Which of the following functions of time represent (a) simple harmonic motion and (b) periodic but not simple harmonic motion? Give the period for each case.
(i)` sinomegat-cosomegat` (ii) `sin^(2)omegat` (iii) `cosomegat+2sin^(2)omegat`

Options

A. a simple harmonic motion with a period `(pi)/(omega)`.
B. a simple harmonic motion with a period `(2pi)/(omega)`
C. a periodic, but not simple harmonic motionn with a period `(pi)/(omega)`
D. a periodic, but not simple harmonic motion with a period `(2pi)/(omega)`

Explanation

To determine which of the given functions represent simple harmonic motion (SHM) and which represent periodic but not SHM, we will analyze each function step by step.

### Step 1: Analyze the first function `sin(ωt) - cos(ωt)`

1. **Rewrite the function**: We can rewrite the function using the sine subtraction formula:
\[ y = \sin(ωt) - \cos(ωt) \]
We can factor out \(\sqrt{2}\) to express it in the form of a sine function:
\[ y = \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin(ωt) - \frac{1}{\sqrt{2}} \cos(ωt) \right) \]
This can be rewritten as:
\[ y = \sqrt{2} \sin\left(ωt - \frac{\pi}{4}\right) \]
This is in the standard form of SHM, \(A \sin(ωt + φ)\).

2. **Determine the period**: The period \(T\) of SHM is given by:
\[ T = \frac{2\pi}{ω} \]

### Step 2: Analyze the second function `sin²(ωt)`

1. **Rewrite the function**: We can use the double angle identity:
\[ y = \sin^2(ωt) = \frac{1 - \cos(2ωt)}{2} \]
This function oscillates but does not represent SHM because it does not have a linear relationship with time.

2. **Determine the period**: The period of \(\cos(2ωt)\) is:
\[ T = \frac{2\pi}{2ω} = \frac{\pi}{ω} \]

### Step 3: Analyze the third function `cos(ωt) + 2sin²(ωt)`

1. **Rewrite the function**: We can express \(2\sin^2(ωt)\) using the identity we derived earlier:
\[ y = \cos(ωt) + 2\left(\frac{1 - \cos(2ωt)}{2}\right) = \cos(ωt) + 1 - \cos(2ωt) \]
This function is a combination of cosine terms and does not fit the form of SHM.

2. **Determine the period**: The dominant term for the period is \(\cos(2ωt)\), which has a period of:
\[ T = \frac{2\pi}{2ω} = \frac{\pi}{ω} \]

### Summary of Results

- **Function (i)**: `sin(ωt) - cos(ωt)`
- Represents: Simple Harmonic Motion
- Period: \(T = \frac{2\pi}{ω}\)

- **Function (ii)**: `sin²(ωt)`
- Represents: Periodic but not SHM
- Period: \(T = \frac{\pi}{ω}\)

- **Function (iii)**: `cos(ωt) + 2sin²(ωt)`
- Represents: Periodic but not SHM
- Period: \(T = \frac{\pi}{ω}\)