Units & Dimensions

JEE Main level, single correct, medium

29 of 149

Question 29 of 149

Consider a modified Bernoulli equation.
(P+ABt2)+ρg(h+Bt)+12ρV2=\left(\mathrm{P}+\frac{A}{B t^{2}}\right)+\rho g(h+B t)+\frac{1}{2} \rho V^{2}= constant
If tt has the dimension of time then the dimensions of AA and BB are ____\_\_\_\_, ____\_\_\_\_ respectively.

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Answer: (D) [ML0 T−1]\left[\mathrm{ML}^{0} \mathrm{~T}^{-1}\right] and [M0LT−1]\left[\mathrm{M}^{0} \mathrm{LT}^{-1}\right]

All terms must have dimension of pressure [ML−1T−2][ML^{-1}T^{-2}].
For (h+Bt)(h + Bt): Since ρgh\rho gh has dimension of pressure, [h]=[L][h] = [L].
For dimensional consistency: [Bt]=[L][Bt] = [L], so [B]=[LT−1]=[M0LT−1][B] = [LT^{-1}] = [M^0LT^{-1}]
For ABt2\frac{A}{Bt^2}: [A][B][T2]=[ML−1T−2]\frac{[A]}{[B][T^2]} = [ML^{-1}T^{-2}]
[A]=[B][T2][ML−1T−2]=[LT−1][T2][ML−1T−2]=[ML0T−1][A] = [B][T^2][ML^{-1}T^{-2}] = [LT^{-1}][T^2][ML^{-1}T^{-2}] = [ML^0T^{-1}]
[A]=[ML0T−1][A] = [ML^0T^{-1}] and [B]=[M0LT−1][B] = [M^0LT^{-1}]

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