Surface tension has the same dimensions as that of

Options

A. coefficient of viscosity
B. impulse
C. momentum
D. spring constant

Answer

To determine which quantity has the same dimensions as surface tension, we will analyze the definition and dimensions of surface tension, and then compare it with the dimensions of other physical quantities.

### Step-by-Step Solution:

1. **Understanding Surface Tension**:
- Surface tension (ST) is defined as the force (F) acting along a line of unit length (L).
- Mathematically, it can be expressed as:
\[ \text{Surface Tension} = \frac{\text{Force}}{\text{Length}} = \frac{F}{L} \]

2. **Finding the Dimensions of Force**:
- The dimension of force (F) is given by Newton's second law:
\[ F = m \cdot a \]
where \( m \) is mass and \( a \) is acceleration.
- The dimensions of mass (m) is [M], and acceleration (a) is given by:
\[ a = \frac{\text{change in velocity}}{\text{time}} = \frac{L}{T^2} \]
- Therefore, the dimension of force becomes:
\[ [F] = [M] \cdot \left[\frac{L}{T^2}\right] = [M L T^{-2}] \]

3. **Calculating the Dimensions of Surface Tension**:
- Now substituting the dimensions of force into the surface tension formula:
\[ [ST] = \frac{[F]}{[L]} = \frac{[M L T^{-2}]}{[L]} = [M T^{-2}] \]

4. **Comparing with Other Quantities**:
- We need to find which of the given options has the same dimensions as surface tension, which we found to be [M T^{-2}].
- Let's analyze the options:
- **Coefficient of Viscosity (η)**:
\[ [η] = \frac{[F]}{[L][T]} = \frac{[M L T^{-2}]}{[L][T]} = [M L^{-1} T^{-1}] \]
(Not equal to [M T^{-2}])
- **Impulse**:
\[ [Impulse] = [F] \cdot [T] = [M L T^{-2}] \cdot [T] = [M L T^{-1}] \]
(Not equal to [M T^{-2}])
- **Spring Constant (k)**:
\[ [k] = \frac{[F]}{[x]} = \frac{[M L T^{-2}]}{[L]} = [M T^{-2}] \]
(This is equal to [M T^{-2}])

5. **Conclusion**:
- The quantity that has the same dimensions as surface tension is the **spring constant (k)**.

### Final Answer:
The surface tension has the same dimensions as the spring constant.