Which of the following is a dimensional constant?

Options

A. Magnification
B. Relative density
C. Gravitational constant
D. Relative error

Answer

To determine which of the given options is a dimensional constant, we will analyze each option step by step.

### Step 1: Analyze Magnification
- **Definition**: Magnification (M) is defined as the ratio of the distance of the image (di) to the distance of the object (do).
- **Formula**: \( M = \frac{d_i}{d_o} \)
- **Dimensions**: The dimension of distance is [L]. Therefore, the dimensions of magnification are:
\[ M = \frac{[L]}{[L]} = [L^0] = M^0 L^0 T^0 \]
- **Conclusion**: Magnification is dimensionless.

### Step 2: Analyze Relative Density
- **Definition**: Relative density (RD) is the ratio of the density of a substance (ρ_substance) to the density of water (ρ_water).
- **Formula**: \( RD = \frac{\rho_{substance}}{\rho_{water}} \)
- **Dimensions**: The dimension of density is [M L^{-3}]. Therefore, the dimensions of relative density are:
\[ RD = \frac{[M L^{-3}]}{[M L^{-3}]} = [L^0] = M^0 L^0 T^0 \]
- **Conclusion**: Relative density is dimensionless.

### Step 3: Analyze Gravitational Constant (G)
- **Definition**: The gravitational constant (G) relates the force between two masses to the distance between them.
- **Formula**: \( G = \frac{F \cdot R^2}{M_1 \cdot M_2} \)
- **Dimensions**:
- Force (F) has dimensions [M L T^{-2}].
- Distance (R) has dimensions [L], and thus \( R^2 \) has dimensions [L^2].
- Masses \( M_1 \) and \( M_2 \) have dimensions [M].

Therefore, the dimensions of G are:
\[ G = \frac{[M L T^{-2}] \cdot [L^2]}{[M] \cdot [M]} = \frac{[M L^3 T^{-2}]}{[M^2]} = [M^{-1} L^3 T^{-2}] \]
- **Conclusion**: G has dimensions and is a dimensional constant.

### Step 4: Analyze Relative Error
- **Definition**: Relative error is the ratio of the mean absolute error to the true value.
- **Formula**: \( \text{Relative Error} = \frac{\text{Mean Absolute Error}}{\text{True Value}} \)
- **Dimensions**: Both the mean absolute error and the true value have the same dimensions, so:
\[ \text{Relative Error} = \frac{[X]}{[X]} = [L^0] = M^0 L^0 T^0 \]
- **Conclusion**: Relative error is dimensionless.

### Final Conclusion
After analyzing all options, we find that:
- Magnification: Dimensionless
- Relative Density: Dimensionless
- Gravitational Constant (G): Dimensional constant
- Relative Error: Dimensionless

Thus, the correct answer is **Gravitational Constant (G)**.

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