If G be the gravitational constant and u be the energy density then which of the following quantity have the dimensions as that of the √(uG): (a) Gravitational potential (b) pressure gradient per unit mass (c) Energy per unit mass (d) Force per unit mass

Answer: (d). [u] = ML⁻¹T⁻² and, from F = Gm²/r², [G] = M⁻¹L³T⁻². [uG] = L²T⁻⁴, so [√(uG)] = LT⁻² (acceleration). Force per unit mass = MLT⁻²/M = LT⁻²; potential and energy per mass are L²T⁻², pressure gradient per mass is L⁻²T⁻². So the answer is (d) Force per unit mass.

If G be the gravitational constant and u be the energy density then… | JEE Main 2024 (5 Apr, Shift 1)

Q27JEE Main 2024 · 5 APR · SHIFT 1Easy

Units & Dimensions

If $\mathrm{G}$ be the gravitational constant and $\mathrm{u}$ be the energy density then which of the following quantity have the dimensions as that of the $\sqrt{\mathrm{uG}}$ :

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Answer: (d)

$[u] = \mathrm{ML^{-1}T^{-2}}$ and, from $F = Gm^2/r^2$ , $[G] = \mathrm{M^{-1}L^3T^{-2}}$ .
$[uG] = \mathrm{L^2T^{-4}}$ , so $[\sqrt{uG}] = \mathrm{LT^{-2}}$ (acceleration).
Force per unit mass $= \mathrm{MLT^{-2}}/\mathrm{M} = \mathrm{LT^{-2}}$ ; potential and energy per mass are $\mathrm{L^2T^{-2}}$ , pressure gradient per mass is $\mathrm{L^{-2}T^{-2}}$ .
So the answer is (d) Force per unit mass.

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