If Planck’s constant h, gravitational constant G, and

Solution

Let the dimension of time [T] be expressed in terms of Planck's constant [h], gravitational constant [G], and speed of light [v] as: [T] = [h]^a[G]^b[v]^c We need to find the values of a, b, and c. Let's write down the dimensions of h, G, and v in terms of mass [M], length [L], and time [T]: [h] (Planck's constant) has dimensions of energy × time, so: [h] = [M L² T^-2][T]=[M L² T^-1] [G] (Gravitational constant) is obtained from Newton's law of gravitation F = (Gm1m2)/r² so G = (Fr²/​ m1m2): [G] = [M^-1 L³ T^-2] [v] (speed of light) has dimensions of length per time: [v] = [L T^-1] Now, substitute these dimensions into the equation for [T]: [T] = [M L² T^-1]^a [M^-1 L³ T^-2]^b [L T^-1]^c [T] = [M^a L^2a T^-a] [M^-b L^3b T^-2b] [L^c T^-c] [T] = [M^a-b L^2a+3b+c T^-a-2b-c] For the dimensions to be equal on both sides, the powers of M, L, and T must be the same: For M: a − b = 0 ⟹ a = b (1) For L: 2a + 3b + c = 0 (2) For T: −a  − 2b – c =1 (3) Substitute a = b from (1) into (2) and (3): From (2): 2a  + 3a +c = 0 ⟹ 5a + c = 0 ⟹ c = −5a (4) From (3): −a−2a−c = 1 ⟹ −3a − c = 1 (5) Now substitute c = −5a from (4) into (5): −3a−(−5a)=1 −3a+5a=1 2a=1 a = 1/2 Since a = b, we have b=1/2​. Now substitute the value of a into equation (4) to find c: c = -5a = -5/2 The dimension of time is [h^1/2 G^1/2 v^-5/2] Therefore, the correct answer is option (C).