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 problem using the principle of homogeneity of dimensions, we need to analyze each of the given equations and check if both sides of the equations have the same dimensions.

Let's denote the variables:
- \( t \) = time
- \( r \) = distance (radius)
- \( g \) = gravitational constant
- \( m \) = mass

### Step 1: Analyze the first equation
The first equation is:
\[ t^2 = \frac{4\pi^2 r^3}{g} \]
We need to check the dimensions on both sides.

**Left Side:**
- The dimension of time \( t \) is \( [T] \), so \( t^2 \) has the dimension \( [T^2] \).

**Right Side:**
- The dimension of \( r \) (length) is \( [L] \), so \( r^3 \) has the dimension \( [L^3] \).
- The gravitational constant \( g \) has the dimension \( [M^{-1} L^3 T^{-2}] \) (derived from \( F = g m_1 m_2 / r^2 \)).

Now, substituting the dimensions into the right side:
\[ \text{Right Side} = \frac{[L^3]}{[M^{-1} L^3 T^{-2}]} = [L^3] \cdot [M] \cdot [T^2] = [M L^3 T^2] \]

Now we can simplify:
\[ \frac{[L^3]}{[M^{-1} L^3 T^{-2}]} = [M L^3 T^2] \]

After canceling \( [L^3] \):
\[ = [M T^2] \]

Thus, the right side simplifies to \( [T^2] \) after canceling out \( [M] \).

Since both sides equal \( [T^2] \), the first equation is dimensionally consistent.

### Step 2: Analyze the second equation
The second equation is:
\[ t^2 = 4\pi r^2 \]
**Left Side:**
- The dimension of \( t^2 \) is \( [T^2] \).

**Right Side:**
- The dimension of \( r^2 \) is \( [L^2] \).

Thus, the right side has the dimension \( [L^2] \).

Since \( [T^2] \neq [L^2] \), the second equation is not dimensionally consistent.

### Step 3: Analyze the third equation
The third equation is:
\[ t^2 = 4\pi r^2 r^3 \]
**Left Side:**
- The dimension of \( t^2 \) is \( [T^2] \).

**Right Side:**
- The dimension of \( r^2 r^3 \) is \( [L^2] \cdot [L^3] = [L^5] \).

Since \( [T^2] \neq [L^5] \), the third equation is not dimensionally consistent.

### Step 4: Analyze the fourth equation
The fourth equation is:
\[ t = \frac{4\pi^2 r^3}{g} \]
**Left Side:**
- The dimension of \( t \) is \( [T] \).

**Right Side:**
- The dimension of \( r^3 \) is \( [L^3] \) and \( g \) has the dimension \( [M^{-1} L^3 T^{-2}] \).

Thus, the right side has the dimension:
\[ \frac{[L^3]}{[M^{-1} L^3 T^{-2}]} = [M L^3 T^2] \]

After canceling \( [L^3] \):
\[ = [M T^2] \]

Since \( [T] \neq [M T^2] \), the fourth equation is not dimensionally consistent.

### Conclusion
The only dimensionally consistent equation is the first one:
\[ t^2 = \frac{4\pi^2 r^3}{g} \]

### Final Answer
The correct option is the first equation.