000t\):
\[ 2 \cos(t) \sin(1000t) = \sin(1001t) - \sin(999t) \]

6. **Combine All Terms**:
Now, combine all the terms:
\[ y = 2 \sin(1000t) + \sin(1001t) - \sin(999t) \]

7. **Identify the Superposition**:
The final expression can be viewed as the superposition of three different SHMs:
- \(y_1 = 2 \sin(1000t)\)
- \(y_2 = \sin(1001t)\)
- \(y_3 = -\sin(999t)\)

Thus, the expression represents the superposition of **three particles**.

### Final Answer:
The expression may be considered to be a result of the superposition of **three particles**.

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