03 Accurate determination with inaccurate measuremetns

Aim¶

To show how variables can still be accurately determined with somewhat inaccurate measurements

Subjects¶

• 1A20 (Error and Accuracy)

Diagram¶

Equipment¶

• Very long pendulum with variable length

• Many students

Presentation¶

The idea is to determine both the acceleration due to gravity and the height of the lecture hall, without knowing the precise length of the pendulum.

Hang the pendulum at the top of the ceiling, make sure you can still change the length of the pendulum. Tell the students that they have to record the time (without any aid) of two periods. Investigate the percentage of students taking the integer of the recorded time, a rought estimate is T = \Sum_i \%_i \cdot T_i

Change the length of the pendulum by stretching both arms, this length is roughly the length of you (see Pols (2024)). Again, ask students to record the time for two periods.

Take at least measurements at 5 different lengths.

The code below can be build during as part of the total data-analysis, it shows students how to do a proper analysis using Python.

Explanation¶

The equation for a simple pendulum is given by $T=2\pi\sqrt{\frac{L}{g}}$. For our experiment, however, $L$ cannot be measured - it is part of the task to do so. Hence, we set an initial guess and know that our ‘true’ value can deviate from this. Hence, there is a known systematic error with an unknown value. In our data-analysis we can make use of this by fitting to the equation $T=2\pi\sqrt{\frac{L+\Delta L}{g}}$, where $\Delta L$ is our systematic error.

Remarks¶

• This demonstration is shown in our Introductory Laboratory Course.