Question

If unit of length and time is doubled, the numerical value of 'g' (acceleration due to gravity) will be:

• A. doubled
• B. halved
• C. four times
• D. remain same

Answer 1

Answer: (A)

doubled

Answer 2

The dimensions of acceleration due to gravity 'g' are $[L T^{-2}]$. Let the original numerical value be $N$ with units $u_L$ and $u_T$, so $g = N \ u_L^1 u_T^{-2}$. If the units of length and time are doubled, the new units are $u'_L = 2u_L$ and $u'_T = 2u_T$. Let the new numerical value be $N'$. Then $g = N' (u'_L)^1 (u'_T)^{-2} = N' (2u_L)^1 (2u_T)^{-2} = N' \cdot 2 \cdot \frac{1}{4} \ u_L^1 u_T^{-2} = \frac{N'}{2} \ u_L^1 u_T^{-2}$. Equating the two expressions for $g$, we get $N = N'/2$, which implies $N' = 2N$. Thus, the numerical value of 'g' is doubled.