Which of the following is dimensionless ?

Options

A. `v^2/(rg)`
B. `(v^2g)/r`
C. `(vg)/r`
D. `v^2rg`

Answer

To determine which of the given quantities is dimensionless, we need to analyze the dimensions of each option provided. The options given are V², R, and G. Let's break down the steps to find the dimensionless quantity.

### Step-by-Step Solution:

1. **Identify the Dimensions of Each Quantity**:
- **Velocity (V)**: The dimension of velocity is given by the formula \( V = \frac{distance}{time} \). Therefore, the dimension of velocity is:
\[ [V] = \frac{L}{T} \quad \text{(where L is length and T is time)} \]
- **V² (Velocity Squared)**: Squaring the velocity gives:
\[ [V^2] = \left(\frac{L}{T}\right)^2 = \frac{L^2}{T^2} \]
- **Radius (R)**: The radius is a measure of length, so:
\[ [R] = L \]
- **Gravitational Acceleration (G)**: The dimension of gravitational acceleration is:
\[ [G] = \frac{L}{T^2} \]

2. **Combine the Quantities to Check for Dimensionlessness**:
- We can form a dimensionless quantity by combining \( V^2 \), \( R \), and \( G \). One possible combination is:
\[ \frac{V^2}{RG} \]
- Substituting the dimensions we found:
\[ [RG] = [R] \cdot [G] = L \cdot \frac{L}{T^2} = \frac{L^2}{T^2} \]
- Now substituting in our expression:
\[ \frac{[V^2]}{[RG]} = \frac{\frac{L^2}{T^2}}{\frac{L^2}{T^2}} = 1 \]
- Since the result is 1, this indicates that the combination \( \frac{V^2}{RG} \) is dimensionless.

3. **Conclusion**:
- Therefore, the combination \( \frac{V^2}{RG} \) is dimensionless. Thus, the correct answer to the question "Which of the following is dimensionless?" is:
\[ \text{Option A: } \frac{V^2}{RG} \]