Question

Dimensions of \(G\) (Universal gravitational constant) in terms of \(h\) (Planck's constant), \(m\) (mass), \(t\) (time) and \(L\) (length) will be:

• A. \(h^{-1}Lm^{-2}t\)
• B. \(hL^{-1}m^{2}t\)
• C. \(hLm^{-2}t^{-1}\)
• D. \(h^{-1}L^{-1}m^{2}t^{-1}\)

Hint:

When expressing one physical constant in terms of others: \begin{itemize} \item Write dimensional formulas \item Assume power relation \item Equate powers of \(M, L, T\) \end{itemize}

Answer 1

The Correct Option is C

Concept: Dimensional formula of gravitational constant: \[ [G] = M^{-1}L^{3}T^{-2} \] Dimensional formula of Planck's constant: \[ [h] = ML^{2}T^{-1} \] Step 1: Assume relation.} \[ G = h^a m^b L^c t^d \] Step 2: Write dimensions of RHS.} \[ [h]^a [m]^b [L]^c [t]^d \] \[ (ML^2T^{-1})^a \cdot M^b \cdot L^c \cdot T^d \] \[ = M^{a+b}L^{2a+c}T^{-a+d} \] Step 3: Compare with dimensions of \(G\).} \[ M^{-1}L^{3}T^{-2} \] Equating powers: Mass: \[ a+b=-1 \] Length: \[ 2a+c=3 \] Time: \[ -a+d=-2 \] Step 4: Solve equations.} From time equation: \[ d = a-2 \] Choose \(a=1\) Then \[ b=-2 \] \[ c=1 \] \[ d=-1 \] Step 5: Substitute values.} \[ G = h^1 L^1 m^{-2} t^{-1} \] \[ G = hLm^{-2}t^{-1} \]