Mechanics Lab: Simple Pendulum ($T \propto \sqrt{L}$)
Investigate the relationship between the **Time Period ($T$)** and **Length ($L$)** of a simple pendulum. This confirms the theoretical prediction that the period is proportional to the square root of the length.
Key Equations & Concepts
â–¼$$ T = 2\pi\sqrt{\frac{L}{g}} $$
Since $2\pi$ and $g$ are constants, the relationship simplifies to: $$ T \propto \sqrt{L} $$
In a real lab, you plot a graph of $T^2$ vs $L$. The resulting straight line confirms the relationship, and its slope ($4\pi^2/g$) is used to find $g$.
Experiment 1: $T$ vs. $L$ Measurement Simulation
Adjust the length ($L$) and measure the resulting period ($T$) for a single oscillation (simulated).
Experiment 2: Determining Length Challenge ($L = gT^2 / 4\pi^2$)
Find the required **Length ($L$)** given a target period ($T$) at a specified location ($g$).
The period of a simple pendulum is virtually independent of its **amplitude** (provided the amplitude is small, typically less than $10^\circ$). This property is known as **isochronism**, and it makes the pendulum a reliable timekeeper.
Sources of Error and Correction (11th Std)
The length $L$ must be measured from the point of suspension to the **center of gravity (CG)** of the bob, not just the string length. Failure to account for the bob's radius introduces systematic error.
The period must be calculated by measuring the time taken for a large number of oscillations (e.g., 20) and dividing by that number to minimize the **reaction time error** of the observer.
In reality, **air friction** and pivot friction cause the amplitude to decrease (damping). This is ignored in the ideal formula, but is crucial in physical pendulum analysis.
