In which of the following pairs, the two physical quantities have different dimensions?

Options

A. Reynold's number and coefficient of friction
B. Curie and frequency of light
C. Latent heat and gravitation constant
D. Planck's constant and angular momentum

Answer

To determine which pair of physical quantities has different dimensions, we will analyze each option provided in the question step by step.

### Step 1: Analyze Option A - Reynolds Number and Coefficient of Friction
- **Reynolds Number (R)**: It is defined as \( R = \frac{V \cdot D}{\nu} \)
- Where:
- \( V \) = velocity (dimension: \( [L T^{-1}] \))
- \( D \) = diameter (dimension: \( [L] \))
- \( \nu \) = kinematic viscosity (dimension: \( [L^2 T^{-1}] \))

Substituting the dimensions:
\[ R = \frac{[L T^{-1}] \cdot [L]}{[L^2 T^{-1}]} = \frac{[L^2 T^{-1}]}{[L^2 T^{-1}]} = 1 \]
Hence, Reynolds number is dimensionless.

- **Coefficient of Friction**: It is defined as the ratio of the frictional force to the normal force, both of which have the same dimensions (force).
\[ \text{Coefficient of Friction} = \frac{\text{Force}}{\text{Force}} = 1 \]
Hence, the coefficient of friction is also dimensionless.

**Conclusion for Option A**: Both quantities are dimensionless.

### Step 2: Analyze Option B - Curie and Frequency of Light
- **Curie (Ci)**: It is a unit of radioactivity defined as \( 1 \text{ Ci} = 3.7 \times 10^{10} \text{ decays per second} \) (dimension: \( [T^{-1}] \)).

- **Frequency (f)**: Defined as the number of cycles per second (dimension: \( [T^{-1}] \)).

**Conclusion for Option B**: Both have the same dimension \( [T^{-1}] \).

### Step 3: Analyze Option C - Latent Heat and Gravitational Constant
- **Latent Heat (L)**: It is defined as the heat absorbed or released per unit mass.
\[ L = \frac{Q}{m} \]
Where \( Q \) is heat (dimension: \( [ML^2T^{-2}] \)) and \( m \) is mass (dimension: \( [M] \)).
\[ \text{Dimension of Latent Heat} = \frac{[ML^2T^{-2}]}{[M]} = [L^2T^{-2}] \]

- **Gravitational Constant (G)**: From the formula for gravitational force \( F = \frac{G m_1 m_2}{r^2} \):
\[ G = \frac{F \cdot r^2}{m_1 \cdot m_2} \]
Where:
- \( F \) (force) has dimension \( [MLT^{-2}] \)
- \( r \) (distance) has dimension \( [L] \)

Thus, the dimension of \( G \):
\[ \text{Dimension of } G = \frac{[MLT^{-2}] \cdot [L^2]}{[M^2]} = [M^{-1}L^3T^{-2}] \]

**Conclusion for Option C**: Latent heat has dimension \( [L^2T^{-2}] \) while the gravitational constant has dimension \( [M^{-1}L^3T^{-2}] \). They are different.

### Step 4: Analyze Option D - Planck's Constant and Angular Momentum
- **Planck's Constant (h)**: From \( E = h \nu \):
\[ h = \frac{E}{\nu} \]
Where:
- \( E \) (energy) has dimension \( [ML^2T^{-2}] \)
- \( \nu \) (frequency) has dimension \( [T^{-1}] \)

Thus, the dimension of \( h \):
\[ \text{Dimension of } h = \frac{[ML^2T^{-2}]}{[T^{-1}]} = [ML^2T^{-1}] \]

- **Angular Momentum (L)**: Defined as \( L = r \cdot p \) (where \( p \) is momentum):
\[ L = [L] \cdot [M][LT^{-1}] = [ML^2T^{-1}] \]

**Conclusion for Option D**: Both Planck's constant and angular momentum have the same dimension \( [ML^2T^{-1}] \).

### Final Conclusion:
The pair of physical quantities that have different dimensions is **Option C: Latent Heat and Gravitational Constant**.

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