Demonstration

Observe the behaviour of light from a point source. Develop the idea of an inverse square relationship and then consider Newton's gravitational law.

Apparatus and Materials

• Light meter (or photographic exposure meter)
• Lamp, 12 V 24 W
• Lamp holder on base
• Power supply for lamp

Health & Safety and Technical Notes

Use standard tungsten lamps. Avoid halogen lamps unless suitably filtered for UV light.

With some exposure meters it may be necessary to reflect back some of the light to the reverse side of the meter in order to make the readings: a piece of white card will do this effectively.

Some exposure meters have scales marked so that adjacent markings represent light values increasing by a factor of 2 for each stop. Care must be taken when interpreting these.

Read our standard health & safety guidance

Procedure

1. Switch the lamp on in a darkened room and direct the meter towards it from distances of 30, 60, 90 cm. Note the meter readings in each case. The readings will be seen to fall off in the proportion 1: 1/4: 1/9, etc.

Teaching Notes

• This is intended as a quick simple demonstration. The units of measurement are not important.
• Newton was worrying about how to explain Kepler’s Laws using gravity. If the Earth’s gravitational field strength did not vary with distance from the Earth, the time for the Moon’s orbit would be about 11-12 hours.
• By reducing the field strength at greater distances, the time predicted for the orbit can be extended. Using the simplest scheme of halving gravity when the distance doubles, does not fit the results for the time of orbit of the Moon around the Earth. (It would give 1/60 of our g at the Moon since the Moon is 60 Earth radii away from Earth.)
• I began to think of gravity extending to the orb of the Moon ... From Kepler's rule ... I deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centres about which they revolve; and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the Earth, and found them to answer pretty nearly. (Newton, recalling his work of 1666.)
• But then he hesitated. It is sometimes said that this was because he did not have an accurate value for the radius of the Earth but that is probably untrue. It is more likely because he saw a difficulty about the gravitational pull of the whole Earth. He put the work aside for many years and turned his attention to developing mathematics and experimenting on light and colour.
• The inverse-square law is a sensible one to try because it is the way in which anything thins out if it sprouts straight lines from a source and continues out without getting lost. Light from a small lamp does that.
• At double the distance, the same light spreading out through clear air without being absorbed, falls on four times the area at double the distance but gives only a quarter of the illumination.

• Another crude example is the ‘butter gun’. Suppose the owner of a restaurant invents a gadget to butter many slices of toast efficiently. It is a small, motor-driven sprayer that squirts out a fine spray of melted butter from its muzzle, in straight lines in a wide cone.
• Suppose the spray just covers one piece of toast at a distance of 30 cm in a standard time. Then four pieces of toast could be placed at 60 cm from the sprayer but the toast would have to sprayed for four times as long, or be more thinly spread. This is the inverse-square law of buttering.

Demonstration

Using a scale drawing to predict the time for a satellite close to the Earth. The results can later be compared with mathematical calculations based on v²/R.

Apparatus and Materials

• Brown paper sheet about 1.5 m long, 15 to 20 cm wide - or a roll of long paper
• Thin wire, strong, about 3.5 m
• Large mass or hook to anchor one end of the wire on the floor or near it
• Metre rule with cm and mm markings

Health & Safety and Technical Notes

Be careful with the thin wire, and avoid getting cuts to the hand.

Read our standard health & safety guidance

This works best on the floor, or the side bench of a laboratory, with small groups of students.

The arc drawn is twice as long as needed for 2 minutes' travel, but the double length enables a symmetrical drawing which will yield a good estimate more easily.

Procedure

1. Using a thin wire anchored on the floor at one end and held taut with a pencil at the other end, draw an arc about 1.3 m long and with a radius of 3.3 m. The centre of the circle is not on the paper but arrange the paper so that the arc is. This represents part of a circular orbit for a satellite at an altitude of 200 km. (A scale drawing with 0.5 mm = 1 km - see Teaching Notes.)


2. On that arc, XAY, mark the mid point A and part of the radius to A. Drop down along that radius the calculated fall 72 km (to scale) AM. (On the same scale 0.5 x 72 = 36 mm.)
3. Draw the tangent at A, symmetry helping. And draw a chord XMY parallel to the tangent, with mid point at M.
4. Transfer the fall AM out to the place where it should be shown as a fall from the tangent to the orbit, a fall NY. Measure the travel distance AY, which is covered in their chosen time 120 seconds (973 km).
5. Calculate the time for a complete orbit, knowing the total travel distance 2π R, which is 2π x 6,600 metres.

Teaching Notes

• A scale of 0.5 mm to 1 km is best for the drawing - a smaller drawing would be difficult to measure. On this scale, R = 6,400 + 200 km is represented by a radius of 0.5 x 6,600 = 3,300 mm = 3.3 m.
• Assume the satellite is always falling inward with acceleration g (≈ 10 m/s/s). First calculate how far the satellite will fall from the tangent in 1 second (5 m). It is clearly difficult to draw a scale diagram with 5 m drawn on it and the radius of the orbit, 6,600 km, and so a longer time needs to be chosen. the satellite's free fall from the tangent to its orbit in a time of 120 seconds will be s = 1/2 gt² = 72 km.
• This will give a travel distance AY of about 973 km. The time for a complete orbit, T, will then be 120/ T =973 / 2π x 6,600, giving a value of T of about 85 minutes. Students are generally very surprised to find that near-Earth satellites orbit with such short periods. Such satellites are used for astronomy, earth-observation (mapping and spying) and weather-forecasting. (Note that communications satellites must remain geostationary, and so orbit at much greater distances from the Earth.)
• It helps to provide illustrative date about a variety of near-Earth satellites. For example, the first artificial satellite, Sputnik 1, had an orbit period of 96.2 minutes. NASA has a website that offers real-time tracking of satellites, including the Hubble Space Telescope and the International Space Station.

NASA website

• Likewise the European Space Agency has a satellites in orbit webpage, where you can find orbit information for its Earth Observation missions. Note that ESA Space Science missions do not have near-earth orbits.
• You could go on to calculate the Moon's orbital time, using the same scale. See the guidance note:

Estimating the Moon's orbit time

Demonstration

Angular momentum is conserved when the radius of orbit is changed.

Apparatus and Materials

• Air table
• V channel
• Steel balls, 2
• Gardening gloves, heavy duty

Health & Safety and Technical Notes

Take care when handling dry ice. Wear eye protection and use thick gardening gloves to handle it.

Read our standard health & safety guidance

Two pieces of V-channel should be cut and mounted on a base in a V arrangement.

The demonstration can also be performed using a solid CO₂ system, mounting the channel on a glass plate and with a small piece of solid CO₂ between it and the table so that there is a frictionless bearing for spinning.

Procedure

1. Load the V with two steel balls to act as planets, holding them high up on each arm by a piece of light metal tubing placed horizontally between the balls as a spacer.

2. Set the device spinning and snatch the metal tube away. When the balls run down to the bottom of the V, much closer to the axis, the system spins much faster.

Teaching Notes

• When the radius of the rotating mass is changed the angular momentum is conserved. Newton gave a geometrical proof of Kepler’s Second Law, using changes of momentum as vectors, because the Law is simply a statement of the conservation of angular momentum around the Sun when the only force acting on the planet is a central one passing through the Sun.
• A much simpler alternative: have a student, standing upright, spin on one heel for a short time while holding a heavy book in one hand. Moving the book towards or away from the body will show a change in angular velocity.
• It helps to raise the toes of the pivot foot. With a little practice, it is possible to make two or more revolutions spinning on one heel without losing balance and having the other foot coming to the rescue.

Demonstration

Simple model to demonstrate the ‘flattening’ of the poles as the Earth rotates.

Apparatus and Materials

• Large hollow rubber ball (e.g. from a toy shop)
• Metal rod
• Hand or cordless power drill

Health & Safety and Technical Notes

Read our standard health & safety guidance

Procedure

1. Drill small holes through the rubber ball at each end of a diameter and slide the metal rod through the ball. The rod should slide freely at one end but tightly at the other by making the diameter of one hole about half that of the rod and the other slightly larger than the rod.
2. Hold the end of the rod in the chuck of a hand or a cordless power drill. Support the drill so that the axis of rotation is upright. Fix a stop at the upper end of the rod (e.g. a small rubber bung).

3. A small solid ball of sponge rubber will show the effect if high speeds are used. Do this behind a safety screen. If the ball disintegrates, pieces could fly off in all directions.
4. Rotate the ball and observe its behaviour.

Teaching Notes

• As the ball is rotated the ‘equator’ will stretch and the ‘poles’ will flatten. Until Newton’s day the Earth was thought to be a perfect sphere. Newton predicted that it must be flattened at the poles and bulging at the equator, in other words, an oblate spheroid.
• The force, F , needed to keep an object in circular motion is given by F = mω²r where m is its mass, ω its angular velocity, and r its orbit radius.
• The angular velocity is the same everywhere on the Earth’s surface. The distance of the Earth’s surface from its spin axis, r, varies with latitude and is greatest at the Equator. The Earth’s bulging near the Equator contributes to the larger force needed there.
• Newton’s argument ran as follows...
• Imagine a pipe of water running through a spherical Earth from the North Pole, to the centre and out to the Equator. If this were filled with water, just to the Earth’s surface at the North Pole, where would the water surface be in the equatorial branch of the pipe?

• At the centre of the Earth, the water pressure at the bottom of the polar pipe is due to the weight of the water. The pressure pushes around the ‘elbow’ at the bottom and out along the equatorial branch, trying to push that column of water outward. The weight of water in the equatorial branch pulls it in. But these two forces on the equatorial branch must be unequal. They must differ by enough to provide an inward centripetal acceleration to act on the water in that pipe, which is being carried around with the spinning Earth.
• The weight of water in the equatorial branch must exceed the outward push from the water at the elbow by the amount needed for mv²/R forces. Therefore the water column in the equatorial pipe must be longer than that in the polar pipe. Newton calculated the extra height and found that 14 miles (23.5 km) would be required. He argued that the Earth at an early pasty stage would bulge out by about this distance. The bulge he predicted was later confirmed.
• A simple version of the model can be made by students using a strip of very thin paper about 20 cm long and 2 cm wide. Make the strip into a loop by joining the two ends together with tape or paste. Push the point of a pencil carefully through the join and then push the pencil through this tightly fitting hole and across the loop. Where the point of the pencil meets the other side of the loop, make a hole just a little wider than the pencil. Push the pencil through this hole.
• Roll the pencil quickly to and fro between the palms of your hand and see the shape of the paper ‘Earth’.

Demonstration

Using the model celestial sphere to show the precession of the equinoxes.

Apparatus and Materials

Model of the celestial sphere from the experiment

Model of the celestial sphere

Health & Safety and Technical Notes

Take care with glassware. Keep the model out of sunlight as flask and water can act as a convex lens and produce localized heating.

Mark a new axis on the sphere, perpendicular to the ecliptic. This can be marked on the flask with small circles of coloured sticky paper indicating where the axis meets the surface. (An ideal arrangement is to fix two small suction caps at these points.)

Read our standard health & safety guidance

Procedure

1. Hold the model so that it can revolve very, very slowly about that axis, whilst the whole model is imagined to be spinning very rapidly (10 million times faster) round the Pole Star axis.

Teaching Notes

• The precession of the equinoxes, as described by early astronomers (from a geocentric - Earth-centred - point of view), was a very obscure creeping motion of the whole system of stars around a special axis (the axis of the ecliptic). It was as though the whole Zodiac belt slipped very slowly round the celestial sphere, carrying all the stars with it and leaving the celestial equator attached to a fixed Earth. In that model it is difficult to describe, but Copernicus made it much simpler.
• The precession of the equinoxes is a slow rotation of the whole pattern of stars around the ecliptic axis, one revolution taking 26 000 years. This motion was discussed by Hipparchus (~190 BC to ~120 BC).

This experiment was safety-tested in April 2007

Related experiments

Model of the celestial sphere

Demonstration

To show how the spin axis of the Earth rotates or precesses.

Apparatus and Materials

• Large gyroscope
• Eye protection

Health & Safety and Technical Notes

Read our standard health & safety guidance

A suitable gyroscope is available from

ASCOL

Alternatively, a gyroscope can be made from a bicycle wheel; a front wheel, 46 cm x 3 cm is satisfactory. Two 10 cm lengths of alkathene rod (or similar) can be used for handles. Drill out the two lengths of plastic rod axially to a depth of 2.5 cm with a 6 mm drill. Screw them firmly on to the axle of the wheel to provide the two handles. Wear eye protection when drilling. The bicycle wheel is even better if it is loaded with a tyre of lead strip. Make sure the lead is firmly fixed to the wheel.

Procedure

1. Drill a 3 mm hole through a diameter of one of the handles about 1 cm from the end. Pass a knotted string through this hole.
2. If the large gyroscope is not available, a toy gyroscope could be used.
3. Hold the wheel vertically in one hand and set it rotating - a series of tangential swipes with the palm of the other hand will do, or spin it rapidly by holding the rim against a motor-driven wheel. Be careful that clothing does not become entangled in the rotating motor.
4. When the wheel is rotating rapidly it can be supported by the string. It will remain vertical and will precess.

Teaching Notes

• The wheel spinning rapidly represents the spinning Earth and the massive rim of the wheel represents the equatorial bulge.
• Precession is the slow, conical motion of the Earth’s spin-axis round the axis of the ecliptic; that is, around a line through the Sun perpendicular to the Earth’s orbit, an axis at 23.5 ° from the Earth’s polar axis.
• Newton showed that this motion is a consequence of gravitation and the Earth’s spin. A spherical Earth, whether spinning or not, would keep its axis pointing in a constant direction among the stars as it followed its orbit around the Sun.
• An Earth with an equatorial bulge will suffer extra gravitational pulls exerted by the Sun (and Moon) on the bulge. These extra forces applied to a spinning body show that rocking forces applied to a spinning body do not succeed in rocking the body, but produce precession instead.

• The Earth, a spinning top, and a ‘mysterious gyroscope’ all precess in the same way, for the same reason. In the sketches above ‘torque axis’ means the axis around which the tilting force tries to rock the spinning object.

• Crude explanation with Newton’s Law II, considering only the top and bottom of rim, A and B. If wheel rocks ever so little in response to torque, A gains momentum Δ mv to the right; B gains Δ mv to the left. Combining these with previous momenta, we find wheel must be precessing.

Demonstration

This demonstration imitates the Earth’s precession, with a spinning flywheel ‘rocked’ by the pull of a taut rubber band.

Apparatus and Materials

• Large gyroscope
• Mount for large gyroscope

Health & Safety and Technical Notes

Read our standard health & safety guidance

A small massive flywheel, which can be set spinning in a frame, is held by another frame hung by a long thread.

Nylon is suitable for the long thread used to support the apparatus because it does not make things spin when used for suspending an object like ordinary strings do. Any torque from the suspension will be unwelcome.

Procedure

1. Hang the frame carrying the rotating flywheel on a long thread, preferably from the ceiling.

2. Spin the flywheel rapidly: this can be done by hand by bringing it against a rapidly rotating wheel attached to a fractional horse-power motor. Be careful that clothing does not become entangled in the rotating motor.

3. Set the wheel so that its axis points in a definite direction, say 45 % from the vertical and across the pupils’ line of sight.

4. Slip an elastic band over the hooks on the frame and wheel axle. Precession will start. The moment the band is slipped off, precession stops immediately.

Teaching Notes

• Precession is the slow, conical motion of the Earth’s spin-axis round the axis of the ecliptic; that is, around a line through the Sun, perpendicular to the Earth’s orbit; an axis at 23.5 ° from the Earth’s polar axis. Newton showed that this motion is a necessary consequence of gravitation and the Earth’s spin.
• A spherical Earth, whether spinning or not, would keep its axis pointing in a constant direction among the stars as it followed its orbit around the Sun. An Earth with an equatorial bulge will suffer extra gravitational pulls exerted by the Sun (and Moon) on the bulge. These extra forces applied to a spinning body show that rocking forces applied to a spinning body do not succeed in rocking the body, but produce precession instead.
• This is a modification of ordinary gyroscope demonstrations, specially arranged for teaching the way in which precession of the equinoxes is caused.
• Initially the wheel spins with its axis pointing in an unchanging direction. When a rubber band is installed, between the outer frame and a hook on the inner frame, precession starts, the spin axis moving slowly round a cone. When the rubber band is unhooked (without interrupting the spin) precession stops.
• The pull of the rubber band represents the Sun’s net gravitational pull on the Earth’s equatorial bulge.