Which one of the following combinations of co | vedclass

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(A) The dimensions of the given constants are:$G = [M^{-1} L^{3} T^{-2}]$$h = [M L^{2} T^{-1}]$$c = [L T^{-1}]$Let the expression be $T = G^{a} h^{b}

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$\left[\frac{G h}{c^{5}}\right]^{\frac{1}{2}}$

Vedclass · Answer

(A) The dimensions of the given constants are:$G = [M^{-1} L^{3} T^{-2}]$$h = [M L^{2} T^{-1}]$$c = [L T^{-1}]$Let the expression be $T = G^{a} h^{b} c^{d}$.Substituting the dimensions:$[T^{1}] = [M^{-1} L^{3} T^{-2}]^{a} [M L^{2} T^{-1}]^{b} [L T^{-1}]^{d}$$[M^{0} L^{0} T^{1}] = [M^{-a+b} L^{3a+2b+d} T^{-2a-b-d}]$Comparing the powers on both sides:For $M$: $-a + b = 0 \implies a = b$For $L$: $3a + 2b + d = 0 \implies 3a + 2a + d = 0 \implies d = -5a$For $T$: $-2a - b - d = 1 \implies -2a - a - (-5a) = 1 \implies 2a = 1 \implies a = 1/2$Thus,$a = 1/2, b = 1/2, d = -5/2$.The expression is $G^{1/2} h^{1/2} c^{-5/2} = \left[\frac{G h}{c^{5}}\right]^{\frac{1}{2}}$.

Which one of the following combinations of constants has the dimensions of time? $[G=$ gravitational constant,$h=$ Planck's constant,$c=$ velocity of light$]$

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