Answer:
Dimensional analysis is a powerful tool used in physics and engineering to analyze and verify the correctness of equations. Let's break down the given equation Q = kpr⁴/ml using dimensional analysis:
On the left-hand side (LHS), we have Q, which represents a quantity that we need to determine its dimensions.
On the right-hand side (RHS), we have k, a constant that doesn't have any units associated with it.
Next, let's break down the dimensions of each term on the RHS:
pr⁴:
- p represents pressure with units of force per unit area (N/m² or Pa).
- r represents distance with units of meters (m).
- r⁴ represents distance raised to the power of four, so its units will be (m^4).
ml:
- m represents mass with units of kilograms (kg).
- l represents length with units of meters (m).
- Therefore, ml represents volume with units of cubic meters (m³).
So, combining the dimensions on the RHS, we have:
kpr⁴/ml = (dimensionless) * (m) * (m^4) / (m³)
Simplifying the equation further:
kpr⁴/ml = (m^(4+1-3))
kpr⁴/ml = (m²)
Comparing the LHS and the RHS, we see that the dimensions of Q are (m²). Therefore, the equation Q = kpr⁴/ml is dimensionally consistent.
By utilizing dimensional analysis, we can confirm that the equation is valid and the dimensions on both sides of the equation match, indicating a correct relationship between the variables.