The displacement of a particle is represented by the equation `y=3cos((pi)/(4)-2omegat).`
The motion of the particle is

Options

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

Explanation

To determine the type of motion represented by the equation \( y = 3 \cos\left(\frac{\pi}{4} - 2\omega t\right) \), we can follow these steps:

### Step 1: Identify the form of the equation
The given equation is in the form of \( y = A \cos(\phi - \omega t) \), where \( A \) is the amplitude, \( \phi \) is the phase constant, and \( \omega \) is the angular frequency.

### Step 2: Rewrite the equation
We can rewrite the equation as:
\[ y = 3 \cos\left(-2\omega t + \frac{\pi}{4}\right) \]
Using the property of cosine that \( \cos(-\theta) = \cos(\theta) \), we can simplify this to:
\[ y = 3 \cos\left(2\omega t - \frac{\pi}{4}\right) \]

### Step 3: Determine the type of motion
The equation \( y = A \cos(\omega t + \phi) \) indicates that the motion is simple harmonic motion (SHM). The presence of the cosine function and the linear time dependence confirms that the particle undergoes oscillatory motion.

### Step 4: Calculate the period of motion
The angular frequency \( \omega \) is related to the period \( T \) of the motion by the formula:
\[ T = \frac{2\pi}{\omega} \]
In this case, since the angular frequency is \( 2\omega \), we can find the period as follows:
\[ T = \frac{2\pi}{2\omega} = \frac{\pi}{\omega} \]

### Conclusion
The motion of the particle is simple harmonic motion with a period of \( \frac{\pi}{\omega} \).