Dimensional Analysis Application

Question

Using dimensional analysis, determine which of the following cannot be the formula for the time period T of a simple pendulum (where l is length, g is acceleration due to gravity, and m is mass)?

RankGuru · Accepted Answer

1. **Goal**: Check for dimensional homogeneity. Dimension of $T$ must be $[T^1]$.
2. **Term Analysis**:
   - $\sqrt{l/g} = \sqrt{\frac{[L]}{[LT^{-2}]}} = \sqrt{[T^2]} = [T^1]$. This is consistent with $T$.
   - $\sqrt{gl} = \sqrt{[LT^{-2}][L]} = \sqrt{[L^2T^{-2}]} = [LT^{-1}]$. This is the dimension of **Velocity**, not time.
3. **Dimensionless constants**: $2\pi$, $1/2\pi$, and $(m/m)$ are dimensionless and do not affect the validity relative to units.
4. **Conclusion**: Formula **$2\pi \sqrt{gl}$** is dimensionally incorrect.

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Dimensional Analysis Application

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