Tag Archive for: error

Simple harmonic motion

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A-Level Physics Required Practicals:
Simple Harmonic Motion

(and a bit of error analysis)

Supporting notes for this film will follow shortly.

Assessment

— to follow

Student Worksheet

— to follow

Comments & Feedback

As ever, no single film can encompass everything one might wish to say about a practical. Please, leave comments with your thoughts about the approach we’ve taken, and your suggestions for alternatives or improvements.


Measuring g via Free Fall

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A-Level Physics Required Practicals:
Measuring g using a free fall method

A-level specifications from all the exam boards include “measure the acceleration due to gravity of a freely falling body” as one of the practicals. Students might be a bit uninterested in measuring the value of a constant with which they are already familiar. However, this practical is likely to be undertaken close to the beginning of an A level course. As such, it can be used to make a number of valuable points, each of which is worth introducing our students to at this stage:

  • By comparing more than one method, students can practice thinking critically about experimental methods.
  • Learn to assess and reduce uncertainty.
  • Consider how to present and process data.
  • Discuss what is meant by ‘constant.’

What’s in the Film

The film shows four different methods of measuring g using a falling object:

  1. Drop a ball and time its fall with a stopwatch.
  2. Drop a ‘g-ball’, which times its own fall.
  3. Drop a ball through light gates.
  4. Use an electromagnetic switch to release a ball bearing, with triggered timing.

Which of these methods you use might depend on your students’ skill, your own preference, apparatus availability, ease of data collection or processing, and class size. It’s good to compare at least two methods (even if one is shown as a demonstration only), to prompt and inform discussion about precision, accuracy and uncertainty.

What’s not in the film

Since we anticipate this practical being used early in the A level course, we’ve not included comments about how to do a full error analysis. There’s more error handling in some of the other films in the series, and we’re planning to complete a film specifically about error as the series continues.

An aside about language

The ASE book The Language of Measurement (£13.50/£8.50 members) refers to precision as:

A measurement is precise if values cluster closely.

In the film, at about 2:22, the word ‘precision’ is used to mean the timer’s smallest scale division. This is us showing our age! A better term might be ‘resolution’.

Other ASE definitions:

Accuracy: a result is accurate if it is close to the true value.

Uncertainty: the interval within which the true value can be expected to lie.


Calculating g from h and t


Methods 1, 2 and 4 give values of the time \(t\) for a ball to fall from rest at a height \(h\). From the equation:

\(s = ut + \frac{1}{2}at^2\)

we have:

\(g = \frac{2h}{t^2}\)

Measuring \(t\) for different values of \(h\) allows a graph to be drawn of \(h\) against \(t^2\). The gradient of that graph is \(g/2\).

In method 3 the ball falls through two light gates separated by a distance \(h\). Each gate gives a value for the average speed of the ball as it passes through, so we can use \(v^2 = u^2 + 2as\) to find its acceleration between the gates.


Uncertainties


Uncertainties arise in four ways:

  1. Starting the timer.
  2. During the ball’s descent, due to air resistance or other factors.
  3. Stopping the timer.
  4. Determining the height of the fall.

Students can think about these as they compare the different methods. They can try to assess whether each uncertainty will make the values of \(t\) and \(h\) too big or too small, or just more uncertain. They can also discuss how each factor will affect the value of \(g\).

For example: if the measured value of \(t\) is too big, the calculated value of \(g\) will be too small, because t is on the bottom line of \(g = 2h/t^2\).


Error in g


Eventually, students must be able to assess the overall uncertainty in their measurements. For example, they might state their measured value as \(g = 8.7 \pm 1~\mathrm{ms^{-2}}\). For now, however, it is a good start to be able to calculate the percentage error, e.g.:

\(\frac{9.8 – 8.7}{9.8} \times 100\% = 11\%\)

Method 1: Dropping a ball

Drop a ball from a measured height \(h\); start and stop a timer to find \(t\).

Students should be able to comment on the problems with starting and stopping the timer at the correct instants. Reaction time here is not the same as the human response to a random stimulus, since we can watch the ball falling and anticipate the correct moment to stop the timer. It’s still, however, a poor way to measure \(g\)!

Repeated measurements for a fixed value of \(h\) will give a range of values for \(t\). Some students may be better at using a consistent technique, which will give a smaller spread in the values of \(t\). This reduces the random error, but there may still be systematic error, which itself may be revealed by repeating the experiment for different values of \(h\).

Questions to ask your students:

  • What’s a better way to release the ball?
  • What’s a better way to measure the time?
  • Why don’t we try to reduce the uncertainty in measuring \(h\)?

Method 2: The g-ball

These cost about £20 + VAT from education suppliers, one of which is Timstar (as of 2020-08-03 Timstar appear to have stopped selling the G Ball. We’ve found other UK suppliers: Philip Harris, Better Equipped, Breckland Scientific. Search for ‘G Ball’ to find others. There appear to be at least two manufacturers, Unilab and Mollic).

The g-ball starts timing when released and stops timing as soon as it hits the floor. Like most stopclocks, it measures to a resolution of 0.01 seconds. The switch release can limit accuracy, but overall the g-ball is a quick way to collect a large number of data points. In the film, Alom and Christina use an L-shaped bracket clipped to a metre rule to press the release switch, to aid a clean release.

Questions to ask your students:

  • How is this better than using a stopwatch?
  • How can they ‘average’ their data?

Your students could just repeatedly drop the g-ball from the same height and average all their data, then substitute their average \(t\) into \(s = \frac{1}{2}gt^2\) to find \(g\). Much better would be to exclude anomalous values first, and better still would be to plot a graph of \(h\) against \(t^2\) to spot those anomalies.

If you’re going to plot a graph, however, you might as well have two meaningful variables. In the film, Christina and Alom drop the ball from different heights and collect a lot of (not very good!) data quickly.

This approach:

  • Helps to spot anomalies visually – an indication of random error.
  • Shows that \(g\) is (more-or-less!) constant: the graph is a straight line.
  • Allows \(g\) to be determined from a gradient, an important skill.
  • Might lead on to a discussion about systematic errors (should the graph pass through the origin?).

Systematic errors can often be eliminated by plotting a graph. For example, if the height is measured 1 cm too short each time, the line on the graph will be shifted downwards. The gradient will be unchanged, and the systematic error should be easily detected.

Another approach, incidentally, would be to plot \(2s\) against \(t^2\), giving a simpler gradient of \(g\). This is a matter of personal preference.

Language:

Random error: a measurement error due to results varying in an unpredictable way.

Systematic error: A measurement error where results differ from the true value by a consistent amount each time.

Method 3: Light gates & data logger

Using light gates is a great way to get students familiar with data loggers. Watch out for the common misconception that using a computer will automatically give a better result! In this case, the timing typically does have a higher resolution, and it’s possible to collect lots of data quickly – both good reasons for using the apparatus.

With two light gates, there are three possibilities:

  • Display values of \(u\) and \(v\), and calculate g using \(v^2 = u^2 + 2as\).
  • Display values of \(u\), \(v\) and \(t\), and calculate g using \(v = u – at\).
  • Position the top light gate just below where the ball is released so the initial velocity is close to zero, then proceed as above.

Note that the measured speeds are always average values, because the ball is accelerating during the time it takes to pass through the light gate.

Questions to ask your students:

  • Can we be sure that the data logger determines accurate times?
  • If we vary \(h\), what graph should we plot to determine \(g\)?
  • How can a graph help to reveal problems with the experimental technique?

Apparatus

If you don’t have multiple sets of data loggers, you could have one set up and have students use it in turn. However, the experience of setting up the apparatus is itself valuable, as it prompts the student to think through the role of each piece of equipment rather than to approach the configured apparatus as a ‘black box.’

Method 4: Electronic Timer

The commercially available apparatus that Alom and Ronan discuss in the film is available from Philip Harris, for about £100+VAT. Instructions for its use are available from the Nuffield Foundation, which also includes circuit diagrams for a DIY version. (update 2020-08-03: it looks like the old practicalphysics.org site has been taken down. The content has moved to spark.iop.org/practical-physics, and the resources on this particular experiment are: Acceleration due to gravity, and Measurement of g using an electronic timer).

Ronan’s version addresses several subtle issues, including that of releasing the ball cleanly, and using a plumb-line to confirm that \(h\) is being measured vertically. We’ll update this article with more details of Ronan’s apparatus when we have them.

Questions to ask your students

  • Between which two points should we measure the height of fall \(h\)?
  • Can we be sure that the timer starts and stops at the exact moments we want it to?
  • If we vary \(h\), what graph should we plot to determine \(g\)?

Assessment

Common Practical Assessment Criteria

At the time of writing, the exam boards appear to agree that this practical might be used to address, in whole or in part:

  • CPAC 2: Applies investigate approaches and methods when using instruments and equipment.
  • CPAC 4: Makes and records observations.

Apparatus & Techniques

Each exam board has published a list of apparatus and techniques with which students much be familiar, along with suggestions as to which elements might be addressed by each practical. For example, Edexcel’s guidance for this practical suggests:

  • 1. Use appropriate analogue apparatus to record a range of measurements (to include length/distance, temperature, pressure, force, angles, volume) and to interpolate between scale markings
  • 2. Use appropriate digital instruments, including electrical multimeters, to obtain a range of measurements (to include time, current, voltage, resistance, mass).
  • 4. Use stopwatch or light gates for timing.
  • 11. Use ICT such as computer modelling, or datalogger with a variety of sensors to collect data, or use of software to process data.

Check your exam board’s resources: there should be a mapping document to help you decide which criteria to assess on each practical.

Student Worksheet

We’ve drafted a student worksheet for this practical, which you may find useful:

Comments & Feedback

As ever, no single film can encompass everything one might wish to say about a practical. Please, leave comments with your thoughts about the approach we’ve taken, and your suggestions for alternatives or improvements.