The equation of motion of a simple harmonic motion is not

Options

A. `x=A sin (omega t + phi)`
B. `x= A cos (omega t -phi)`
C. `x= a sin omega t + bcos omega t`
D. `x=A sin (omega t + phi)`+ B sin (2 omega t + phi)`

Answer

To determine which equation does not represent simple harmonic motion (SHM), we will analyze each option based on the fundamental equation of SHM, which is given by:

\[ a = -\omega^2 x \]

where \( a \) is the acceleration, \( \omega \) is the angular frequency, and \( x \) is the displacement.

### Step-by-Step Solution:

1. **Option A: \( x = A \sin(\omega t + \phi) \)**
- Differentiate \( x \) with respect to time \( t \) to find velocity \( v \):
\[ v = \frac{dx}{dt} = A \omega \cos(\omega t + \phi) \]
- Differentiate \( v \) to find acceleration \( a \):
\[ a = \frac{d^2x}{dt^2} = -A \omega^2 \sin(\omega t + \phi) \]
- Substitute \( x \) back into the equation \( a = -\omega^2 x \):
\[ a = -\omega^2 (A \sin(\omega t + \phi)) \]
- This satisfies the SHM condition. Thus, **Option A is valid**.

2. **Option B: \( x = A \cos(\omega t - \phi) \)**
- Differentiate \( x \) to find velocity \( v \):
\[ v = \frac{dx}{dt} = -A \omega \sin(\omega t - \phi) \]
- Differentiate \( v \) to find acceleration \( a \):
\[ a = \frac{d^2x}{dt^2} = -A \omega^2 \cos(\omega t - \phi) \]
- Substitute \( x \) back into the equation \( a = -\omega^2 x \):
\[ a = -\omega^2 (A \cos(\omega t - \phi)) \]
- This satisfies the SHM condition. Thus, **Option B is valid**.

3. **Option C: \( x = A \sin(\omega t) + B \cos(\omega t) \)**
- Differentiate \( x \) to find velocity \( v \):
\[ v = \frac{dx}{dt} = A \omega \cos(\omega t) - B \omega \sin(\omega t) \]
- Differentiate \( v \) to find acceleration \( a \):
\[ a = \frac{d^2x}{dt^2} = -A \omega^2 \sin(\omega t) - B \omega^2 \cos(\omega t) \]
- Substitute \( x \) back into the equation \( a = -\omega^2 x \):
\[ a = -\omega^2 (A \sin(\omega t) + B \cos(\omega t)) \]
- This satisfies the SHM condition. Thus, **Option C is valid**.

4. **Option D: \( x = A \sin(\omega t + \phi) + B \sin(2\omega t + \phi) \)**
- Differentiate \( x \) to find velocity \( v \):
\[ v = \frac{dx}{dt} = A \omega \cos(\omega t + \phi) + 2B \omega \cos(2\omega t + \phi) \]
- Differentiate \( v \) to find acceleration \( a \):
\[ a = \frac{d^2x}{dt^2} = -A \omega^2 \sin(\omega t + \phi) - 4B \omega^2 \sin(2\omega t + \phi) \]
- Substitute \( x \) back into the equation \( a = -\omega^2 x \):
\[ a = -\omega^2 \left( A \sin(\omega t + \phi) + B \sin(2\omega t + \phi) \right) \]
- The term \( -4B \omega^2 \sin(2\omega t + \phi) \) does not match the form \( -\omega^2 x \) because it introduces a term with \( \sin(2\omega t + \phi) \), which is not part of the SHM equation. Thus, **Option D is not valid**.

### Conclusion:
The equation of motion of a simple harmonic motion is not represented by **Option D**.