Force constant and surface tension have the same dimensions in

Options

A. Mass
B. length
C. Time
D. all the above

Answer

To solve the question, we need to analyze the dimensions of both the force constant and surface tension.

### Step-by-Step Solution:

1. **Understanding Force Constant (k)**:
- The force constant (k) is defined as the ratio of force (F) to displacement (x).
- Mathematically, it is expressed as:
\[ k = \frac{F}{x} \]
- The dimensions of force (F) are given by:
\[ [F] = [M][L][T^{-2}] \]
where M is mass, L is length, and T is time.
- The dimensions of displacement (x) are:
\[ [x] = [L] \]
- Therefore, the dimensions of the force constant (k) can be calculated as:
\[ [k] = \frac{[F]}{[x]} = \frac{[M][L][T^{-2}]}{[L]} = [M][L^{0}][T^{-2}] = [M][T^{-2}] \]

2. **Understanding Surface Tension (T)**:
- Surface tension (T) is defined as the force (F) acting along a length (A).
- Mathematically, it is expressed as:
\[ T = \frac{F}{A} \]
- The dimensions of area (A) are:
\[ [A] = [L^{2}] \]
- Thus, the dimensions of surface tension (T) can be calculated as:
\[ [T] = \frac{[F]}{[A]} = \frac{[M][L][T^{-2}]}{[L^{2}]} = [M][L^{-1}][T^{-2}] \]

3. **Comparing Dimensions**:
- From the calculations, we have:
- Dimensions of force constant (k): \([M][L^{0}][T^{-2}]\)
- Dimensions of surface tension (T): \([M][L^{-1}][T^{-2}]\)
- We can see that both have the dimensions of mass (M) and time (T) in common, but the dimensions of length (L) differ.

4. **Conclusion**:
- The force constant and surface tension do not have the same dimensions. The correct answer to the question is that they have the same dimensions in mass and time, but not in length.

### Final Answer:
The force constant and surface tension have the same dimensions in mass and time, but not in length.