Question

Find the dimensions of √(G h c⁵).

• A. \(ML^2T^{-1}\)
• B. \(M^0L^0T^{1}\)
• C. \(M^0LT^{-1}\)
• D. \(M^0L^0T^{-1}\)

Answer 1

The Correct Option is B

Concept:
To find the dimensions of a given expression, substitute the dimensional formulas of each physical quantity and simplify using laws of indices. \[ \text{Dimension of } \sqrt{\frac{Gh}{c^5}} \] Step 1: Write dimensions of each quantity. Gravitational constant: \[ [G] = M^{-1}L^{3}T^{-2} \] Planck's constant: \[ [h] = ML^{2}T^{-1} \] Speed of light: \[ [c] = LT^{-1} \] Step 2: Substitute into the expression. \[ \sqrt{\frac{Gh}{c^5}} = \sqrt{ \frac{(M^{-1}L^{3}T^{-2})(ML^{2}T^{-1})}{(LT^{-1})^{5}} } \] Step 3: Simplify numerator. \[ (M^{-1}L^{3}T^{-2})(ML^{2}T^{-1}) = M^{0}L^{5}T^{-3} \] Step 4: Simplify denominator. \[ (LT^{-1})^{5} = L^{5}T^{-5} \] Step 5: Combine numerator and denominator. \[ \frac{M^{0}L^{5}T^{-3}}{L^{5}T^{-5}} = M^{0}L^{0}T^{2} \] Step 6: Apply square root. \[ \sqrt{M^{0}L^{0}T^{2}} = M^{0}L^{0}T^{1} \] \[ \boxed{M^{0}L^{0}T^{1}} \]