A : If displacement y of a particle executing simple harmonic motion depends upon amplitude a angular frequency `omega` and time t then the relation `y=asinomegat` cannot be dimensionally achieved.
R : An equation cannot be achieved by dimensional analysis, if it contains dimensionless expressions.

Options

A. If both Assertion & Reason are true and the reason is the correct explanation of the assertion, then mark (1).
B. If both Assertion & Reason are true but the reason is not the correct explanation of the assertion, then mark (2).
C. If Assertion is true statement but Reason is false, then mark (3).
D. If both Assertion and Reason are false statements, then mark (4).

Answer

To solve the problem, we need to analyze the assertion (A) and the reason (R) provided in the question.

### Step 1: Understand the Assertion (A)
The assertion states that the relation \( y = a \sin(\omega t) \) cannot be dimensionally achieved.

**Explanation:**
- In this equation, \( y \) is the displacement, \( a \) is the amplitude, \( \omega \) is the angular frequency, and \( t \) is the time.
- The sine function, \( \sin(\omega t) \), is a dimensionless quantity. This means that whatever is inside the sine function must also be dimensionless.

### Step 2: Analyze the Dimensions
To analyze the dimensions:
- The dimensions of displacement \( y \) is [L] (length).
- The dimensions of amplitude \( a \) is also [L].
- The dimensions of angular frequency \( \omega \) is [T\(^{-1}\)] (inverse time).
- The dimensions of time \( t \) is [T].

Now, let's look at the term \( \omega t \):
- The dimensions of \( \omega t \) are [T\(^{-1}\)] × [T] = dimensionless.

### Step 3: Conclusion on Dimensional Analysis
Since \( \sin(\omega t) \) is dimensionless, we cannot relate the dimensions of \( y \) to the dimensions of \( a \) through dimensional analysis. Therefore, the assertion that the relation cannot be dimensionally achieved is correct.

### Step 4: Understand the Reason (R)
The reason states that an equation cannot be achieved by dimensional analysis if it contains dimensionless expressions.

**Explanation:**
- This is true because dimensional analysis relies on the dimensions of the quantities involved. If a term is dimensionless, it does not contribute to the dimensional relationship, making it impossible to derive a meaningful dimensional equation.

### Final Conclusion
Both the assertion (A) and the reason (R) are correct, and the reason correctly explains why the assertion is true.

### Summary
- Assertion (A) is correct: The relation \( y = a \sin(\omega t) \) cannot be dimensionally achieved.
- Reason (R) is correct: An equation cannot be achieved by dimensional analysis if it contains dimensionless expressions.