The displacement of a particle along the x- axis it given by `x = a sin^(2) omega t` The motion of the particle corresponds to

Options

A. simple harmonic motion of frequency `omega t pi`
B. simple harmonic motion of frequency `3omega 2 pi`
C. simple harmonic motion
D. simple harmonic motion of frequency `omega//2 pi`

Explanation

To determine whether the motion of a particle described by the displacement equation \( x = a \sin^2(\omega t) \) corresponds to simple harmonic motion (SHM), we need to analyze the relationship between displacement, velocity, and acceleration.

### Step-by-Step Solution

1. **Identify the Displacement Equation**:
The displacement of the particle is given by:
\[ x = a \sin^2(\omega t) \]

2. **Differentiate to Find Velocity**:
To find the velocity \( v \), we differentiate the displacement \( x \) with respect to time \( t \):
\[ v = \frac{dx}{dt} = \frac{d}{dt}(a \sin^2(\omega t)) \]
Using the chain rule, we have:
\[ v = a \cdot 2 \sin(\omega t) \cdot \cos(\omega t) \cdot \frac{d}{dt}(\omega t) \]
This simplifies to:
\[ v = 2a \sin(\omega t) \cos(\omega t) \cdot \omega \]
Using the identity \( 2 \sin(\theta) \cos(\theta) = \sin(2\theta) \):
\[ v = a \omega \sin(2\omega t) \]

3. **Differentiate to Find Acceleration**:
Now, we differentiate the velocity \( v \) to find the acceleration \( a \):
\[ a = \frac{dv}{dt} = \frac{d}{dt}(a \omega \sin(2\omega t)) \]
Again applying the chain rule:
\[ a = a \omega \cdot 2\cos(2\omega t) \cdot \frac{d}{dt}(2\omega t) \]
This simplifies to:
\[ a = 2a \omega^2 \cos(2\omega t) \]

4. **Check for Simple Harmonic Motion**:
For motion to be classified as simple harmonic motion, the acceleration must be proportional to the negative of the displacement:
\[ a = -\omega^2 x \]
Here, we have:
\[ a = 2a \omega^2 \cos(2\omega t) \]
Since \( \cos(2\omega t) \) does not yield a direct proportionality to \( -x \) (which is \( -a \sin^2(\omega t) \)), the motion does not satisfy the criteria for SHM.

5. **Conclusion**:
Therefore, the motion of the particle described by the equation \( x = a \sin^2(\omega t) \) does not correspond to simple harmonic motion.

### Final Answer:
The motion of the particle corresponds to **non-simple harmonic motion**.