If the units of length, mass, and time are doubled, the unit of energy becomes $n$ times the original unit. Find the value of $n$.

Solution

1. The dimensional formula for energy is $[E]=[M{L}^2{T}^{-2}]$.
2. Let the new units be $M^{\prime }=2M$, $L^{\prime }=2L$, and $T^{\prime }=2T$.
3. Substituting these into the dimensional formula, the new energy $E^{\prime }$ is $E^{\prime }=(2M)(2L{)}^2(2T{)}^{-2}$.
4. Simplifying the expression, we get $$\begin{array}{rl}{E}^{\prime }& =(2M)(4L^2)(\frac{1}{4}{T}^{-2})\\ & =4M{L}^2{T}^{-2}\\ & =4E\end{array}$$.
5. Comparing $E^{\prime }=nE$, we find $n=4$.
6. Hence the answer is (B).