Which of the quantities defined below have the same dimensional formula?

\sqrt{\frac{E n e r g y}{M a s s}}

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\sqrt{\frac{P r e s s u r e}{D e n s i t y}}

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\sqrt{\frac{F o r c e}{L i n e a r d e n s i t y}}

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\sqrt{\frac{A n g u l a r f r e q u e n c y}{R a d i u s}}

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Solution

The correct options are
A $\sqrt{\frac{P r e s s u r e}{D e n s i t y}}$
B $\sqrt{\frac{E n e r g y}{M a s s}}$
C $\sqrt{\frac{F o r c e}{L i n e a r d e n s i t y}}$
$\sqrt{\frac{E n e r g y}{M a s s}} = \sqrt{\frac{- M^{1} L^{2} T^{- 2}}{M^{1}}} = L T^{- 1}$
$\sqrt{\frac{P r e s s u r e}{D e n s i t y}} = \sqrt{\frac{M^{1} L^{- 1} T^{- 2}}{M^{1} L^{- 3}}} = L T^{- 1}$
$\sqrt{\frac{F o r c e}{L i n e a r D e n s i t y}} = \sqrt{\frac{M^{1} L^{1} T^{- 2}}{M^{1} L^{- 1} T^{0}}} = L T^{- 1}$
$\sqrt{\frac{A n g u l a r f r e q}{r a d i u s}} = \sqrt{\frac{T^{- 1}}{L^{1}}} = \sqrt{L^{- 1} T^{- 1}}$
Hence, options A, B, C are correct.

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