The displacement of a particle is represented by the equation `y=sin^(3)omegat.` The motion is

Options

A. non-periodic
B. periodic but not simple harmonic
C. simple harmonic with period `2pi//omega`
D. simple harmonic with period `ppi//omega`

Explanation

To determine the nature of the motion represented by the displacement equation \( y = \sin^3(\omega t) \), we can analyze the equation step by step.

### Step 1: Identify the form of the displacement equation
The given displacement equation is:
\[ y = \sin^3(\omega t) \]
This indicates that the displacement of the particle varies with time according to the sine function raised to the third power.

### Step 2: Check for periodicity
The sine function, \( \sin(\omega t) \), is a periodic function with a period of \( T = \frac{2\pi}{\omega} \). Since \( y \) is a function of \( \sin(\omega t) \), we need to check if \( y = \sin^3(\omega t) \) is also periodic.

- The sine function has a period of \( 2\pi \), and raising it to any power (including the third power) does not change its periodic nature. Therefore, \( y = \sin^3(\omega t) \) is periodic with the same period \( T = \frac{2\pi}{\omega} \).

### Step 3: Determine if the motion is simple harmonic
For motion to be classified as simple harmonic motion (SHM), the acceleration must be directly proportional to the negative of the displacement. Mathematically, this means:
\[ \frac{d^2y}{dt^2} = -k y \]
for some constant \( k \).

#### Step 3.1: Calculate the first derivative
First, we differentiate \( y \) with respect to time \( t \):
\[ \frac{dy}{dt} = 3\sin^2(\omega t) \cdot \omega \cos(\omega t) \]

#### Step 3.2: Calculate the second derivative
Next, we differentiate again to find the acceleration:
\[ \frac{d^2y}{dt^2} = \frac{d}{dt} \left( 3\sin^2(\omega t) \cdot \omega \cos(\omega t) \right) \]
Using the product rule and chain rule, we can simplify this expression. However, we need to check if the resulting expression can be expressed as a constant times \( -y \).

After performing the differentiation and simplification, we find that:
\[ \frac{d^2y}{dt^2} \neq -k y \]
This indicates that the motion is not simple harmonic.

### Conclusion
The displacement \( y = \sin^3(\omega t) \) is periodic, but it does not satisfy the conditions for simple harmonic motion. Therefore, the motion of the particle is periodic but not simple harmonic.

### Final Answer
The motion is periodic but not simple harmonic.

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