The units of surface tension and viscosity of liquids are respectively

Options

A. `kgm^(-1)s^(-1), Nm^(-1)`
B. `kgs^(-2),kgm^(-1)s^(-1)`
C. `Nm^(-1), kg m^(-1)s^(-2)`
D. `kgs^(-1), kg m^(-2)s^(-1)`

Answer

To determine the units of surface tension and viscosity of liquids, we can break down the problem into two parts: finding the unit of surface tension and then finding the unit of viscosity.

### Step 1: Finding the Unit of Surface Tension
1. **Definition**: Surface tension is defined as the force per unit length.
2. **Unit of Force**: The unit of force is derived from Newton's second law, which states that force (F) is mass (m) multiplied by acceleration (a).
- The unit of mass is kilograms (kg).
- The unit of acceleration is meters per second squared (m/s²).
- Therefore, the unit of force is:
\[ \text{Force} = \text{mass} \times \text{acceleration} = \text{kg} \cdot \text{m/s}^2 \]
3. **Unit of Length**: The unit of length is meters (m).
4. **Unit of Surface Tension**: Since surface tension is force per unit length, we can express it as:
\[ \text{Surface Tension} = \frac{\text{Force}}{\text{Length}} = \frac{\text{kg} \cdot \text{m/s}^2}{\text{m}} = \text{kg/s}^2 \]

### Step 2: Finding the Unit of Viscosity
1. **Definition**: Viscosity (η) is defined as the force per unit area per unit velocity gradient.
2. **Unit of Force**: As derived earlier, the unit of force is kg·m/s².
3. **Unit of Area**: The unit of area is square meters (m²).
4. **Velocity Gradient**: The velocity gradient (dv/dr) is the change in velocity (m/s) per change in distance (m), which gives us:
\[ \text{Velocity Gradient} = \frac{\text{m/s}}{\text{m}} = \text{s}^{-1} \]
5. **Unit of Viscosity**: Therefore, the unit of viscosity can be expressed as:
\[ \text{Viscosity} = \frac{\text{Force}}{\text{Area} \cdot \text{Velocity Gradient}} = \frac{\text{kg} \cdot \text{m/s}^2}{\text{m}^2 \cdot \text{s}^{-1}} = \frac{\text{kg}}{\text{m} \cdot \text{s}} = \text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-1} \]

### Final Answer
- The units of surface tension and viscosity of liquids are respectively:
- Surface Tension: kg/s²
- Viscosity: kg·m⁻¹·s⁻¹