On the basis of dimensions, decide which of the following relation for the displacement of a particle undergoing simple harmonic motion is not correct :

Options

A. A.`y=asin((2pit)/(T))`
B. B. `y=acosomegat`
C. C.`y=(a)/(T)sin((t)/(a))`
D. D.`y=asqrt(2)(sin((2pit)/(T))+cos((2pit)/(T)))`

Explanation

To determine which relation for the displacement of a particle undergoing simple harmonic motion (SHM) is not correct based on dimensions, we will analyze each option provided in the question.

### Step-by-Step Solution:

1. **Understanding Displacement in SHM**:
- In simple harmonic motion, the displacement \( y \) can be expressed in terms of amplitude \( A \), angular frequency \( \omega \), time \( t \), and phase difference \( \phi \).
- The general form is:
\[ y = A \sin(\omega t + \phi) \]
- Here, \( A \) is the maximum displacement (amplitude), and its unit is meters (m).

2. **Analyzing Each Option**:
- **Option A**: \( y = A \sin\left(\frac{2\pi t}{T}\right) \)
- Here, \( T \) is the time period (in seconds).
- The term \( \frac{2\pi t}{T} \) is dimensionless (as both \( t \) and \( T \) are in seconds).
- Therefore, \( y \) has the unit of \( A \), which is meters. This option is correct.

- **Option B**: \( y = A \cos(\omega t) \)
- Similar to Option A, \( \omega t \) is dimensionless since \( \omega \) (angular frequency) has units of radians per second and \( t \) is in seconds.
- Thus, \( y \) has the unit of \( A \), which is meters. This option is also correct.

- **Option C**: \( y = \frac{A}{T} \)
- Here, \( A \) is in meters, and \( T \) is in seconds.
- Therefore, the unit of \( \frac{A}{T} \) is \( \frac{\text{meters}}{\text{seconds}} \), which is a unit of velocity (m/s).
- This does not represent a displacement, making this option incorrect.

- **Option D**: \( y = A\sqrt{2} \sin\left(\frac{2\pi t}{T}\right) \)
- The term \( A\sqrt{2} \) is still in meters, and \( \sin\left(\frac{2\pi t}{T}\right) \) is dimensionless.
- Thus, \( y \) has the unit of meters, making this option correct.

3. **Conclusion**:
- Based on the dimensional analysis, the relation that is not correct for the displacement of a particle undergoing simple harmonic motion is **Option C**: \( y = \frac{A}{T} \).

### Final Answer:
**Option C is not correct.**