Which of the following expression does not represent SHM?

Options

A. `A cos omega t`
B. `A sin2 omega t`
C. `A sin omega t + B cos omega t `
D. `A sin^(2) omega t`

Explanation

To determine which expression does not represent Simple Harmonic Motion (SHM), we need to analyze each option based on the standard forms of SHM. The standard equations for SHM can be expressed as:

1. \( x(t) = A \cos(\omega t) \)
2. \( x(t) = A \sin(\omega t) \)

Where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( t \) is time.

Now, let's evaluate each option:

### Step 1: Analyze Option A
**Expression:** \( A \cos(\omega t) \)

This expression is in the standard form of SHM. Therefore, it represents SHM.

### Step 2: Analyze Option B
**Expression:** \( A \sin(2\omega t) \)

This expression can be rewritten using the relationship of angular frequency. The term \( 2\omega \) can be treated as a new angular frequency. Thus, it can be expressed as:

\[ x(t) = A \sin(\omega' t) \]
where \( \omega' = 2\omega \).

This is also in the form of SHM. Therefore, it represents SHM.

### Step 3: Analyze Option C
**Expression:** \( A \sin(\omega t) + B \cos(\omega t) \)

This expression can be converted into a single sine or cosine function using the trigonometric identity:

\[ R \cos(\omega t - \phi) \]

where \( R = \sqrt{A^2 + B^2} \) and \( \phi \) is the phase angle. Thus, this expression can also be represented in the form of SHM. Therefore, it represents SHM.

### Step 4: Analyze Option D
**Expression:** \( A \sin^2(\omega t) \)

This expression is the square of a sine function. It can be rewritten using the identity:

\[ \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \]

Thus, we can express it as:

\[ x(t) = \frac{A}{2} (1 - \cos(2\omega t)) \]

This does not fit the standard forms of SHM since it is not a simple sine or cosine function but rather a function that includes a constant term and a cosine term with double the frequency. Therefore, it does not represent SHM.

### Conclusion
The expression that does not represent SHM is:

**Option D: \( A \sin^2(\omega t) \)**

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