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Which of the following is a good predictor of a period $\text{t}$ of a swinging pendulum based on the factors such as Length (l), Angle ($\theta$), the mass (m) and acceleration due to gravity (g)?

1 Answer

Given the four factors, we can assume the period of the pendulum to be $[t] = [kl^a m^b g^c \theta^d]$, where $a,b,c,d$ are unknown real numbers.
However, $\theta$ is dimensionless, and $k$ is a constant that is assumed to be dimensionless as well, so we get: $T= L^a M^b (LT^{-1})^c$
Equating the powers of $M, L$ and $T$ on both sides of $[t] = T$, we get:
$L: 0 = a+c, M:0=b, T:1=-2c \rightarrow b=0, c = \large\frac{-1}{2}$$, a = -c = \large\frac{1}{2}$
$\Rightarrow t = kl^{\large\frac{1}{2}} g^{\large\frac{-1}{2}} \theta^d$ (d is unresolved and can assume any value)
$\Rightarrow$ The general result is: $t = f(\theta)l^{\large\frac{1}{2}} g^{\large\frac{-1}{2}}$
answered Jul 11, 2014 by balaji.thirumalai

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