Which of the following equations represent S.H.M. ?

Options

A. `y = 2 sin ( omega t +30^(@))`
B. `y =2 cos( omega t - 30^(@))`
C. `y = 2 sin omega t + 3 cosomega t `
D. `y = 2 sin omega t .cosomega t `

Answer

To determine which of the given equations represent Simple Harmonic Motion (S.H.M.), we need to analyze each equation based on the characteristics of S.H.M. The general form of S.H.M. can be expressed as:

\[ y = A \sin(\omega t + \phi) \]
or
\[ y = A \cos(\omega t + \phi) \]

where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.

Let's analyze each option one by one.

### Step-by-Step Solution:

1. **Option 1: \( y = 2 \sin(\omega t + 30^\circ) \)**
This equation is in the form of \( y = A \sin(\omega t + \phi) \) where \( A = 2 \) and \( \phi = 30^\circ \).
**Conclusion:** This represents S.H.M.

2. **Option 2: \( y = 2 \sin(\omega t + 5) \)**
This equation is also in the form of \( y = A \sin(\omega t + \phi) \) where \( A = 2 \) and \( \phi = 5 \).
**Conclusion:** This represents S.H.M.

3. **Option 3: \( y = 2 \cos(\omega t - 30^\circ) \)**
This equation is in the form of \( y = A \cos(\omega t + \phi) \) where \( A = 2 \) and \( \phi = -30^\circ \).
**Conclusion:** This represents S.H.M.

4. **Option 4: \( y = 2 \sin(\omega t) + 3 \cos(\omega t) \)**
We can rewrite this equation using the concept of combining sine and cosine into a single sine function.
- The amplitude \( A \) can be calculated as:
\[ A = \sqrt{(2^2 + 3^2)} = \sqrt{4 + 9} = \sqrt{13} \]
- The phase angle \( \phi \) can be calculated using:
\[ \tan(\phi) = \frac{3}{2} \implies \phi = \tan^{-1}\left(\frac{3}{2}\right) \]
Thus, we can express this as:
\[ y = \sqrt{13} \sin(\omega t + \phi) \]
**Conclusion:** This represents S.H.M.

5. **Option 5: \( y = 2 \sin(\omega t) + \cos(\omega t) \)**
Similar to the previous option, we can combine these terms:
- The amplitude \( A \) is:
\[ A = \sqrt{(2^2 + 1^2)} = \sqrt{4 + 1} = \sqrt{5} \]
- The phase angle \( \phi \) can be calculated as:
\[ \tan(\phi) = \frac{1}{2} \implies \phi = \tan^{-1}\left(\frac{1}{2}\right) \]
Thus, we can express this as:
\[ y = \sqrt{5} \sin(\omega t + \phi) \]
**Conclusion:** This represents S.H.M.

### Final Conclusion:
All the given equations represent S.H.M.