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//pi`
B. simple harmonic of frequency `3 omega//2pi`
C. non-simple harmonic motion
D. simple harmonic motion of frequency `omega//2 pi`

Answer

To determine the type of motion represented by the displacement equation \( x = a \sin^2(\omega t) \), we will analyze the equation step by step.

### Step 1: Rewrite the Displacement Equation
The displacement of the particle is given as:
\[ x = a \sin^2(\omega t) \]
We can use the double angle identity for sine to rewrite this equation:
\[ \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \]
Applying this identity, we have:
\[ x = a \left(\frac{1 - \cos(2\omega t)}{2}\right) = \frac{a}{2} - \frac{a}{2} \cos(2\omega t) \]

### Step 2: Differentiate to Find Velocity
Next, we differentiate \( x \) with respect to time \( t \) to find the velocity \( v \):
\[ v = \frac{dx}{dt} = \frac{d}{dt}\left(\frac{a}{2} - \frac{a}{2} \cos(2\omega t)\right) \]
Using the chain rule, we differentiate:
\[ v = 0 + \frac{a}{2} \cdot 2\omega \sin(2\omega t) = a\omega \sin(2\omega t) \]

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

### Step 4: Analyze the Relationship Between Acceleration and Displacement
For simple harmonic motion (SHM), the acceleration \( a \) must be directly proportional to the negative of the displacement \( x \):
\[ a = -k x \]
Substituting our expressions for \( a \) and \( x \):
\[ 2a\omega^2 \cos(2\omega t) \neq -k\left(\frac{a}{2} - \frac{a}{2} \cos(2\omega t)\right) \]
This shows that the acceleration is not proportional to the displacement in the required manner for SHM.

### Conclusion
Since the acceleration is not proportional to the negative of the displacement, the motion described by the equation \( x = a \sin^2(\omega t) \) does not correspond to simple harmonic motion. Therefore, the correct answer is that it represents **non-simple harmonic motion**.