Question

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

• A. A
• B. B
• C. C
• D. D

Hint:

In simple pendulum experiments, plotting $\dfrac{1}{T^2}$ vs $L$ gives an inverse curve, while plotting $T^2$ vs $L$ gives a straight line.

Answer 1

The Correct Option is D

For a simple pendulum, the time period $T$ is given by: \[ T = 2\pi\sqrt{\frac{L}{g}} \]
Step 1: Express $\dfrac{1{T^2}$ in terms of $L$. Squaring both sides: \[ T^2 = \frac{4\pi^2 L}{g} \] Taking reciprocal: \[ \frac{1}{T^2} = \frac{g}{4\pi^2}\cdot\frac{1}{L} \]
Step 2: Analyze the relationship.
\[ \frac{1}{T^2} \propto \frac{1}{L} \] Thus, $\dfrac{1}{T^2}$ varies inversely with $L$.

Step 3: Identify the correct graph.
An inverse relation between $\dfrac{1}{T^2}$ and $L$ gives a rectangular hyperbola, decreasing with increase in $L$.
Among the given plots, this behavior corresponds to Option (D).

Final Answer: $\boxed{\text{Option (D)}}$