Which of the following represents a SHM?

Options

A. `sin omega t - cos omega t`
B. `sin omega t + cos omega t`
C. `sin omega t + 2 cos omega t`
D. All of these

Answer

To determine which of the given equations represents Simple Harmonic Motion (SHM), we need to check if they satisfy the condition:

\[ \frac{d^2y}{dt^2} \propto -y \]

This means that the second derivative of \(y\) with respect to time \(t\) should be directly proportional to the negative of \(y\).

### Step-by-Step Solution:

**Option 1: \(y = \sin(\omega t) - \cos(\omega t)\)**

1. **Find the first derivative \( \frac{dy}{dt} \)**:
\[ \frac{dy}{dt} = \omega \cos(\omega t) + \omega \sin(\omega t) \]

2. **Find the second derivative \( \frac{d^2y}{dt^2} \)**:
\[ \frac{d^2y}{dt^2} = -\omega^2 \sin(\omega t) + \omega^2 \cos(\omega t) \]

3. **Express \( \frac{d^2y}{dt^2} \) in terms of \(y\)**:
\[ \frac{d^2y}{dt^2} = -\omega^2 y \]
This confirms that this equation represents SHM.

**Option 2: \(y = \sin(\omega t) + \cos(\omega t)\)**

1. **Find the first derivative \( \frac{dy}{dt} \)**:
\[ \frac{dy}{dt} = \omega \cos(\omega t) - \omega \sin(\omega t) \]

2. **Find the second derivative \( \frac{d^2y}{dt^2} \)**:
\[ \frac{d^2y}{dt^2} = -\omega^2 \sin(\omega t) - \omega^2 \cos(\omega t) \]

3. **Express \( \frac{d^2y}{dt^2} \) in terms of \(y\)**:
\[ \frac{d^2y}{dt^2} = -\omega^2 y \]
This confirms that this equation also represents SHM.

**Option 3: \(y = \sin(\omega t) + 2\cos(\omega t)\)**

1. **Find the first derivative \( \frac{dy}{dt} \)**:
\[ \frac{dy}{dt} = \omega \cos(\omega t) - 2\omega \sin(\omega t) \]

2. **Find the second derivative \( \frac{d^2y}{dt^2} \)**:
\[ \frac{d^2y}{dt^2} = -\omega^2 \sin(\omega t) - 2\omega^2 \cos(\omega t) \]

3. **Express \( \frac{d^2y}{dt^2} \) in terms of \(y\)**:
\[ \frac{d^2y}{dt^2} = -\omega^2 y \]
This confirms that this equation also represents SHM.

### Conclusion:
All three options represent SHM. Therefore, the answer is **all of these**.