Using dimensional analysis, derive an expression for the time period $T$ of a simple pendulum, given that $T$ may depend on its length $l$, the mass $m$ of the bob, and the acceleration due to gravity $g$.

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Accepted Answer

Assume $T = k\, l^{a} g^{b} m^{c}$, where $k$ is a dimensionless constant and $a, b, c$ are unknown exponents to be found.

Substituting dimensions — $[T]=[T^1]$, $[l]=[L]$, $[g]=[LT^{-2}]$, $[m]=[M]$ — on the right-hand side:
$$[T] = [L]^{a}[LT^{-2}]^{b}[M]^{c} = M^{c}\,L^{a+b}\,T^{-2b}$$

Comparing exponents of $M$, $L$, $T$ on both sides ($M^0L^0T^1$ on the left):
- Mass: $c = 0$ (so $T$ does not depend on $m$ at all)
- Time: $-2b = 1 \Rightarrow b = -	frac12$
- Length: $a+b = 0 \Rightarrow a = -b = 	frac12$

So $T = k\, l^{1/2} g^{-1/2} m^{0} = k\sqrt{\dfrac{l}{g}}$. …

Q.Using dimensional analysis, derive an expression for the time period $T$ of a simple pendulum, given that $T$ may depend on its length $l$ , the mass $m$ of the bob, and the acceleration due to gravity $g$ .

Concept understanding — Dimensional Analysis

Q.Using dimensional analysis, derive an expression for the time period TTT of a simple pendulum, given that TTT may depend on its length lll, the mass mmm of the bob, and the acceleration due to gravity ggg.

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Q.Using dimensional analysis, derive an expression for the time period $T$ of a simple pendulum, given that $T$ may depend on its length $l$ , the mass $m$ of the bob, and the acceleration due to gravity $g$ .

Unlock everything free for 14 days