Which of the following motions is not simple harmonic?

Options

A. `Asinomegat`
B. `Asinomegat +Bcosomegat`
C. `Asinomegat +Bcos2omegat`
D. `Asinomegat +Bsinomegat`

Answer

To determine which of the following motions is not simple harmonic, we need to analyze each option based on the characteristics of simple harmonic motion (SHM). SHM is defined as a type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. The motion can be represented mathematically by sinusoidal functions (sine or cosine).

Let's analyze the options step by step:

1. **Option A: \( x(t) = A \sin(\omega t) \)**
- This is a standard form of SHM. The acceleration is proportional to the displacement, and it can be expressed as \( a = -\omega^2 x \). Thus, this motion is simple harmonic.

2. **Option B: \( x(t) = A \sin(\omega t) + B \cos(\omega t) \)**
- This can be rewritten using the sine addition formula. It can be expressed as \( R \sin(\omega t + \alpha) \), where \( R = \sqrt{A^2 + B^2} \) and \( \alpha \) is a phase constant. This is also a form of SHM, as it is a sinusoidal function.

3. **Option C: \( x(t) = A \cos(2\omega t) \)**
- The function \( \cos(2\omega t) \) is a cosine function but with a frequency that is double that of the standard SHM. However, it still represents a periodic motion and can be expressed as a sinusoidal function. Therefore, this is also simple harmonic motion.

4. **Option D: \( x(t) = A \sin(\omega t) + B \cos(\omega t) \)**
- Similar to option B, this can also be expressed in terms of a single sinusoidal function. Thus, this is also simple harmonic motion.

Upon analyzing all options, we realize that all options except for one are forms of simple harmonic motion. The only one that does not fit the criteria of SHM is:

**Option C: \( x(t) = A \cos(2\omega t) \)**

### Conclusion:
The motion that is not simple harmonic is **Option C**.