Ncert Exercises & Solutions

39. In the expression $P = {El}^{2} m^{- 5} G^{- 2}$ $P = {El}^{2} m^{- 5} G^{- 2}$ , E, m, l, and G denote energy, mass, angular momentum, and gravițational constant, respectively. Show that P is a dimensionless quantity.

Hint: Dimensions of LHS = Dimensions of RHS

Step 1: Find the dimensions of E, m, l, and G.

$\begin{array}{ll} Given, expression is & P = {El}^{2} m^{- 5} G^{- 2} \\ where E is energy, & [E] = [{ML}^{2} T^{- 2}] \\ m is mass, & [m] = [M] \\ l is angular momentum, & [l] = [{ML}^{2} T^{- 1}] \\ G is gravitational constant, & [G] = [M^{- 1} L^{3} T^{- 2}] \end{array}$ $\begin{array}{ll} Given, expression is & P = {El}^{2} m^{- 5} G^{- 2} \\ where E is energy, & [E] = [{ML}^{2} T^{- 2}] \\ m is mass, & [m] = [M] \\ l is angular momentum, & [l] = [{ML}^{2} T^{- 1}] \\ G is gravitational constant, & [G] = [M^{- 1} L^{3} T^{- 2}] \end{array}$

Step 2: Find the dimensions of P.

Substituting dimensions of each term in the given expression,

$\begin{array}{l} [P] = [{ML}^{2} T^{- 2}] \times [{ML}^{2} T^{- 1}]^{2} \times [M]^{- 5} \times [M^{- 1} L^{3} T^{- 2}]^{- 2} \\ = [M^{1 + 2 - 5 + 2} L^{2 + 4 - 6} T^{- 2 - 2 + 4}] = [M^{0} L^{0} T^{0}] \end{array}$ $\begin{array}{l} [P] = [{ML}^{2} T^{- 2}] \times [{ML}^{2} T^{- 1}]^{2} \times [M]^{- 5} \times [M^{- 1} L^{3} T^{- 2}]^{- 2} \\ = [M^{1 + 2 - 5 + 2} L^{2 + 4 - 6} T^{- 2 - 2 + 4}] = [M^{0} L^{0} T^{0}] \end{array}$

Therefore, P is a dimensionless quantity.

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