Class practical

A simple method for drawing an ellipse, using the two foci.

Apparatus and Materials

For each student or group of students

• Drawing board
• Drawing pins, 2
• Loop of thread, about 40 cm long
• Pencil

Health & Safety and Technical Notes

Read our standard health & safety guidance

Procedure

1. Stick two drawing-pins into a board (though not pressed too far in) about 10 cm apart and slip the loop of thread over them. Insert a pencil point in the loop and position it so that the thread is tight. Move the pencil round the pins, always keeping the thread tight and thus trace out an ellipse.

2. Students could be encouraged to draw ellipses with the drawing pins closer or farther apart.

Teaching Notes

• Drawing ellipses is a useful skill for students to master.
• The Earth's orbit around the Sun is not perfectly circular. It follows a slightly elliptical path, with the Sun at one focus.

Demonstration

Three methods of visualising elliptical orbits.

Apparatus and Materials

Either Method 1:

• Retort stand with rod and clamp, or tripod
• Small steel ball approximately 5 mm diameter, or a marble
• Large glass funnel at least 20 cm diameter

Or Method 2:

• Rubber sheeting
• Toy hoop or embroidery frame
• Retort stand, rods, clamps, boss head

Or Method 3:

• Alpha-particle scattering analogue apparatus

Health & Safety and Technical Notes

Read our standard health & safety guidance

For method 2, stretch the rubber sheeting over the rigid horizontal circular frame and secure it tightly with tape. It should be stretched a little, equally in all directions.

Fix a vertical metal rod over the sheet so that it is pushing the sheet into a curved well, thus imitating, roughly, an inverse-square-force potential.

For method 3, the apparatus is available from a number of suppliers: Philip Harris, Griffin Education, or ASCOL.

Procedure

Method 1: Firmly hold the glass funnel vertically and let the ball fall into it. Friction will affect the orbit and make it precess, but the motion around the funnel will be elliptic. Select a ball which will fall right through the funnel.

Method 2: Project the small steel ball across the sheet. By choosing suitable initial conditions the ball can be made to describe an oval like an ellipse with one focus on the axis of the well.

Method 3: Balance the plastic or aluminium hill upside down. Wood blocks can be used to hold it in position. Project the ball across the inverted hill so that it will orbit the centre. The elliptical path will be visible.

Teaching Notes

• Method 1 is not a true representation of Kepler's Second Law. The ball starts in a circular orbit, but it gradually moves lower in the funnel because of friction and so the time for a circular orbit gets smaller. The orbit also precesses as a result of friction.
• Method 3 uses a valley with a 1/r profile. This corresponds to a 1r² force of gravitational attraction and so will closely model the movement of a satellite around a larger mass.

Demonstration

Demonstrating the law using air or carbon dioxide pucks – or an ice rink.

Apparatus and Materials

• Air table
• Air or CO₂ puck
• Light spring
• Rope

Health & Safety and Technical Notes

Take care when handling the glass plate.

If taking a class off site, follow your employer's procedures for off-site educational visits.

Read our standard health & safety guidance

Procedure

1. Tether the puck to the air table using a light spring.
2. Project the puck across the surface so that it orbits the tether point. The elliptical path can be observed.
3. Alternatively, if the class is taken to an ice rink, a heavy student, representing the Sun, should be positioned firmly in the centre of the rink, holding a rope, the other end of which is held by another, lighter, student, who skates around the first. As the first student pulls the rope in, the second, the orbiting planet, will speed up.

Teaching Notes

• The change in speed as the puck orbits can be clearly seen and the areas traced out in equal times mapped out.

• When the arm from the Sun to an orbiting planet is long, the planet's speed is slower and the arm is short, the planet's speed is faster.
• Starting with Mars, Kepler discovered that the orbital path of each planet is an ellipse with the Sun at one focus. The line from the Sun to any planet sweeps out equal areas in equal times. All the sectors shown have equal areas; the planetary orbits are much closer to circles in shape than in this figure. This is Kepler's Second Law.
• If you mark the position of a planet once a month on its elliptical orbit, and draw radii from the Sun to those points, the areas of sectors between those radii are all equal. This demonstrates Kepler’s Second Law.
• Later, Newton showed that Kepler's Second Law can be explained by a central gravitational force (directed towards the Sun) which goes as 1r², where r is the planet's average radius of orbit.

This simple class activity demonstrates the increase in speed with a diminishing radius of orbit.

Apparatus and Materials

For each student or group of students

• Rubber bung
• Length of twine

Health & Safety and Technical Notes

Read our standard health & safety guidance

Procedure

1. Attach a length of twine securely to the rubber bung. Whirl it round in a circle.

2. After a few revolutions, allow the twine to wind itself up around the finger. As the length shortens, so the planet moves faster and faster.

Teaching Notes

• Kepler's Second Law states that a planet in its orbit sweeps out equal areas in equal times. The time of orbit of the puck will fall as its radius of orbit decreases.
• You can measure the load by attaching a newton meter to the bottom of the string. As the force on the puck is increased, its orbit gets smaller and its speed increases.
• The arm from the Sun to planet sweeps out equal areas in equal periods of time. More distant planets move more slowly in their orbit. If you mark the position of a planet once a month on its elliptical orbit, and draw radii from the Sun to those points, the areas of sectors between those radii are all equal.
• Kepler replaced constant speed of planetary motion (as with Ptolemy and the Greeks) with constant sweeping out of an area. Developing scientific explanations often involves a search for constant values.

Demonstration

The speed of the puck increases as the radius of orbit of the puck decreases.

Apparatus and Materials

• Glass tube and fishing swivel, or freely swivelling pulley
• Block of solid CO₂ ,
Or

• Carbon dioxide cylinder (syphon type)

• Glass plate and gantry
• Wedges, 4
• Brass ring pucks, 2

Health & Safety and Technical Notes

Dry ice is very cold. Wear thermal gloves to handle it, and wear safety spectacles.

Take care when handling the glass plate.

Read our standard health & safety guidance

Procedure

1. The simplest technique is to use a swivelling furniture castor for the pulley. Drill a hole through the shank and cut a groove around the periphery of the wheel (see diagram). Support the pulley centrally just above the large glass plate so that the wheel can turn freely about the two axes. Use oil, if necessary, to allow the system to move freely.

2. Fix a cup hook, or other device, to a double puck, so that a length of light string can be fastened to its side. Place the puck on the glass plate, supported on a block of solid carbon dioxide about 1 cm thick and of external diameter about 2 cm larger than the metal block. Pass the string over the groove and through the pulley shank, so that it can be held by the hand.

3. An alternative arrangement uses the bar across the gantry to support a glass tube, which acts as the flexible pulley. Connect one end of the twine to a specially shaped puck, and the other end to a fishing swivel, which enables the twine to rotate freely. Connect this in turn to a spring balance which is held by hand. The puck must be specially shaped to give it a wider base, but the same mass as the usual CO₂ pucks.
4. The glass plate should be carefully cleaned with ethanol before it is used.
5. Keep the carbon dioxide puck in motion before starting the experiment, so that it does not freeze to the plate.
6. Start the puck moving in a circular orbit on the glass plate, applying the centrally directed force through the string. Increase this force and observe the effect.

Teaching Notes

As the force on the puck is increased, its orbit gets smaller and its speed increases. Kepler's Second Law states that a planet in its orbit sweeps out equal areas in equal times.

Class practical

The simple kit demonstrates the increase in speed with a diminishing radius of orbit.

Apparatus and Materials

For each student or group of students

• Centripetal force kit

Health & Safety and Technical Notes

Ensure that the bungs are securely attached and that each student has sufficient space to whirl the bung safely.

The centripetal forces kit is available from:

Technology Supplies Limited

Procedure

1. Set the bung whirling around the top of the apparatus.
2. Pull on the thread to increase the load.

Teaching Notes

• Kepler's Second Law states that a planet in its orbit sweeps out equal areas in equal times. The time of orbit of the puck will fall as its radius of orbit decreases.
• You can measure the load by attaching a newton meter to the bottom of the string. As the force on the puck is increased, its orbit gets smaller and its speed increases.

Demonstration

Apparatus and Materials

• Rotating stool or platform
• Masses, 2 kg, 2

Health & Safety and Technical Notes

Supervise students so they don't spin too quickly, or they may topple over.

A freely spinning music stool (or office chair) may be used for these demonstrations. It should be oiled before use.

Procedure

1. Sit a student on the stool and ask them to hold a massive object, about 2 kg , in each hand with their arms stretched out sideways. Turn the stool so that it rotates fairly rapidly.
2. Ask the student on the stool to draw in their arms, so holding the ‘planets’ tightly to their chest.
3. Ask them to stretch their arms out again.

Teaching Notes

• This demonstration shows the conservation of angular momentum when the radius of the rotating mass is changed. As some parts of a spinning object are pulled in to make it more compact, then the object spins faster.
• Newton gave a geometrical proof of Kepler’s Second Law, using changes of momentum (a vector). The Law is simply a statement of the conservation of angular momentum. The only force acting on a planet (gravity) is always directed towards the Sun.
• Other examples of Kepler’s Second Law include the following...
• Suppose a high diver doing a somersault discovers half way down in his flight that he is going to hit the water feet first, when this is not his plan. He can double up, knees against the chest, and spin faster for a carefully judged time. To spin slower, the arms are spread out wide.
• A skater starts spinning slowly on one toe with arms stretched out and the body tilted. She then pulls herself upright with arms folded and spins very fast.
• The much more complicated movement of a falling cat, which always lands on its feet.

This experiment was safety-tested in March 2008