Using the principle of homogeneity of dimensions, which of the following is correct?

Options

A. `T^(2)=(4pi^(2)r^(3))/(GM)`
B. `T^(2)=4pi^(2)r^(2)`
C. `T^(2)=(4pi^(2)r^(3))/(G)`
D. `T=(4pi^(2)r^(3))/(G)`

Explanation

To solve the question using the principle of homogeneity of dimensions, we need to analyze each option given in the question. Let's go through the solution step by step.

### Step 1: Understand the Principle of Homogeneity of Dimensions
The principle of homogeneity of dimensions states that in a valid equation, the dimensions on both sides must be the same. This means that we can equate the dimensions of physical quantities to check if the equation is valid.

### Step 2: Analyze Option A
The first option states:
\[ t^2 = 4\pi^2 R^3 \]

### Step 3: Determine Dimensions
- The dimension of time \( t \) is denoted as \( [T] \).
- The dimension of radius \( R \) (which is a length) is denoted as \( [L] \).

### Step 4: Write Dimensions for Both Sides
- Left-hand side (LHS):
\[ [t^2] = [T^2] \]

- Right-hand side (RHS):
\[ [4\pi^2 R^3] = [L^3] \]
(Note: The constants \( 4 \) and \( \pi^2 \) are dimensionless and do not affect the dimensions.)

### Step 5: Compare Dimensions
Now we compare the dimensions:
- LHS: \( [T^2] \)
- RHS: \( [L^3] \)

### Step 6: Check for Homogeneity
Since \( [T^2] \) (from LHS) is not equal to \( [L^3] \) (from RHS), this indicates that the equation is not dimensionally homogeneous. Therefore, option A is incorrect.

### Step 7: Conclusion
Based on the analysis, we conclude that option A is not correct. We would need to analyze the other options (2, 3, and 4) similarly, but since the question only asks for the correctness of the provided option, we can summarize that option A does not satisfy the principle of homogeneity of dimensions.