Which of the following is not simple harmonic function ?

Options

A. `y = a sin 2omegat + b cos^(2) omegat`
B. `y = a sin omega t + b cos 2 omega t`
C. `y = 1 - 2 sin^(2) omega t`
D. `y = (sqrt(a^(2) +b^(2)))sin omega t cos omega t`

Explanation

To determine which of the given equations is not a simple harmonic function, we need to analyze each equation based on the standard forms of simple harmonic motion (SHM). The standard equations for SHM are:

1. \( x = a \sin(\omega t + \phi) \)
2. \( x = a \cos(\omega t + \phi) \)

Now, let's evaluate each option one by one:

### Step 1: Analyze Each Equation

1. **Equation A: \( y = a \sin(2\omega t) + B \cos^2(\omega t) \)**

We can rewrite \( \cos^2(\omega t) \) using the double angle identity:
\[ \cos^2(\omega t) = \frac{1 + \cos(2\omega t)}{2} \]
Substituting this back into the equation gives:
\[ y = a \sin(2\omega t) + B \left(\frac{1 + \cos(2\omega t)}{2}\right) \]
This can be expressed as:
\[ y = \left(a + \frac{B}{2}\right) \sin(2\omega t) + \frac{B}{2} \]
Since the frequency is the same (2ω), this can be considered a simple harmonic function.

2. **Equation B: \( y = a \sin(\omega t) + B \cos(2\omega t) \)**

Here, we have two different frequencies: \( \omega \) and \( 2\omega \). Since the frequencies are different, we cannot express this equation in the standard SHM form. Therefore, this equation is **not** a simple harmonic function.

3. **Equation C: \( y = 1 - \sin^2(\omega t) \)**

Using the identity \( \sin^2(\omega t) = \frac{1 - \cos(2\omega t)}{2} \), we can rewrite this as:
\[ y = 1 - \left(\frac{1 - \cos(2\omega t)}{2}\right) = 1 - \frac{1}{2} + \frac{\cos(2\omega t)}{2} = \frac{1}{2} + \frac{\cos(2\omega t)}{2} \]
This can be expressed as:
\[ y = \frac{1}{2} + \frac{1}{2} \cos(2\omega t) \]
This shows that it is a simple harmonic function with frequency \( 2\omega \).

4. **Equation D: \( y = \sqrt{a^2 + b^2} \sin(\omega t) \cos(\omega t) \)**

Using the identity \( \sin(\omega t) \cos(\omega t) = \frac{1}{2} \sin(2\omega t) \), we can rewrite this as:
\[ y = \frac{\sqrt{a^2 + b^2}}{2} \sin(2\omega t) \]
This is also a simple harmonic function with frequency \( 2\omega \).

### Conclusion

After analyzing all the equations, we find that:

- **Equation A** is a simple harmonic function.
- **Equation B** is **not** a simple harmonic function.
- **Equation C** is a simple harmonic function.
- **Equation D** is a simple harmonic function.

Thus, the answer is **Equation B**.

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