Simple pendulum
Get to grips with the variables behind the motion of a simple pendulum
Age
16 to 18
Challenge level
Problem
A simple pendulum consists of a string of length $l$ and a bob of mass $m$. We release the bob from a small angle with the vertical, it swings back and forth with a period $T$. The period $T$ could depend on the length $l$ of the pendulum, the mass of the bob $m$ and the acceleration of gravity $g$.
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- Find a combination of $l$, $m$ and $g$ which has the dimensions of time.
- Test your relationship of the period $T$ on the length $l$ by measuring the period of the pendulum for a few different values of $l$.
The correct formula relating $T$ with other quantities involves a constant that cannot be obtained by dimensional analysis but it can be found by experiment if $g$ is known.
- Using your experimental results and the value $g = 9.81 m/s^2$ find a constant.
- Divide this constant by $\pi$. Does your result is close to an integer?
- Find the formula for $T$.
Getting Started
- Read an article on dimensional analysis.
- The period could be expressed as $T = \alpha \times m^{A}g^{B}l^{C}$ where we need to choose A, B, C such that the dimensions on both sides of equation agrees. $\alpha$ is a constant which can be found by experiment.
- The acceleration of gravity has dimensions $L/T^2$ (meter per second squared).