As the expression in ivolving sine function , which of the following equations does not represent a simple harmonic motion ?

Options

A. `y = a sin omegat`
B. `y=a tan omegat`
C. `y=a cos omegat`
D. `y= a sin omegat +b cos omegat`

Answer

To determine which of the given equations does not represent simple harmonic motion (SHM), we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding SHM**:
- A motion is said to be simple harmonic if the acceleration of the object is directly proportional to its displacement from the mean position and is directed towards the mean position. Mathematically, this can be expressed as:
\[ \frac{d^2y}{dt^2} = -\omega^2 y \]
- Here, \(y\) is the displacement, and \(\omega\) is the angular frequency.

2. **Analyzing Option A: \(y = a \sin(\omega t)\)**:
- Differentiate \(y\):
\[ \frac{dy}{dt} = a \omega \cos(\omega t) \]
- Differentiate again:
\[ \frac{d^2y}{dt^2} = -a \omega^2 \sin(\omega t) = -\omega^2 y \]
- This satisfies the SHM condition.

3. **Analyzing Option B: \(y = a \tan(\omega t)\)**:
- Differentiate \(y\):
\[ \frac{dy}{dt} = a \omega \sec^2(\omega t) \]
- Differentiate again:
\[ \frac{d^2y}{dt^2} = 2a \omega^2 \sec^2(\omega t) \tan(\omega t) \]
- The second derivative does not yield a simple proportionality to \(y\), hence this does not satisfy the SHM condition.

4. **Analyzing Option C: \(y = a \cos(\omega t)\)**:
- Differentiate \(y\):
\[ \frac{dy}{dt} = -a \omega \sin(\omega t) \]
- Differentiate again:
\[ \frac{d^2y}{dt^2} = -a \omega^2 \cos(\omega t) = -\omega^2 y \]
- This satisfies the SHM condition.

5. **Analyzing Option D: \(y = a \sin(\omega t) + b \cos(\omega t)\)**:
- This can be rewritten as:
\[ y = R \sin(\omega t + \phi) \]
- Where \(R = \sqrt{a^2 + b^2}\) and \(\phi\) is a phase constant.
- Differentiating will yield:
\[ \frac{d^2y}{dt^2} = -\omega^2 y \]
- This also satisfies the SHM condition.

6. **Conclusion**:
- From the analysis, we see that options A, C, and D represent SHM. However, option B does not represent SHM.

### Final Answer:
The equation that does not represent simple harmonic motion is **Option B: \(y = a \tan(\omega t)\)**.