What it shows
A loop of string round two pins and a pencil draws an ellipse, because the pencil keeps the sum of its distances to the two pins constant. With pins 6 cm apart and a loop giving a major axis of 10 cm, the minor axis comes out as 8 cm and the eccentricity as 0.6, exactly as the formulas predict.
The science
An ellipse is the set of points whose distances to two fixed points, the foci, add to a constant 2a. The string loop fixes this sum. Half the distance between the foci is c, and the semi-minor axis b satisfies b squared = a squared minus c squared, so with a = 5 and c = 3, b = 4. The eccentricity e = c divided by a measures how squashed the ellipse is: e = 0 is a circle, and as the pins move apart e approaches 1 and the ellipse becomes a thin sliver. Planets move on ellipses with the Sun at one focus.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A soft board or thick cardboard 30 cm square
- Two drawing pins
- A loop of thread tied to a measured length
- A pencil and a ruler
- Graph paper to draw on
Safety
For this project: Drawing pins are sharp; keep them in a closed box.
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.
Fix graph paper to the board and push two pins in 6 cm apart on a horizontal line, so c = 3 cm.
Tie a thread loop so that, when pulled taut round both pins into a triangle, the pencil point is 5 cm from the midpoint along the axis: the loop length is 2a + 2c = 16 cm.
Keeping the thread taut with the pencil, draw all the way round. Keep the pencil upright or the curve will wobble.
Measure the longest width (major axis, 10 cm) and the shortest (minor axis, 8 cm), and compare with b = square root of 25 minus 9 = 4.
Pick any point on the curve, measure its distance to each pin and add them: always 10 cm.
Move the pins to 8 cm apart with the same loop and draw again; measure the new minor axis, which is about 4.9 cm, and compute e = 0.8.
Put both pins in the same hole and draw: a circle with e = 0.
Mount the three drawings with a table of a, b, c and e for each.
What you should see
The first ellipse measures 10 cm by 8 cm with the distance sum 10 cm at every point, and the three drawings show e rising from 0 to 0.8 as the foci separate.
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 defining property of an ellipse?
The sum of the distances from any point on it to the two foci is constant.
2What is the eccentricity of the first ellipse?
0.6, which is c divided by a with c = 3 and a = 5.
3What happens when the two foci coincide?
The ellipse becomes a circle, with eccentricity 0.
4Where is the Sun in Earth's orbit?
At one focus of the ellipse, not at the centre.
Take it further
Draw a hyperbola with a ruler pivoted at one focus and a string tied at the other, keeping the difference of distances constant.
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.
Free ebooks for this project
Science projects
Maths Working Models and Activities
Make maths visible: models for number, geometry, algebra and data
Science projects
Investigatory Projects for Class 11 and 12
Topics, method and report structure for physics, chemistry and biology