# Investigating the time of oscillation of a pendulum

##### Class practical

Students make a timer using a simple pendulum.

#### Apparatus and materials

*For each student group*

Pendulum (e.g. Plasticine bob on string/thread)

Stopclock

Chemical balance (0-100 g)

Stand, clamp and boss

Protractor

#### Health & Safety

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** You may need to explain what one oscillation for a pendulum is (motion *'there and back again'*). Variables to investigate include the mass of the pendulum bob, length of the pendulum (best measured to centre of bob), and initial amplitude (angle or displacement).

**2** A discussion following students' first attempts might lead to the following ideas for improving their measured value.

Reduce the uncertainty in a measurement of periodic time by:

- measuring many oscillations to calculate the average time for one oscillation
- increasing the total time measured for multiple swings

There is some uncertainty when measuring both the start time and also the stopping time, resulting from the experimenter's reflex time (as much as 0.2 s each, i.e. totalling 0.4 s). The percentage uncertainty which this 0.4 s represents decreases as the total time measured increases. 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.

Improve the accuracy of a measurement of periodic time 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.

**3** The periodic time for a swinging pendulum is constant only when amplitudes are small. Its period of oscillation is then *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})

Students investigating the effect of bob mass or pendulum length should keep the maximum angle of swing under 5^{o}.

**4 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.

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 as one of the simplest units of time, 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 0007214227), which provides further background about the development of clocks and their use in navigation. It also has some examples of the struggles that can happen in the development of science and technology.

*This experiment was safety-checked in January 2007*