If the units of length, mass, and time are doubled, the unit of energy becomes how many times the original unit?

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. Substitute these into the formula: $E^{\prime }=(2M)(2L{)}^2(2T{)}^{-2}$.
4. Simplify the expression: $$\begin{array}{rl}{E}^{\prime }& =(2M)(4L^2)(\frac{1}{4}{T}^{-2})\\ & =2(M{L}^2{T}^{-2})\\ & =2E\end{array}$$.
5. Hence the answer is (B).