# Investigation of a simple pendulum

Students investigate factors affecting the oscillation time for a simple pendulum.

#### Apparatus and materials

*For each student group*

string, 2m length

metre rule

bob (e.g. hanger with slotted masses, 10g, or lump of Plasticine)

Stopclock or stopwatch

Chemical balance (0-1 g)

Stand, clamp and boss

Protractor

Metal strips used as jaws, 5 cm, 2

G-clamp

#### Health & Safety and Technical notes

Put something on the floor to prevent damage should the mass fall.

Avoid large amplitude oscillations.

If large masses are used then the stands may need to be clamped to the bench.

#### Procedure

**a** Show a demonstration pendulum and ask students to think about the variables that may affect the time period for one oscillation.

**b** Ask students to select one independent variable, collecting a set of data to investigate its effect on the oscillation time.

**c** After students have completed an initial investigation and drawn conclusions, ask them to evaluate their method in terms of its accuracy and improve on it.

####

Teaching notes

**1** Given the right attitude, students can really enjoy these investigations. Choose how far to take them, to suit your students’ age and experience. You may need to explain what one oscillation for a pendulum means (motion “there and back again” i.e. moving in the original direction).

Variables to investigate include:

- the mass of the pendulum bob
- length of the pendulum (best measured to centre of bob)
- initial amplitude (angle or displacement).

The periodic time for a swinging pendulum is constant only when amplitudes are small. Students investigating the effect of bob mass or pendulum length should keep the maximum angle of swing under 5°.

**2** Timing the oscillation period for various lengths can be quite tedious. You could arrange it so that pairs of students contribute their results to a communal graph and table of results for the whole class.

**3** A discussion following students’ first attempts at measuring the periodic time might lead to the following ideas for improving their measured value:

- measure many oscillations to calculate the average time for one oscillation
- increase the total time measured for multiple swings.

There will be some measurement uncertainty both when starting the clock and when stopping the clock, dependent on the experimenter’s reflexes in operating the stopwatch (as much as 0.2 s at each time, i.e. totalling 0.4 s). The percentage of the total time measured which this uncertainty represents will vary. If more swings are counted and the total time is greater, then 0.4 s will be a smaller percentage of that total time. Students could carry out simple error calculations to discover, for example, the effect of a human reaction time of 0.2 seconds on timings of 2 s, 20 s and 200 s.

[You may wish to get them to estimate the human reaction time or measure it as a separate activity. There are many web-based activities freely available.]

They can improve the accuracy of their measurements by:

- making timings by sighting the bob past a fixed reference point (called a ‘fiducial point’)
- sighting the bob as it moves fastest past a reference point. The pendulum swings fastest at its lowest point and slowest at the top of each swing.

**4** Students can first plot a graph of periodic time, *T*, against length, *l*, getting a curve (a parabola). They could try a few quick calculations to see whether the graph to plot is *T*, *l / T*, *√T* or *T ^{2}* against

*l*rather than just telling them it is

*T*against

^{2}*l*.

The period of oscillation of a simple pendulum is

*T*= 2π√(l /

*g*) where:

*T*= time period for one oscillation (s)

*l*= length of pendulum (m)

*g*= acceleration due to gravity (m/s

^{-2})

A graph of

*T*against

^{2}*l*should be a straight line graph, showing that

*T*. This line may indicate that more readings are needed as the plotted points may be too close together.

^{2}∝ lFrom the graph of

*T*against

^{2}*l*the value of

*g*can be found because the slope of the graph is

*4π*.

^{2}/g**5 How Science Works Extension:**

This provides an excellent opportunity for planning, carrying out and evaluating an investigation using multiple skills. The number of variables is limited but there is enough scope to allow students to develop an approach and select appropriate ranges and intervals.

Students often assume that any sensibly selected independent variable will always have an effect upon the dependent variable. Many may decide to investigate the effect of the mass of the bob, which yields an unexpected (counter-intuitive) result: the mass has no effect on periodic time. Proving that there is no link between two variables can be just as significant as finding one.

To investigate the (misleading) power of suggestion, issue printed instructions to the class. Give half the class instructions with the hypothesis that amplitude has an effect on periodic time, and the other half the class instructions with the hypothesis that amplitude has

*no effect*. Their task is to test the hypothesis, by collecting and analysing data. The whole class will think they have identical instructions. Afterwards, compare what each half of the class has found.

The introductory discussion can put the pendulum into a scientific and historical context by describing the development of timing devices. Start with the hours of a day easily measured with a sundial. Use this to introduce Galileo Galilei (1564-1642) and the (possibly apocryphal) story that his understanding of the behaviour of pendulums was spurred by observing the bronze chandelier or incense burner in the cathedral at Pisa. Galileo’s pendulum introduced a method of measuring short periods of time that improved on the use of the human pulse. You could extend this timeline by describing further developments in timing devices, right up to the atomic clock (usually containing caesium) which is accurate to within 10

^{-9}seconds per day.

For students who take a particular interest in the measurement of time, suggest the book

*Longitude*by Dava Sobel (ISBN 978 000 721422 7), which provides further background about the development of clocks and their use in sea navigation. It also has examples of the personal and social struggles that can occur in the development of science and technology.