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Forces 2: 2.4 Determining and explaining the centre of mass of an object and other aspects of weight and gravity phenomena including  weightlessness and comparing objects falling in air or a vacuum

Doc Brown's Physics exam study revision notes

2.4A The centre of mass

What is the centre of mass of an object?

An important note on terminology

The centre of mass is defined as the point at which the distribution of mass is equal in all directions and does not depend on gravitational field.

For calculation purposes, the centre of mass of an object is a single point in the object through which the whole weight of an object is considered to act.

The centre of gravity is defined as the point at which the distribution of weight is equal in all directions and it does depend on gravitational field.

However, in terms of spatial position, they are actually the in the same position.

How can it be determined by experiment?

For some calculations, and, of great importance to structural engineers, you may need to know where the centre of mass (sometimes wrongly called the centre of gravity).

The centre of mass is a single point in the object through which the whole weight of an object is considered to act.

Its quite easy to envisage where it is for a regular shape e.g. a rectangular block - shown in profile in the diagram below.

The stability of a free standing object

A standing object becomes unstable when the vertical line through its centre of mass falls outside its base, which effectively acts as a base.

The weight of the object causes a turning effect about the pivotal base. The idea is illustrated by the diagram below of a regular shaped block, shown in profile, and tilted at various angles (but it could be a bus!).

The position of centre of mass of a regular object is easy to determine e.g. the symmetrical centre of a rectangular block illustrated in the 3D and 2D diagrams below.

For a rectangular block, the centre of mass is found at the intersection of four lines drawn from opposite corners.

1. The vertical line from the centre of mass passes right through the centre of the object's base.

The object is completely stable - no moment (turning force) is generated.

2. The vertical line from the centre of mass still passes through the base, but not its centre, and the object is unstable, so it will wobble a bit from side to side, and eventually settle down in an stable upright position as in 1.

The edge of the object touching the ground acts as a pivot point.

The weight of the object creates an anticlockwise moment (turning force) that makes the object fall back in an anticlockwise direction, but not sufficient to topple the object over.

3. The vertical line from the centre of mass passes outside of the object's base. The object won't even wobble, it is highly unstable and will just topple over on its side (to become stable!).

Again, the edge of the object touching the ground acts as a pivot point.

Again, the weight of the object creates a clockwise moment (turning force) that makes the object fall over in a clockwise direction, and sufficient to topple the object over on its side.

Tests on stability in terms of the centre of mass are important e.g. road vehicles like buses are safety tested to see the maximum angle allowed when tilted over without toppling over in an accident.

For regular shaped objects of uniform density its quite easy to figure it out. e.g. the centre of mass of a cube will be at the centre, equidistant from the 8 vertices.

For a rectangular block, the centre of mass point is defined by the co-ordinates H/2, B/2 and L/2. The same argument applies if H = B = L for a cube.

For a sphere of uniform density, the centre of mass will be at its dead centre.

For an irregular shaped object like yourself its a bit more tricky!

However, if it is a 'flat' object like a sheet of thick cardboard, wood or metal you can do quite a simple experiment to determine the centre of mass (centre of gravity).

In the school/college laboratory it quite easy. You pin the object at various points, ensuring it can hang freely under its own weight and hang a weighted plumb line down from the same point. The pin holes should be as near to the edge of the irregular object as possible.

When the object is quite stationary you mark another point on the other side of the object so you can then join them both up to give a locating line (e.g. line A). You then repeat this, choosing another point further round the object giving line B and then C etc.

You should find that all the lines intersect at point G, the centre of mass.

The method works because when the object hangs freely, there is equal mass (weight) on either side of the plumb line and this is independent of the pin point.

You can determine the centre of mass of a teachers by pinning them up by the tips of their ears, fingers or toes!

2.4B Other aspects of weight and gravity phenomena

This page will answer questions such as ...  Why can a feather and iron bar fall at the same rate in a vacuum?

Weightlessness

If an object is 'weightless' it is apparently not being subject to any gravitational field force.

The most obvious example is an object out in deep space well away from any gravitational field of a star or planet.

Why does a feather and a hammer fall at the same velocity in vacuum?

Experiment A: When you drop from a few metres height, a heavy object like a hammer and a light object like a feather at the same time, your experience will tell you to expect the hammer to fall quite rapidly and hit the floor first and the feather to follow on far more slowly.

Experiment B: If you repeat the experiment e.g. with a small but heavy weight and a small feather at one end of an enclosed large glass sealed tube from which all the air is pumped out, you see a very different result. Both objects fall at the same rate. (Its the same if you do the experiment on the moon!)

The reason for the result you see in experiment B is because all objects, whatever their mass, experience the same accelerating force due to the gravitational force field (of the Earth).

The acceleration is actually ~10m/s2.

In experiment A the feather is much lighter and, more importantly, a much greater surface area to mass ratio and the friction effect (drag) of it passing through the air is much greater than what the hammer experiences.

So the descent of the feather is slowed down. If the air is removed, there is no drag effect on either object and they accelerate to Earth at the same rate.

You would see exactly the same effect on our moon, which has virtually no atmosphere and the first men on the moon did a similar B experiment with a hammer and feather.

In the old 'black and white' footage from the first moon landing by US astronauts, you could clearly make two important observations:

(i) On dropping, the hammer and feather hit the moon's surface at the same time - no air resistance.

(ii) The hammer, quite plainly, fell with much less acceleration than on Earth because of the much smaller gravitational field force on the moon, due to its much smaller mass.

On the Earth's surface, the gravitational acceleration is 9.8 m/s2 and on the Moon it is only 1.6 m/s2.

See also Part 3.6 Other situations of falling objects e.g. feather or hammer in different gravitational fields, thought experiments described and explained!

Keywords, phrases and learning objectives for forces

Know what the centre of mass of an object  is.

Know how to determine the centre of mass of a regular object.

Be able to describe an experiment to determining the centre of mass of an irregular object.

Know what is meant by weightlessness and comparing objects falling in air or a vacuum

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