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 assertion and reason question, we will analyze both the assertion (A) and the reason (R) step by step.

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

- **Displacement \( y \)**: The dimension of displacement is [L] (length).
- **Amplitude \( A \)**: The amplitude is also a measure of length, so its dimension is [L].
- **Angular frequency \( \omega \)**: The angular frequency has dimensions of [T]⁻¹ (inverse time).
- **Time \( t \)**: The dimension of time is [T].

### Step 2: Analyzing the Expression
The term \( \sin(\omega t) \) is a sine function, which is a dimensionless quantity. This is because the argument of the sine function must be in radians, and radians are considered dimensionless.

### Step 3: Dimensional Analysis
Now, we can perform dimensional analysis on the equation \( y = A \sin(\omega t) \):

- The left-hand side \( y \) has dimensions [L].
- The right-hand side consists of \( A \) (which has dimensions [L]) multiplied by \( \sin(\omega t) \) (which is dimensionless).

Thus, the right-hand side also has dimensions [L]. Therefore, the equation \( y = A \sin(\omega t) \) is dimensionally consistent.

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

This is true because dimensional analysis focuses on the dimensions of the quantities involved. If a term is dimensionless, it does not contribute to the dimensional consistency of the equation.

### Conclusion
Both the assertion and reason are true, and the reason correctly explains the assertion. Therefore, the correct answer is that both assertion and reason are true, and the reason correctly explains the assertion.

### Final Answer
Both assertion and reason are true, and the reason correctly explains the assertion.

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