# Model of the oblate Earth

##### 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

In using a cordless drill, avoid spinning the ball at high speeds.

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.

#### Procedure

Rotate the ball and observe its behaviour.

#### Teaching notes

1 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.

2 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’.