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!
For a more detailed discussion see also
Turning forces & moments - spanners
to wheelbarrows, levers, gears & equilibrium situations
INDEX physics notes FORCES
section 2 on mass, weight and gravity
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.
INDEX physics notes FORCES
section 2 on mass, weight and gravity
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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INDEX physics notes FORCES
section 2 on mass, weight and gravity
