COMEDK2024PhysicsUnits and DimensionsActual
If units of mass, length and gravitational constant are chosen to fundamental units, the dimensions of time would be
Options
- AM ^ -1 / 2 ~L ^ 3 / 2 G ^ -1 / 2
- BM ^ 1 / 2 ~L ^ 1 / 2 G ^ -1 / 2
- CM ⁻¹ ~L ^3 G ⁻¹
- DM ^1 ~L ^ 1 / 2 G ^2
Correct answer
A. M ^ -1 / 2 ~L ^ 3 / 2 G ^ -1 / 2
Step-by-step solution
The dimension of the gravitational constant G is given by the formula F = G m₁ m₂ r^2 , which implies [G] = [F][r^2] [m^2] = MLT⁻² L^2 M^2 = M⁻¹ L^3 T⁻² . We express time T in terms of mass M , length L , and gravitational constant G as T = M^a L^b G^c . Substituting the dimensions, we get [T] = M^a L^b (M⁻¹ L^3 T⁻²)^c = M^ a-c L^ b+3c T^ -2c . Comparing the powers of T on both sides: -2c = 1 c = -1/2 . Comparing the powers of M on both sides: a - c = 0 a = c = -1/2 . Comparing the powers of L on both sides: b + 3c