Understanding the Equation
The given equation is \( P = \frac{a - t^2}{bx} \). To analyze the dimensions of \(\frac{a}{b}\), we need to understand the dimensions of each term in the equation.
Identifying Dimensions of Each Variable
- Pressure (P):
- The dimension of pressure is given by the formula \( \text{Pressure} = \frac{\text{Force}}{\text{Area}} \).
- Thus, \( [P] = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \).
- Distance (x):
- The dimension of distance is simply:
- \( [x] = [L] \).
- Time (t):
- The dimension of time is:
- \( [t] = [T] \).
Analyzing the Terms a, b, and t²
1. Term \( t^2 \):
- The dimension of \( t^2 \) is:
- \( [t^2] = [T^2] \).
2. Term \( a \):
- Rearranging the equation:
- \( a - t^2 = P \cdot bx \)
- The dimensions of \( a \) must match those of \( P \cdot bx \):
- \( [a] = [P][b][x] \).
3. Term \( b \):
- Therefore, we can express \( b \) in terms of the other dimensions:
- \( [b] = \frac{[P]}{[x]} = \frac{[M][L^{-1}][T^{-2}]}{[L]} = [M][L^{-2}][T^{-2}] \).
Final Calculation of Dimensions \(\frac{a}{b}\)
- Now, substituting the dimensions:
- \( [\frac{a}{b}] = \frac{[P][b][x]}{[b]} = [P][x] \).
- Therefore:
- \( [\frac{a}{b}] = [M][L^{-1}][T^{-2}][L] = [M][L^{0}][T^{-2}] = [M][T^{-2}] \).
Conclusion
The dimensions of \(\frac{a}{b}\) in the equation \( P = \frac{a - t^2}{bx} \) are \( [M][T^{-2}] \).

The dimension of a/b is MT^-2