What it shows
The height of a point on a turning wheel, measured above the centre, follows a sine curve in time. Tracing it on moving paper gives amplitude, period and a smooth wave.
The science
Imagine a point at the end of a radius r which turns through an angle theta from the horizontal. Its height above the centre is r sin theta, and the sideways distance is r cos theta. When theta increases steadily with time, a graph of height against theta is a sine curve with amplitude r, and the period is one full turn, 360 degrees or 2 pi radians. Cutting 15 degree steps and plotting the heights gives the same wave that a pen on moving paper would trace.
Materials
Cost: under Rs 50. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A cardboard disc of radius 10 cm
- A pushpin or a nail through the centre to a board
- A pencil or a coloured pen as a pointer
- A long strip of paper, 60 cm by 10 cm, or a roll of receipt paper
- A ruler and a protractor
- A set square
- A second sheet of graph paper for plotting
- Tape
Safety
For this project: An adult must help with the pin or the nail so that fingers are not pricked.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Draw a circle of radius 10 cm on the board and mark the centre. Divide it with the protractor into 24 equal parts of 15 degrees each.
On a long paper strip, draw a horizontal line through the middle and mark 24 equal gaps of 2 cm along it, each gap matching 15 degrees of turn.
At each division of the circle, use the set square to measure the height of the point above the centre line. Heights above are positive and below are negative.
Mark each height on the strip at the matching position, going up for positive and down for negative.
Join the points with a smooth curve. This is the sine curve for 0 to 360 degrees.
Check some readings with a calculator: at 30 degrees the height should be 10 x 0.5 = 5 cm, at 90 degrees 10 cm and at 210 degrees minus 5 cm.
Make a second model: fix the pen to the disc rim, turn the disc slowly and pull the paper at a steady speed under the pen to draw the wave by hand.
Change the radius to 5 cm and the speed of the paper and see how the amplitude and period of the wave change.
What you should see
A smooth wave with the greatest height 10 cm, one full wave for each turn, and a table showing that measured heights agree with r sin theta.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1What is the amplitude of the wave?
The radius of the wheel, which is the greatest height from the centre line.
2What is the period?
One full turn, 360 degrees, after which the wave repeats.
3How does a cosine wave differ?
It has the same shape but is shifted by 90 degrees to start at its maximum.
4Where do sine waves occur in nature?
In sound, light, alternating current and the swing of a pendulum.
Take it further
Fix two pens at 90 degrees on the disc to trace sine and cosine waves together on the same paper.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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