JEE Main · 2023 · 01 Feb 2023 · Shift 2 · Units & Measurement

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If the velocity of light c, universal gravitational constant G and planck's constant h are chosen as fundamental quantities. The dimensions of mass in the new system is:

(A) $[h^{\frac{1}{2}} c^{- \frac{1}{2}} G^{1}]$
(B) $[h^{1} c^{1} G^{- 1}]$
(C) $[h^{- \frac{1}{2}} c^{\frac{1}{2}} G^{\frac{1}{2}}]$
(D) $[h^{\frac{1}{2}} c^{\frac{1}{2}} G^{- \frac{1}{2}}]$ ✓ Correct

Answer: (D)

Solution

Writing mass as

M = h^{x} c^{y} G^{z}

and substituting the dimensions

[h] = M L^{2} T^{- 1}

[c] = L T^{- 1}

and

[G] = M^{- 1} L^{3} T^{- 2}

gives, on matching powers:

x - z = 1

for mass,

2 x + y + 3 z = 0

for length and

- x - y - 2 z = 0

for time. Solving these together yields

x = \frac{1}{2}

y = \frac{1}{2}

and

z = - \frac{1}{2}

. The dimensions of mass are therefore

[h^{1/2} c^{1/2} G^{- 1/2}]

JEE Main · 2023 · 01 Feb 2023 · Shift 2 · Single correct option

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JEE Main · 2023 · 01 Feb 2023 · Shift 2 · Units & Measurement

Solution

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