Which of the following equation does not represent a simple harmonic motion

Options

A. `y=asinomegat`
B. `y=acosomegat`
C. `y=asinomegat+bcosomegat`
D. `y=a tan omegat`

Explanation

To determine which of the given equations does not represent simple harmonic motion (SHM), we need to analyze each equation based on the fundamental characteristics of SHM. In SHM, the restoring force is directly proportional to the negative of the displacement from the equilibrium position. This means that the acceleration of the object should also be directly proportional to the negative of its displacement.

Let's analyze each equation step by step:

1. **Equation 1: y = a sin(ωt)**
- This equation represents a sinusoidal function, which is characteristic of SHM.
- The acceleration can be derived as follows:
- First derivative (velocity): \( \frac{dy}{dt} = aω \cos(ωt) \)
- Second derivative (acceleration): \( \frac{d^2y}{dt^2} = -aω^2 \sin(ωt) \)
- Since acceleration is proportional to \(-y\), this equation represents SHM.

2. **Equation 2: y = a cos(ωt)**
- Similar to the first equation, this is also a sinusoidal function.
- The acceleration can be derived as follows:
- First derivative (velocity): \( \frac{dy}{dt} = -aω \sin(ωt) \)
- Second derivative (acceleration): \( \frac{d^2y}{dt^2} = -aω^2 \cos(ωt) \)
- Again, acceleration is proportional to \(-y\), so this equation also represents SHM.

3. **Equation 3: y = a e^(ωt)**
- This is an exponential function.
- The acceleration can be derived as follows:
- First derivative (velocity): \( \frac{dy}{dt} = aω e^{ωt} \)
- Second derivative (acceleration): \( \frac{d^2y}{dt^2} = aω^2 e^{ωt} \)
- Here, acceleration is not proportional to \(-y\), which means this equation does not represent SHM.

4. **Equation 4: y = a tan(ωt)**
- This equation represents a tangent function, which is not periodic like sine or cosine.
- The acceleration can be derived as follows:
- First derivative (velocity): \( \frac{dy}{dt} = aω \sec^2(ωt) \)
- Second derivative (acceleration): \( \frac{d^2y}{dt^2} = 2aω^2 \sec^2(ωt) \tan(ωt) \)
- The acceleration is not proportional to \(-y\), indicating that this equation does not represent SHM.

### Conclusion:
The equation that does not represent simple harmonic motion is **Equation 3: y = a e^(ωt)**.