Given that `v` is speed, r is the radius and `g` is the acceleration due to gravity. Which of the following is dimensionless

Options

A. (a)`v^(2)g//r`
B. (b)`v^(2)rg`
C. (c)`vr^(2)g`
D. (d)`v^(2)//rg`

Answer

To determine which of the given expressions is dimensionless, we need to analyze each option step by step. We will use the following dimensional formulas:

- Speed (v) has dimensions of \( [L T^{-1}] \)
- Radius (r) has dimensions of \( [L] \)
- Acceleration due to gravity (g) has dimensions of \( [L T^{-2}] \)

Let's analyze each option:

### Step 1: Analyze Option A: \( \frac{v^2}{g \cdot r} \)

1. **Calculate \( v^2 \)**:
\[ v^2 = (L T^{-1})^2 = L^2 T^{-2} \]

2. **Calculate \( g \cdot r \)**:
\[ g \cdot r = (L T^{-2}) \cdot (L) = L^2 T^{-2} \]

3. **Combine the dimensions**:
\[ \frac{v^2}{g \cdot r} = \frac{L^2 T^{-2}}{L^2 T^{-2}} = 1 \]
This is dimensionless.

### Step 2: Analyze Option B: \( v^2 \cdot r \cdot g \)

1. **Calculate \( v^2 \cdot r \cdot g \)**:
\[ v^2 \cdot r \cdot g = (L^2 T^{-2}) \cdot (L) \cdot (L T^{-2}) = L^4 T^{-4} \]
This has dimensions.

### Step 3: Analyze Option C: \( v^2 \cdot r^2 \cdot g \)

1. **Calculate \( v^2 \cdot r^2 \cdot g \)**:
\[ v^2 \cdot r^2 \cdot g = (L^2 T^{-2}) \cdot (L^2) \cdot (L T^{-2}) = L^5 T^{-4} \]
This has dimensions.

### Step 4: Analyze Option D: \( \frac{v^2}{r \cdot g} \)

1. **Calculate \( r \cdot g \)**:
\[ r \cdot g = (L) \cdot (L T^{-2}) = L^2 T^{-2} \]

2. **Combine the dimensions**:
\[ \frac{v^2}{r \cdot g} = \frac{L^2 T^{-2}}{L^2 T^{-2}} = 1 \]
This is dimensionless.

### Conclusion:

From the analysis, we find that both Option A and Option D are dimensionless. However, since the question asks for which of the following is dimensionless, we can conclude that:

**Correct Options: A and D are dimensionless.**