Consider a modified Bernoulli equation.
$$\left(P + \frac{A}{Bt^2}\right) + \rho g(h + Bt) + \frac{1}{2}\rho V^2 = \text{constant}$$
If $t$ has the dimension of time then the dimensions of $A$ and $B$ are _____, _____ respectively.
Answer: (D) $[\mathrm{ML^0T^{-1}}]$ and $[\mathrm{M^0LT^{-1}}]$
Concept. Only like quantities can be added. In the first bracket $\dfrac{A}{Bt^2}$ is added to $P$, so it is a pressure. In the second bracket $Bt$ is added to the height $h$, so it is a length. Find $B$ first.
Step 1. Dimensions of $B$.
$$[Bt]=[h]=L\ \Rightarrow\ [B]=\frac LT=M^0LT^{-1}$$
Step 2. Dimensions of $A$.
$$\left[\frac{A}{Bt^2}\right]=[P]=ML^{-1}T^{-2}\ \Rightarrow\ [A]=ML^{-1}T^{-2}\cdot LT^{-1}\cdot T^2=MT^{-1}$$
Answer. $[ML^0T^{-1}]$ and $[M^0LT^{-1}]$ (Option D).
Solution by Sreeraj P, M.Sc Physics
