JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Units & Measurement

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Planck's constant h , speed of light c and gravitational constant G are used to from a unit of length L and a unit of mass M . Then the correct options(s) is (are)

(A) $M \propto c$ ✓ Correct
(B) $M \propto G$
(C) $L \propto h$ ✓ Correct
(D) $L \propto G$ ✓ Correct

Answer: (A), (C), (D)

Solution

Using dimensional analysis, let

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

. Writing dimensions:

M = (M L^{2} T^{- 1})^{x} (L T^{- 1})^{y} (M^{- 1} L^{3} T^{- 2})^{z}

.

Equating powers:

x - z = 1

2 x + y + 3 z = 0

- x - y - 2 z = 0

. Solving gives

x = 1/2

y = 1/2

z = - 1/2

. Thus

M \propto h c / G

, so (A) is correct.

For length L:

x - z = 0

2 x + y + 3 z = 1

- x - y - 2 z = 0

. Solving gives

x = 1/2

y = - 3/2

z = 1/2

. Thus

L \propto h G / c^{3/2}

, so (C) and (D) are correct.

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JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Units & Measurement

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