Question

Consider a modified Bernoulli equation. \[ P + \frac{A}{Bt^2} + \rho g(h + Bt) + \frac{1}{2} \rho v^2 = \text{constant} \] If t has the dimension of time then the dimensions of A and B are_________ respectively.

• A. \( [MLT^{-1}] \) and \( [M^0LT] \)
• B. \( [ML^0T^{-2}] \) and \( [M^0LT^{-1}] \)
• C. \( [ML^0T^{-2}] \) and \( [M^0LT^{-2}] \)
• D. \( [MLT^{-1}] \) and \( [M^0LT^{-1}] \)

Hint:

When a question on dimensions seems inconsistent, check for alternative interpretations. Standard terms like \( \rho g h \) (pressure) can be parts of force equations if multiplied by an area. Always start by analyzing the simplest parts of the equation first, like terms added in parentheses, e.g., \( (h + Bt) \), to find the dimensions of one constant.

Answer 1

The Correct Option is B

The given equation is a modified form of Bernoulli’s equation:

\[ P + \frac{A}{B t^2} + \rho g (h + Bt) + \frac{1}{2}\rho v^2 = \text{constant} \]

Since this is a Bernoulli-type equation, every term must have the dimensions of pressure.

\[ [P] = [ML^{-1}T^{-2}] \]

---

Step 1: Dimensions of \( \dfrac{A}{Bt^2} \)

This term must have dimensions of pressure:

\[ \frac{[A]}{[B][T^2]} = [ML^{-1}T^{-2}] \]

Rearranging:

\[ [A] = [B]\,[T^2]\,[ML^{-1}T^{-2}] \]

---

Step 2: Dimensions of \( B \) from the term \( \rho g (h + Bt) \)

The quantity inside the brackets must have dimensions of length:

\[ [h] = [Bt] = L \]

Since \( t \) has dimensions \( [T] \), we get:

\[ [B] = [LT^{-1}] \]

---

Step 3: Dimensions of \( A \)

Substitute \( [B] = [LT^{-1}] \) into the expression for \( [A] \):

\[ [A] = [LT^{-1}] \times [T^2] \times [ML^{-1}T^{-2}] \]

Simplifying:

\[ [A] = [ML^0T^{-1}] \]

---

Step 4: Verification of remaining terms

• \[ [\rho g h] = (ML^{-3})(LT^{-2})(L) = ML^{-1}T^{-2} \]
• \[ \left[\frac{1}{2}\rho v^2\right] = (ML^{-3})(L^2T^{-2}) = ML^{-1}T^{-2} \]

All terms are dimensionally consistent with pressure.

---

Final Answer:

\[ [A] = ML^0T^{-1}, \qquad [B] = LT^{-1} \]