Which pair has the same dimensions?

Options

A. work and power
B. density and relative density
C. momentum and impulse
D. stress and strain

Explanation

To determine which pair has the same dimensions, we will analyze the dimensions of each quantity step by step.

### Step 1: Analyze Work
- **Definition**: Work (W) is defined as force (F) multiplied by displacement (d).
- **Formula**: \( W = F \cdot d \)
- **Dimensions of Force**: The dimension of force is given by \( [F] = MLT^{-2} \).
- **Dimensions of Displacement**: The dimension of displacement is \( [d] = L \).
- **Dimensions of Work**:
\[ [W] = [F] \cdot [d] = (MLT^{-2}) \cdot (L) = ML^2T^{-2} \]

### Step 2: Analyze Power
- **Definition**: Power (P) is defined as work done per unit time.
- **Formula**: \( P = \frac{W}{t} \)
- **Dimensions of Time**: The dimension of time is \( [t] = T \).
- **Dimensions of Power**:
\[ [P] = \frac{[W]}{[t]} = \frac{ML^2T^{-2}}{T} = ML^2T^{-3} \]

### Step 3: Analyze Density
- **Definition**: Density (ρ) is defined as mass (m) per unit volume (V).
- **Formula**: \( \rho = \frac{m}{V} \)
- **Dimensions of Volume**: The dimension of volume is \( [V] = L^3 \).
- **Dimensions of Density**:
\[ [\rho] = \frac{[m]}{[V]} = \frac{M}{L^3} = ML^{-3} \]

### Step 4: Analyze Relative Density
- **Definition**: Relative density is the ratio of the density of a substance to the density of a reference substance (usually water).
- **Dimensions of Relative Density**: Since it is a ratio of two densities, it is dimensionless.
- **Dimensions**: \( [\text{Relative Density}] = 1 \) (dimensionless)

### Step 5: Analyze Momentum
- **Definition**: Momentum (p) is defined as mass (m) multiplied by velocity (v).
- **Formula**: \( p = m \cdot v \)
- **Dimensions of Velocity**: The dimension of velocity is \( [v] = LT^{-1} \).
- **Dimensions of Momentum**:
\[ [p] = [m] \cdot [v] = M \cdot (LT^{-1}) = MLT^{-1} \]

### Step 6: Analyze Impulse
- **Definition**: Impulse (I) is defined as force (F) multiplied by time (t).
- **Formula**: \( I = F \cdot t \)
- **Dimensions of Impulse**:
\[ [I] = [F] \cdot [t] = (MLT^{-2}) \cdot (T) = MLT^{-1} \]

### Step 7: Compare Dimensions
- **Work**: \( ML^2T^{-2} \)
- **Power**: \( ML^2T^{-3} \)
- **Density**: \( ML^{-3} \)
- **Relative Density**: Dimensionless
- **Momentum**: \( MLT^{-1} \)
- **Impulse**: \( MLT^{-1} \)

### Conclusion
From the analysis, we can see that:
- Momentum and Impulse both have the same dimensions: \( MLT^{-1} \).

Thus, the pair that has the same dimensions is **Momentum and Impulse**.