Using a simple pendulum experiment g is determind by measuring its time period T. Which of the following plots represent the correct relation between the pendulum length L and time period T?

From simple pendulum relation

$$T=2\pi\sqrt{\frac{L}{g}}$$

Squaring,

$$T^2=\frac{4\pi^2L}{g}$$

Rearrange:

$$\frac{1}{T^2}=\frac{g}{4\pi^2L}$$

$$\frac{1}{T^2}\propto\frac{1}{L}$$

or equivalently

$$L=\frac{g}{4\pi^2}\cdot\frac{1}{(1/T^2)}$$

Thus L is inversely proportional to $$\frac{1}{T^2}$$

So graph of $$L$$ versus $$1/T^2$$ is a rectangular hyperbola.

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